Statistical Quality Control - Coursework Example

Introduction In the current world of business the quality of products and services is considered to be a major decision factor. Whether a consumer is being looked at as an individual, a corporation, a retail store or a military defense program, when a decision is being mad by the consumer with regard to making a purchase, quality will definitely be of great importance just as cost and schedule. It is with his regard that quality improvement has become a major concern to many U.S. corporations. Statistical quality control as a field may be defined as being statistical and engineering methods that are employed in measurement, control, and monitoring as well as improving quality. This field can be traced back to 1920s with Dr. W alter A Shewhart being the pioneers of the field. Quality improvement and statistics In order to have quality improvement there is need to have there is need to statistical quality control which is a collection of tools. Quality refers to fitness for use. Quality is determined by both design quality and quality of conformance. Here quality of design refers to different levels or grades of performance, serviceability, reliability and functions that come as a result of engineering and management decisions being made. On the other hand quality of conformance refers to having systematic reduction in the level of variability with defects being eliminated up to a point where every unit that is produced is identical with no defect. Quality improvements are sometimes misunderstood as being gold plating of some products or simply spending some extra in development of a product or a process. Introduction to control chart The ability to quickly detect the occurrence of assignable causes or process shifts with the aim of investigating the process and to undertake the relevant corrective measure is seen as the main objective of statistical process control. The control chart comes in handy as an online process-monitoring technique which is widely applied specifically for this purpose. Figure 1 Control charts are also applicable in the estimation of the parameters of production process and with such information; there is determination of the capability level in meeting the specification. The chart is also important in providing information that used making improvement to the process. Even though complete elimination of variability is not achievable, control charts are useful in reduction of the variability as much as possible. In figure 1 a clear illustration of typical control chart where the quality characteristics that is measured or obtained through computation is plotted against time or the sample number. The usual practice is to take samples at periodic intervals such as an hour. In the chart CL is the centre line and it gives the average value of the quality characteristics that corresponds to the in-control state. The UCL (upper control limit) and the LCL (lower control limit) are also given on the control chart. The choosing of the control limits is such that when the process is in control; there will be very high probability that almost all sample points will lie between the two limits. The guiding principal is that when the points are within the control limits, the assumption is that the process is in control and no need to take any action. On the other hand having a point laying outside the limits would be an indication that the process is out of control warranting an investigation and corrective measure to be applied so as to eliminate the assignable cause that is bringing about the behaviour. Figure to is an illustration of what actions need to be taken a response to assignable causes of variation. Figure 2 In order to give a general model for control chart we take W to be a sample statistic which is a measure of a quality characteristic and taking the mean of W as  and with a standard deviation of  . This case the quantities UC, UCL, and LCL will be given by where k represents the distance of the control limit from the center line mw represents the mean of some sample statistic, W. While sw represents standard deviation of some statistic, W. X-bar & R or S control chart In addressing the issue of quality characteristics in products or services where it can be expressed as a measurement, usually what is monitored is the mean and variability of quality characteristic. Controlling the average quality is achieved by use of control charts for averages usually referred to as the chart. Control of process variability is achievable by use of the range chart (R Chart) or a standard deviation chart popularly known as S chart this depending on the manner of estimation of the population standard deviation. Where the standard deviation  and mean of the process are known and there can be assumption that the quality characteristic is normally distributed. Considering the X bar () chart where we know that  can be used to represen5t the centre line in the control chart with the upper and lower limits being placed at    eqn1 For the case where the parameters and  are not known, then they will be estimated using the preliminary samples as the basis of the estimation where the samples are taken when the process is assumed to be in control. A preliminary sample of 20 to 25 is usually recommended and in case we have m a the number of preliminary samples each of size n. typical cases involve n being 4, 5 or 6 where the small sample is usually used often arise when rational subgroups are constructed. Letting the sample mean in the ith sample to be , there can be estimation of the population mean  using the grand mean Such that  eqn 2 In this case the centre line can be taken as  on a a X bar control chart. The estimation of  can be achieved by use of either the standard deviation or use of observations in each of the samples. Having sample size that is relatively small ensures little loss efficiency wise in the estimation of  in the sample ranges. There is need to establish the relationship between a sample range in a normal distribution population which has known parameters and standard deviation. With R being a random variable the relative range is a quantity given by  eqn 3 which is also random with the parameters of the distribution of W being