QUME507_M8

QUME 507 Module 8: Quality Control

Introduction On Topic 8.1 we will study the different statistical and probabilistic methods applied to quality control. The objective of quality control is to ensure products achieve company expectations. Probability techniques and dispersion analysis will be used. Students will be taught how to use statistical programs for the application of these models.. Topic 8.1. Statistical Control of Processes and Total Quality Quality management, commonly called quality control (QC), is fundamental in every organization. One of the most important functions of managers is to make sure their company may deliver a quality product to the right place, at the right moment, and for the right price. Quality not only concerns manufactured products, but it is also important in services, from bank services to hospital care and education. It should be pointed out that a product’s or a service’s quality is the degree to which the product or service complies with specifications. Each time, the definitions of quality include an additional emphasis on satisfying client needs. According to Demings (2000): “Improvement of quality transfers wasted portions of man-hours and machine-time into the manufacturing of good products and better services. The result is a chain reaction, eventually lowering costs, improving the competitive position of the product, making customers happier, more sales, and more jobs.” (p.2) Total quality management (TQM) encompasses the entire organization from the supplier to the client. As so, meeting client expectations requires an emphasis on TQM so that the company may be a leader in world markets. Statistical Control of Processes Statistical control processes help to establish standards. They may also supervise, measure, and correct quality problems. One of the tools used in the process are control graphs. Control graphs show superior and inferior limits for the process wanting to control. A control graph is a graphic presentation of data through time, and it is elaborated in a way so as the new data may be quickly compared with the previous performance.

Superior and inferior limits on a control graph may be represented by units of temperature, pressure, weight, longitude, etc. We take exit samples from the processes and we graph the averages of those samples in a graph containing the limits . For additional information you may watch the following video at https://youtu.be/E08XBauXPXU Learn & Apply (2017). Control Chart: History, Concept & Nelson rules. [Video]. Recuperado de https://youtu.be/E08XBauXPXU Figure 1 presents useful information that may be represented in a contol graph. As it can be seen, when the sample averages fall within the superior and inferior control limits and there is no discernible pattern, it is said the process is under control; if not, the process is out of control or out of adjustment. Figure 1. Patterns one should look for on control graphs Taken from Render et al. (2018) for educational purposes. Process Variability In the decade of the 1920s, Walter Shewhart from Bell Laboratories, made the distinction between common causes and special causes of variation. The key is to keep

variations under control. From now on we will discuss how to elaborate control graphs which help managers and workers develop a process capable of producing within the established limits. Firstly, when elaborating control graphs, small sample averages are used, instead of the data from individual part. This is due to the fact that individual pieces tend to be very erratic in order to quickly visualize tendencies. The finality of control graphs is to help distinguish between natural variation and variations due to assignable causes. Natural variations affect almost all production processes and must be expected. They are random and uncontrollable. When the distribution of this variation is normal it will have two parameters. 1. Mean,  (the central tendency measure, in this case the average value) 2. Standard deviation,  (variation, the amount by which the small values differ from the large ones) It needs to be pointed out that while the distribution is kept within the specified limits, it is said to be in control and modest variations are tolerated. On the other hand, assignable variations are related when a process is not under control and the special (assignable) causes of variation must be detected and eliminated. In this case, variations are not random and may be controlled. Control panels help identify where there might be a problem. The objective of a control system of processes is to provide a statistical sign when assignable variation causes are present. Assignable variations in a process are traced to a specific problem. Control Graphs for Variables Control graphs for the mean x and the range, R, are used to watch processes measured in continuous units. On the other hand, the x graph (x-bar graph) indicates us if there have been changes in the central tendency of a process. Values on the R graph indicate a gain or a loss in uniformity. Both graphs must be used when monitoring variables. The statistical basis for x graphs is the central limit theorem. The theorem establishes that the distribution of sample means will follow a normal distribution as the size of the

sample grows. Even with small size samples, the distribution is almost normal. It also establishes that: 1. The mean of the sample distribution will be equal to the population mean. 2. The standard deviation of the sample distribution will be equal to the standard deviation of the population divided by the square root of the size of the sample, as shown in the following equation: We frequently estimate and µ with the average of all sample means (identified as x). Figure 2 shows three possible population distributions, each one with its own mean, µ and standard deviation. Figure 2. Sample and population distributions Taken from Render et al. (2018) for educational purposes Since it is a normal distribution we can establish that:

