In previous classes we mentioned statistics at two moments: when we talked about control charts in basic quality tools and when we talked about the beginnings of quality. Statistical process control (SPC) replaced, for example, the old inspection of processes, whether in purchasing, materials receiving, final product compliance, etc., with inspection based on sampling.Objectives of Statistical Process ControlOriginally, SPC aimed to monitor a product or service during its production process, because if it showed problems, the procedure would be stopped so that the failures could be corrected and everything would return to its normal condition.Nowadays, more than a statistical tool, SPC is seen as a management philosophy and a set of techniques and skills, originating from statistics and production engineering, aimed at ensuring the stability and continuous improvement of a production process.Thus, SPC has as its main function the control and improvement of the process, its fundamental principles being:• thinking and deciding based on data and facts;• thinking by separating cause from effect, always seeking to know the root cause of problems;• recognizing the existence of variability in production and managing it;• instantly identifying sources and locations of dysfunction and correcting problems in time.SPC can be applied to any process. For this purpose, there is a set of methods and tools that can help with application, such as: histograms, checklists for verification and control, Pareto charts, cause-and-effect diagrams (Ishikawa), and control charts.Control ChartA control chart involves regular chronological records of one or more characteristics calculated from samples obtained from production. These values are entered, in chronological order, into a chart that has a center line and two limits, called control limits (lower and upper).We can verify that process variability can arise from common causes or special causes.Common causes of variationThese are considered random and unavoidable. When the process presents only common causes of variation, the process variables follow a normal distribution. For example, the weight of bagged rice from a food distributor will follow a normal distribution if the process presents only common causes of variation that are within the control limits.Special causes of variationThese occur for clearly identifiable reasons and can be eliminated. Special causes change the process parameter, mean, and standard deviation, because they are outside the control limits. For example, equipment whose calibration is incorrect will present special variation. Adjusting it will make the variation disappearHow to use the control chartYou may be wondering what a control chart is for. The answer is simple: to check the causes of variation and act on them whenever necessary, in order to avoid losses in the production process. Thus, the variable control process must follow five phases:1. Determine the type of control chart to be used.2. Establish a plan for taking samples from what is being produced.3. Determine the limits of the control chart and the mean for each variable that will be controlled.4. Place the values found on the charts, checking whether these values are within the chart limits, in which case the process will be under control.5. Analyses and situations: the results obtained must be analyzed, checking whether there is a need for some type of action.PHASE I – Determine the type of control chart to be usedThere are two basic types of control charts.Quantitative variable chartThese are used when the samples can be represented by quantitative units of measurement (weight, height, length, etc.). The best known are the charts of the mean and range (X and R), the mean and standard deviation (X and σ), and individual values and range (X, R).• (X and R): these are the charts of the mean and range. They are the most widely used. The charts of mean and range complement each other and should be implemented simultaneously. The chart aims to control variability at the process average level and any change that occurs in it. It is also very important to check the dispersion of a process, which may undergo changes due to assignable causes. This increase in variability will be detected by the R chart of ranges. In this case, we are speaking specifically of special variations, such as an unadjusted piece of equipment.• (X and σ): these are the charts of the mean and standard deviation. They are an option to the mean and range charts, and later on we will learn how to calculate and interpret them.• (X, R): charts of individual values and range. In some cases, it may be more convenient to control the process based on individual readings rather than samples. This occurs particularly when inspection and measurement are expensive, the test is destructive, or when the characteristic being examined is relatively homogeneous (such as the pH of a chemical solution).Qualitative variable chartare used when there are situations in which quality characteristics cannot be measured numerically. For example, a light bulb is classified as “works” or “does not work.” In other words, there are cases in which quality characteristics are better represented by the presence or absence of an attribute, rather than by some measurement. The best known are:• p charts: for controlling the proportion of defective units in each sample;• np charts: for controlling the number of defective units per sample;• c charts: for controlling the number of defects per sample;• u charts: for controlling the number of defects per unit of product.Another example is using parameters from other companies to assess the situation of a product. For example, checking a bank’s classification regarding investing in a certain company: "do not invest", "invest with caution", "invest".PHASE