determined in any sample size n. The mean for the distribution W is referred to as d2 where we have tabulation of the d2 in various n. W will have a standard deviation d3 and with   eqn 4 Taking Ri to be the range of ith sample and having the range being expressed as  eqn 5 In which case  is taken as the estimator of  and considering equation 4 an unbiased estimator of  may be obtained from  eqn 6 The upper and lower limits for the X bar chartr may be applicable such that   eqn 7 The constant can thus be defined as  eqn 8 Now having done the computation of the sample values  and  , we can define the X bar control chart as  eqn 9 Where there is tabulation of constant A2 for various sample sizes. The R chart parameters can also be determined where we have a CL being . in this case when in order to determine UCL and LCL there is need to have an estimate of  as well as the standard deviation R. Here the assumption is also that the process will be in control and W which is the distribution of the relative range is also important. The estimation of  can obtained from  eqn 10 And this will give as UCL and LCL as  eqn 11 With  and  the R chart definition is given by  eqn 12 In which  is the sample average range and there is a tabulation of constants D3 and D4 In the R chart there is a possibility of the LCL being a negative and where that happens the LCL is set at zero. With the points plotted on an R chart not taking negative values, there is elimination of any chance for a point falling below an LCD of zero. In cases where there is use of preliminary samples in the construction of limits for control charts the limits will customarily be taken as being trial values. In which case the m sample means and ranges are supposed to be plotted on the required charts, and any points that goes beyond the control limits need investigation. Where there is discovery of assignable causes then there will be need for their elimination with new limits for the control charts being determined. By taking this step the process is likely to come to statistical control and then there may be assessment of the inherent capabilities. The R chart are studied first because if there is lack of consistency in process variability over time then the calculated control limits may not be reliable. Use of S chart may be adopted where there is calculation of standard deviations of each of the subgroups with the standard deviations being used in monitoring of the process standard deviation. When it comes to S chart, the common practice I utilization of the standard deviations in development of the control limits in the chart. Usually the sample size being used for the subgroups is smaller with less than 10 items and this there is not appreciable difference in the chart that is generated from the standard deviations or the ranges. But with computer softwares being often being applied I the implementation of the control charts, it follows that S charts are the most common. With S being a biased estimator of  such that  (eqn 13) with c4 being a constant that I close to but not equal to 1 similar calculation to that used in E(S) can be used in deriving the standard deviation of the statistic S such that  (eqn 14) and this gives the center line and the the three-sigma control limits for S as    (eqn 15) Assuming the presence of m preliminary samples available, each being of size n, and leting Si to denote standard deviation of the ith sample.  (eqn 16) With , an unbiased estimator of  will be  (eqn 17) Thus a control chart for a standard deviation will have    (eqn 18) The LCL in a S chart can take a negative value for which case the normal practice is setting LCL to zero. In using an S chart the estimate of  given in eqn 15 is applied in the calculation of the control limits for X bar chart. This results to the control limits being    (eqn 19) Control chart for individual measurement More often we have situations where the sample size applied in sample control will be such that we have n =1which translates to the sample consisting of an individual unit. Typical examples where this is experienced include the following Where there is use of automated inspection and measurement technology with each of the units being inspected In cases where a very slow production rate are involved thus making it an inconvenience if sizes of n>1 were to be allowed to accumulate before effecting analysis Where repeat measurements encountered on a process are believed to record some differences only as a result of laboratory errors or errors due to the analysis as experienced in most chemical processes Process plants including measurements made in papermaking involving parameters such as the thickness of coat on a roll will exhibit a very small difference with a very small standard deviation in the case where the main objective to have control over the coating thickness on the roll. In some of the cases individuals control chart is found to be of great use. The individuals control chart involves the using a moving range in two successive observations so as to come up with an estimation of process variability. Process capability Having some information on process capability is important where the level of performance of a process when its operation is control is evaluated. The tolerance chart also referred to as tier chart and the histogram are put into use in the assessment of process capability. Figure 12 is an example of a tolerance chart involving 20 samp0-les drawn from a vane-manufacturing process where we have the vane opening of 0.50300.0010 as the specifications. With regards to the coded data the USL=40 we have LSL=20 as indicated on the chart. Measurements that are connected with a line indicate they are in the same sub group. A tolerance chart is very useful when it comes to revealing some patterns exhibited over time with regards to individual measurements or the chart may be used in showing that some value of  or r may have been brought about just one or two observation in the sample that are unusual. It is also worth noting that it may be in order plotting specification limits on the chart as it is a chart with focus on individual measurements Figure 13 shows a histogram where measurements of vane opening are displayed. In the histogram there has been