1. 99.7% of the times, sample averages fall within the population mean, if the process only has random variations. 2. 2. 95.5% of the times, sample averages fall within the population mean, if the process only has random variations. 3. If a point in the control graph falls outside the control limits, we are 99.7% sure that the process has changed. To see additional information and download example templates from Excel you may access https://ingenioempresa.com/grafico-de-control/ Ingenio Empresa. (2016). Gráfico de Control. Recovered from https://ingenioempresa.com/grafico-de-control/ Five Steps to use X and R Graphs 1. Collect 20 to 25 samples of n = 4 or n = 5 from a stable process and calculate the mean and range of each one. 2. Calculate the general means (x and R), establish proper control limits, usually at the level of 99.7% and calculate the preliminary superior and inferior control limits. 3. Graph the means and ranges of the samples in their respective control graphs, and determine if they fall outside the acceptable limits. 4. Investigate the points and patterns that inidicate the process is out of control. Try to assign variation causes, and then retry the process. 5. Collect additional samples and, if necessary, revalidate the control limits with the new data. Attribute Graphs Control graphs for X and R are not applied when there are attributes shown, which may be classified as defective or non-defective. Measuring defective articles implies counting them (as, number of bad electric bulbs at a given lot, or number of letters or data for registers mistakenly entered). There are two types of attribute control graphs: 1. Those which measure the percentage of defects in a sample, called p graphs 2. Those which count the number of defects, called c graphs On Topic 8.2 we will be applying the statistical control of processes by using the above mentioned graphs.

Below, there are article or study references which may help you delve into the topic of statistical control of processes and total quality, which are found in the folder of complementary material. Raheem, M. A., Gbolahan, A. T. & Udoada, I. E. (2016). Application of Statistical Process Control in a Production Process. Science Journal of Applied Mathematics and Statistics, 4(1),1-11. Leakemariam, B. & Tesfay. G. (2016). Assessing the Awareness and Usage of Quality Control Tools with Emphasis to Statistical Process Control (SPC) in Ethiopian Manufacturing Industries. Intelligent Information Management. 8. 143-169. Madanhirea, I. & Mbohwa, C. (2016). Application of Statistical Process Control (SPC) in Manufacturing Industry in a Developing Country. Procedia CIRP, 40, 580 – 583. ------------------------------------------------------------------------------- Introduction to Topic 8.2 On Topic 8.2 we will apply statistical methods in the solution of problems applied to the manufacturing industry and client service. The student will be taught to understand and apply variable control graphs and attribute control graphs. The student will also use statistical packages such as Excel QM to solve quality control problems applied to business. Topic 8.2 Applied Quality Control Below, we will be discussing examples for the use and application of variable graphs and attribute graphs. Establishment of Limits on X Bar Graphs If the standard deviation is known from historical data of the population in the process, the superior and inferior limit controls may be established with the following formulas: Sumperior Limit Control (LCS) = Inferior Limit Contro LCI) = Where: x = mean of sample means Z = number of normal standard deviations (2 for 95.5% of trust, 3 for 99.7%) = standard deviation of the sample distribution of sample means = Manufacture: Box Filling Example Let’s say a large production lot of corn flake boxes is sampled each hour. In order to establish control limits which include 99.7% of sample means, 36 random

boxes are selected and weighed. The standard deviation of the total population of boxes is estimated in 2 ounces, through the analysis of previous records. The average mean of all samples taken is 16 ounces. As so, we have x = 16 ounces, = 2 ounces, n = 36 and z = 3. The control limits are: Application Excel QM When you open the program, choose the method Quality Control/ x-bar and R Charts. On thr following screen enter the number of samples, size of sample, and select standard deviation, as shown on image 1. Image 1 When the calculation sheet begins, enter thr size of the sample (36), the standard deviation (2) and the sample mean (16). The superior and inferior limits are displayed immediately. See image 2.

