1. Software Quality Control Methods

2. Introduction Quality control methods have received a world wide surge of interest within the past couple of decades as industries compete to design and produce more reliable products more efficiently. Attention has been focused on the “quality” of a product or service, which is a considered to be a general term denoting how well it meets the particular Demands imposed upon it. The origins of quality control can be traced back to the implementation of control charts by Walter Shewhart in the 1920s. More recently the original quality control ideas together with related management principles and guidelines have formed a subject which is often referred to as Total Quality Management.

3. Introduction Another important quality control tool is Statistical Process Control (SPC), which is discussed. Statistical Process Control utilizes control charts to provide a continuous monitoring of a process. Finally, acceptance sampling produces, which can be used to make decision about the acceptability of batches of items, are discussed.

4. Statistical Process Control Consider a manufacturing organization that is involved in the production of a vast Number of a certain kind of product, such as a metal part, a computer chip, or a chemical solution. These products are manufactured using a process which typically involves the input of raw materials, a series of procedures, and possibly the involvement of one or more operators. Statistical process control concerns the continuous assessment of the various stages of such a process to ascertain the “quality” of the product as it passes through the process. A key component of this assessment is the use of control charts.

5. Control Charts A control chart is a simple quality control tool whereby certain measurements of products at a particular point in a manufacturing process are plotted against time. This simple graphical method allows a supervisor to detect when something unusual is happening to the process. With this continuous assessment, any problem can be fixed as they occur. This is in contrast to a less desirable scheme whereby products are examined only at the end of the process.

6. Piston Head Construction Consider the manufacture of a piston head that is designed to have a radius of 30.00 mm. A control chart can be used to monitor the actual values of the radius of the manufactured piston heads, and to alert a supervisor if any changes in the process occur. For example, the control chart in Figure indicates that there has suddenly been an increase in the average radius of the piston heads. With the continuous monitoring provided by the control chart the supervisor can immediately investigate the reasons behind the radius increase and can take the appropriate corrective measures. In this case due to a sudden rise in the variability of the radius values.

7. Control Limits In order to help judge whether a point on a control chart is indicative of the process having moved out of control, a control chart is drawn with a center line and two control limits. These control limits are the upper control limit (UCL) and the lower control limit (LCL). It is useful to realize that this procedure is essentially performing a hypothesis test of whether the process is in control.

8. Control Limits It is useful to think of the null hypothesis as being H O : process in control with the alternative hypothesis H A : process out of control. When new observations on the process are taken, the null hypothesis is accepted as long as the point plotted on the control chart falls within the control limits. However, if the point lies outside the control limits then the null hypothesis is rejected and there is evidence that the process is out of control.

9. Control Limits Typically, “3-sigma” control limits are used which are chosen to be three standard deviations σ above and below the center line.

10. Example Piston Head Construction Suppose that experience with the process of manufacturing piston heads leads the supervisors to conclude that the in-control process produces piston heads with radius values which are normally distributed with a mean of μ 0 = 30.00 mm and a standard deviation of σ = 0.05 mm. How should a control chart be constructed?

11. Example Suppose that the observations X 1 ……x n represent the radius values of the random sample of n piston heads chosen at a Particular time. The point plotted on the control chart is the observed value of the sample average When the process is in control this sample average is an observation from a distribution with a mean value of  0 = 30.00 mm and a standard deviation of

12. Example A 3-sigma control chart therefore has a certain line at the “control value”  0 = 30.00 mm together with control limits and. If the sample size taken every hour is n = 5 then the control limits are and as shown in Figure.

13. Example The probability of a type one error of this control chart, that is, the probability that a point on the control chart lies outside the control limits when the process is still in control, is where the random variable X is normally distributed with a mean of 30.00 and a standard deviation of.This probability can be evaluated as where the random variable Z has a standard normal distribution, which is 1-0.9974 = 0.0026, as mentioned previously.