elimination of observations linked to samples 6,8,9,11 and 19 which corresponds to out of-control points in the  or R chart. The impression that one would have through examination of the histogram is that the process the process can meet the expected specification but it runs off-center. Figure 3 Figure 4 Attribute control chart P Chart (Control Chart for Proportions) Many a times the desire is to have products classified as either falling into defective group or nondefective group through comparison with a standard. Through this approach simplicity and cost effectiveness is realized. A typical case is where the diameter of a ball bearing may be checked through determination on whether it will go through some circular holes that are cut in a template. When such an approach is taken then there is use of control charts for attributes. One of the characteristics of the attribute control charts is the need to have a large sample size as compared in the case of variable measurements. P chart which may also be referred to as fraction-defective control chart is an example of control charts for attributes. Supposing we have D being the number of units that are defective when a random sample of size n is considered and assuming D being a random variable which is binomial in distribution with an unknown p parameter. In which case the fraction defective given by  (eqn 20) for each of the samples will be plotted on the chart. The variance of the statistic  on the other have is given by  (eqn 21) Thus there can be construction of a p chart by p as the centre line with the control limits being given as  (eqn 22) But more often the true process fraction defective is unknown and it calls for its estimation by use of data obtained from preliminary samples. Taking the scenario where we have m as the preliminary samples each of size n, and with Di being the number of items that are defective in sample i . Then  (eqn 23) can be seen as being the sample fraction defective in ith sample. We will have the average fraction defective as  (eqn 24) In which case  is to be used as an estimator of p as the centre line and in the calculation of the control limit This will result to  (eqn 25) With  representing the observed value in the average fraction defective Control chart performance In the design of a control chart specifying of the control limits is seen as being one of the critical decisions that are to be made. When control limits are moved further there will be a decreased chance of committing type 1 error meaning that the risk that a point will fall beyond the control limits such that it indicates that there is out-of-control condition where there is no presence of assignable condition. On the other hand where we have a wide control limits the results will be increased type II error risk- meaning that there is increased risk that a point will fall between the control limits when a process is out of control. Moving the LCL and UCL nearer the CL will result to the opposite effect. On a Shewhart control chart the control limits often will be located a distance of 3standard deviations of the variable that is being plotted on the chart away from the centre line. This means that the constant k used in eqn 1 is to be assigned a value of 3 with the limits being referred to as 3-sigma control limits. Average run length (ARL) of a control chart is seen as being away of decisions evaluations concerning the sample size and the frequency at which sampling is done. The ARL may be considered as being the average number of points which are necessary to be plotted before a point can be passed as being out of control. Montgomery (2001) observation is that in any Shewhart control chart, the calculation of the ARL can be done by use of the geometric random mean. Taking a t case where we have p as the probability of any point going beyond the control limits then we can have the relationship  (eqn 26) Thus considering an X bar chart having 3-sigma limits with p=0.0027 as the probability of a single point falling beyond the limits for a process that is in control. The average run length associated with the X bar chart for the case when the process is in control is The interpretation of this is that on average there should be an out of control signal generation in every 370 points even when the process remains in control. Time-weighted chart One of the time weighted charts that is considered to be the best alternative to a Shewhart control chart is the cumulative sum control chart otherwise referred to as CUSUM. The chart displays much better performance with regards to ARL when it comes to detection of small shifts when compared to Shewhart chart, but this chart does not result to a significant drop in-control ARL. CUSUM charts involves the plotting of the cumulative sums I the deviations away from a target value of the sample values. Taking an example where we have a sample size have been collected with the average jth sample being  . Suppose we have as being the target process mean, the CUSUM chart is formed by having the quantity  (eqn 27) Being plotted against number i. Here  is referred to as cumulative sum up to ith sample inclusive of the ith sample itself. Other SPC problem solving tools As much as control chart appears to be a powerful tool in the investigation of the causes that could be causing variation in process, its effectiveness can highly be enhanced by use of some other SPC tools. As can be seen in the U chart showing the number of defects where there is a sample consisting of 5 printed circuit boards. In the chart there is exhibition of statistical control but there is need for reduction of the defect number. In the example there is 8/5=1.6 defects on one board a situation that would call intensive level of rework. The first step would involve the construction of a Pareto diagram where individual defects are to be highlighted as can be observed in Figure 6 where it is observed that insufficient solder and solder balls are the defects that have the highest frequency of occurrence accounting for 68% of all the defects. Also a keen look at the chart reveals that the first five defects are solder related defects which is a clear pointer that flow solder process give an opportunity for improvement. Figure 6 Implementation SPC When