Image 2 Manufacture: Super Cola Example Super Cola refreshment bottles are labeled with a “net weight of 16 ounces”. A general average of the process has been found at 16.01 ounces, taking samples from various lots, where each sample contains five bottles. The average range of the process is 0.25 ounces. We want to determine the superior and inferior control limits for the averages of this process. Application with Excel QM From the menu on Excel QM, select Quality Control and specify the X-Bar and R Charts options. Enter the number of samples (1) and choose Range on the start window. The sample size (5) may be entered in this start window or in the calculation sheet. See image 3. Image 3 When the calculation sheet is started, enter the range (0.25) and the sample mean (16.01). The superior and inferior limits appear immediately. See image 4.

Image 4 Application of Attribute Graphs in the Evaluation of Performance and Client Service Below, we will see an example for attribute p and c graphs. P Graph Use: ARCO Example With a popular software package for data bases, ARCO assistants enter daily thousands of secure records. The work samples of 20 assistants are indicated in the following table: Sample Error Defect Sample Error Defect Number Numbe Fractio Number Number Fraction s rs n s s 1 6 0.06 11 6 0.06 2 5 0.05 12 1 0.01 3 0 0.00 13 8 0.08 4 1 0.01 14 7 0.07 5 4 0.04 15 5 0.05 6 2 0.02 16 4 0.04 7 5 0.05 17 11 0.11 8 3 0.03 18 3 0.03 9 3 0.03 19 0 0.00 10 2 0.02 20 4 0.04 80 One hundred entered records by each one were carefully examined to determine if they contain errors; the defect fraction in each sample was calculated. Figure 1 shows the results on the graph for the ARCO example. Figure 1. ARCO Graph

Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Application Excel QM Excel QM may be used to develop the limits for a p graph, to determine which samples exceed the limits, and to elaborate the graph. On the drop down menu, choose Quality Control and specify the p Charts option. Enter the number of samples (20), enter the title if you wish, and select Graph if you wish to see the p graph. When the calculation sheet is initiated, enter the size of each sample (100) and the number of defects in each one of the 20 samples. One sample (number 17) is identified as a sample that exceeds the limits. See images 5 and 6 in which the results and graph of the program are seen. Image 5 Image 6

C Graph Use: Red Top Cab Taxi Company Example Red Top Cab receives various complaints each day about the behavior of its drivers. On a 9 day period (in which days is the measure unit), the owner receives the following number of phone calls from angry passengers: 3, 0, 8, 9, 6, 7, 4, 9, 8, with a total of 54 complaints. Application Excel QM Excel QM serves to develop the c graph limits, as well as to determine which samples exceed the limits and to elaborate the graph. From the drop down menu select Del menú Quality Control and specify the c Charts option. Enter the number of samples (9 days, in this example), enter the title if you wish, and choose Graph if you wish to see the c graph. When the calculation sheet is initiated, enter the number of complaints (defects) in each of the 9 samples. See images 7 and 8. Image 7. Superior Result Sheet

Image 8. Inferior Result Sheet: Graph Below, you will find references of articles or studies that may help you delve into the topic of statistical control of processes and total quality, which are found in the folder of complementary material. layinka, A. & Ikubanni, P. (2017). Application of Statistical Quality Control (SQC) in the Calibration of Oil Storage Tanks. Journal of Production Engineering. 20. 127-132. Korie, C. & Adubisi, O. & Ben, J. O. (2017). Statistical Quality Control (of the Production Materials in Lager Beer, FUW. Trends in Science & Technology Journal, 2. 1-69. Sarina, L. & Jiju, A., Norin, A. & Saja, A. (2015). A systematic review of statistical process control implementation in the food manufacturing industry. Total Quality Management & Business Excellence. 1-14.

Simanováa, L. & Gejdoš, P. (2015). The Use of Statistical Quality Control Tools to Quality Improving in the Furniture Business. Procedia Economics and Finance, 34, 276 – 283.