14. Example It should be mentioned even if a series of points all lie within the control limits there may still be reason to believe that the process has moved out of control. Remember that if the process is in control then the points plotted on the control chart should exhibit random scatter about the center line. Any patterns observed in the control chart maybe indications of an out of control process. For example, the last set of points on the control chart shown in Figure all lie within the control limits but they are all above the center line. This suggests that the process may have moved out of control.

15. Example There is a series of rules which has been developed to help identify patterns in control charts which are symptomatic of an out-of-control process even though no individual point lies outside the control limits. These rules are often called the western electric rules, which is where they were first suggested. Most computer packages will implement these rules for you upon request.

16. Properties of control charts It is also useful to consider the probability that the control chart indicates that the process is out of control when it really is out of control. With in the hypothesis testing framework this is the power which is defined to be power = P(reject H 0 when H 0 is false) =1 – P(Type II error) = 1 -  Another point of interest relates to who long a control chart needs to be run before an out of control process is detected. The expected value of the number of points that need to be plotted on a control chart before one of them lies outside the control limits and the process is determined to be out of control is know as the Average Run Length (ARL).

17. Properties of control charts If the process has moved out of control so that each point plotted has a probability of 1 -  of lying outside the control limits independent of the other points on the control chart, then the number of points which must be plotted before one of them lies outside the control limits has a geometric distribution with success probability 1 - . The expected value of a random variable with a geometric variation is the reciprocal of the success probability, so that in this case the average run length is

18. Example Piston head construction Suppose that the piston head manufacturing process has moved out of control due to a slight adjustment in some part of the machinery so that the piston heads have radius values which are now normally distributed with a mean value  = 30.06 mm, instead of the desired control value 30.00 mm, and with the same standard deviation  = 0.05 mm as before. How good is the control chart at detecting this change?

19. Example The plotted points on the control chart are observations of the random variable X, which is now normally distributed with a mean  = 30.06 mm and a standard deviation = 0.0224 mm. The probability that a point lies within the control limits is therefore which can be written as

20. Example where the random variable Z has a standard normal distribution. This is The probability that a point lies outside the control limits is therefore 1 -  = 1 – 0.622 = 0.378. In other words, once the process has moved out of control in this manner, there is about a 40% chance that each point plotted on the control chart will alert the supervisor to the problem. The average run length in this case is So the problem should be detected within 2 or 3 hours.

21. This section has provided a general introduction to the use of control charts and the motivation behind their use. Specific types of control charts for specific problems are now considered in more detail.

22. Variable Control Charts The X –chart looks for changes in the mean value  and the R-chart looks for changes in the standard deviation  of the variable measured. These control charts can be constructed from a base set of data observations which are considered to be representative of the process when it is in control. This data set is typically consists of a set of samples of size n taken at k different points in time. The sample size n is usually quiet small, perhaps only 3,4, or 5 but it may be as large as 20 in some cases. The control chart should be set up using data from at least k = 20 distinct points in time.

23. X-Charts An X-chart consists of a sample averages plotted against time and monitors changes in the mean value of a variable. The lines on the control chart can be determined from a set of k samples of size n, with the center line being taken as x, the overall average of the sample averages, and with control limits UCL = x + A 2 r and LCL = x – A 2 r, where r is the average of the k sample ranges. In practice, this X-chart is used in conjunction with an R-chart discussed below, which monitors changes in the variability of the measurements.

24. R-Charts The lines on the control chart are calculated from the data set of in control observations, with the center line taken to be, r, The average of the k sample ranges r 1,…..r k, and with control limits UCL = D 4 r and LCL = D 3 r, in Figure.

25. The R-Chart An R-chart consists of sample ranges plotted against time and monitors changes in the variability of a measurement of interest. The lines on the control chart can be determined from a set of k samples of size n, with the center line being taken as r, The average of the k sample ranges, and with control limits UCL = D 4 r and LCL = D 3 r