Statistical process control (SPC) is applied successfully by any company there will be high chance of that there will be significant payback realization in the companies. As much as this method would appear as being made of a collection of problem based tools that have their basis in statistics, more successful use of the method is achievable beyond just learning and use of the tools. The management being involved and showing high commitment in improving of quality in the processes is of vital importance for success realization in SPC application. Management is supposed to serve as an example to other people in the organization who will look upon them for guidance. Having a team approach is another important aspect, for one person attempting to effect process improvement is almost an impossible undertaking. Many of the tools that are used in problem solving may be of great help when it comes to building an improvement team which includes pareto charts, defect concentration diagrams as well as the cause-and effect diagrams. It is desirable that the basic SPC tools to be widely known and applied in the entire organization. It is important to have continuous SPC training and quality improvement is also necessary if there is to be achievement of the widespread knowledge of the tools. It is worth noting that SPC is not to be taken as a program that is to be applied once at the point when business is in trouble but rather improvement based on SPC quality improvement program is to be continuous improvement that is to be undertaken on a regular time intervals that could be weekly, quart5erly or a yearly basis. It is important for SPC to be taken as a culture or the organization with the control chart being a vital tool in process improvement. It should not be expected that any process to operate naturally in-control state, and application of control charts comes out as an important step which when undertaken early enough in SPC program as a measure of elimination of assignable causes would bring down process variability with the process performance being stabilized. In order to have quality and productivity level improvement it calls for management of data and facts but not just to rely on ones judgment. Control charts play a central role in effecting this change in management approach. In the implementation of a SPC program that covers the entire company the elements that are likely to feature include Leadership management Team approach Investing education of employees education Emphasis being placed on continuous improvement Putting in place a mechanism for success recognition Importance of management leadership and having a team approach is something that cannot be overemphasized. Measurement of progress and level of success and knowledge of this success being spread in the entire organization need to be encouraged. Communicating successful improvements in the entire company is likely to be a source of motivation and would serve as incentive for improvement in other processes with continuous improvement being made a normal component of running the business. A framework of implementation of quality and improving productivity is provided by the philosophy by W. Edwards which is summarized in 14 management points. The success in Japan’s industrialization has been linked to recognizing and applying of these management principles and this continues to catalyze the quality and productivity improvement efforts in the country. Case study: Analysis of number of defects in a printed circuit board This case study involves analysis of number of defects in a printed circuit board (PCB) which is one of the components of electronic components of vehicles manufactured by Toyota. PCBs assembly involves a combination of manual and automatic process. A flow solder machine is usually used in making the mechanical and electrical connections for the leaded components. The boards go through a flow solder process in almost a continuous pattern and in an hour there is selection of five boards which are then inspected as a way of process control. The total number of defects in each of the samples is established and Table 234 gives the results of 20 samples. Table 1 Sample Defects no. 2 Defects/unit Sample Defects no. 2 Defects/unit 1 6 1.2 11 9 1.8 2 4 0.8 12 15 3.0 3 8 1.6 13 8 1.6 4 10 2.0 14 10 2.0 5 9 1.8 15 8 1.6 6 12 2.4 16 2 0.4 7 16 3.2 17 7 1.4 8 2 0.4 18 1 0.2 9 3 0.6 19 7 1.4 10 10 2.0 20 13 2.6 The U chart will have a CL given by =1.6 The UCL and LCL are given by The control chart is as shown in figure 7 where with LCL being negative, it is set to 0. A look at the chart clearly shows the process is in control but the worry is that having 8 defects in every group of 5 circuit boards is a worrying trend and this calls for improvement in the process. an investigation is launched at this point so as to establish the specifically what are the defects that are found on PCBs which in turn will reveal areas that calls for improvement. Figure 7 Conclusion In this paper the relevance of SPC has clearly been demonstrated. It was seen that importance of management leadership and having a team approach is something that cannot be overemphasized. Measurement of progress and level of success and knowledge of this success being spread in the entire organization need to be encouraged. Communicating successful improvements in the entire company is likely to be a source of motivation and would serve as incentive for improvement in other processes with continuous improvement being made a normal component of running the business. A framework of implementation of quality and improving productivity is provided by the philosophy by W. Edwards which is summarized in 14 management points. The success in Japan’s industrialization has been linked to recognizing and applying of these management principles and this continues to catalyze the quality and productivity improvement efforts in the country. References Ishikwa, K (1982). Guide to quality control. White plains, NY: unipub-Kraus International. Juran.]. M. (1988). Juran on planning for quality. New York press. Juran.]. M. (1986). Management of quality(course materials). Wilton, CT: Juran Institute, Inc. McConnell, J. (1987). Analysis and control of variation. Australia: Delaware. Read More