II – Establish a plan for taking samples from what is being producedBefore data collection is carried out, it is necessary to choose the sample size, also called rational subgroups, as well as the sampling frequency and the number of samples to be collected.In sampling, it is essential to choose samples that represent subgroups of items that are as homogeneous as possible, aiming to highlight differences between the subgroups. This aims, if present, to make special causes manifest through differences between the subgroups. The goal here is to eliminate bias in sampling.Pay attention to this example: imagine that you want to conduct a survey about voting intentions for the government of the state of Rio de Janeiro. Would you interview people from a single neighborhood? From a single age group? Certainly not, because this sample would not represent the whole population and your data would certainly be biased. With the sampling we are talking about here, it is the same thing: we must seek samples that represent each subgroup. We call them representative samples. For example, we cannot take samples from only one machine, nor only from one work shift.PHASE III - Determine the control chart limitsThe limits will be calculated as if we were calculating outliers. We had a lesson here about what outlier means and how to calculate it.There is a standard for identifying whether something is an outlier, that is, something off the line. Generally, we check whether the probability of it happening is extremely low. The criterion for "extremely low" is whether the probability of occurring is less than 0.1%.According to the normal distribution rule, for something to have less than 0.1% chance of occurring, it must be 3 standard deviations away from the mean.Thus the limits will be calculated as follows: the upper limit will be the mean plus 3 times the standard deviation; the lower limit will be the mean minus 3 times the standard deviation.Case with sampling with meanIf you know the probability of something, such as a product being defective, its standard deviation (σ) in a sample (n) will be given by the following formula (using the Binomial Distribution):\[ σ =\sqrt{np(1-p)} \]And its mean will be p*n.That is, if a product has a 10% chance of being defective and you will take a sample of 100 units. On average, you will find 10% of 100 defective products, that is, 10 defective products. And the standard deviation will be:\[ σ =\sqrt{100 · 10% · (1 - 10%)} \]\[ σ =\sqrt{10 · 90%} \]\[ σ =\sqrt{9} \]\[ σ =3 \]So how would we make the control chart for this sample?\[ \text{Upper limit} = \overline{x} + 3 * σ \]\[ \text{Upper limit} = 10 + 3 * 3 \]\[ \text{Upper limit} = 19 \]This means that defective products are already expected. But we will only really be concerned about the equipment after finding more than 19 defective products in a sample of 100.\[ \text{Lower limit} = \overline{x} - 3 * σ \]\[ \text{Lower limit} = 9 - 3 * 3 \]\[ \text{Lower limit} = 1 \]The same reasoning applies to the lower limit. We will only be concerned if there is a problem if there are fewer than 1 defective product.These two situations (having more than 19 defective products or having fewer than 1) are so unusual and difficult to occur (less than 0.1%) that the machine is probably not set up correctly.Case with sampling with probabilityThe standard deviation (σ) in this case changes slightly. Since the objective is not to calculate the number of defective parts but rather the probability of finding defective parts within the sample.In this case, remember that the larger your sample, the smaller the standard deviation relative to the mean, since everything will tend toward the mean (central limit theorem). According to the T-Student Distribution:\[ σ =\sqrt{\frac{p(1-p)}{n}} \]Recalculate the upper and lower limits now.With qualitative variablesIn qualitative situations, the criterion for creating upper and lower limits must be discussed with the team.Phase IV: Analysis and stability of the limitsStability refers to checking the variability of the process regarding its normal behavior (whether or not it is within control limits, for example).A process is considered stable when its control charts do not indicate signs of abnormality or the presence of special causes. In this way, processes under control present “well-behaved” charts, following a known pattern.To verify process stability, the paired charts must always be analyzed (of the mean and range, or of the mean and standard deviation) in search of signs of abnormality.The characteristics of a natural, stable pattern in a control chart can be summarized as:• the majority of points are close to the center line (about 68% within the ±1σ interval around the mean) without, however, excessive concentration in this interval, where “σ” is the standard deviation;• about 95% of the points (19 out of 20 points) are contained within the ±2σ interval around the mean;• no point falls outside the control limits (because its probability is only 0.27%);• the points are distributed more or less equally above and below the mean;• there are no trends of systematic increase or decrease. For example, the probability of seven consecutive points above the mean is 0.78%. Therefore, when this occurs, it is interpreted as an upward trend in the mean;• there are no cyclical oscillationsConclusionProcess control enables us to take corrective actions before deviations occur that could cause rework costs, scrap, or other types of losses.Always keep in mind that there is no other way to achieve higher productivity, lower costs, and consequently higher profit margins, unless we have production processes under control, that is, without special causes. If these exist, it will be up to you, a future production engineer, to apply your knowledge to eliminate them.Do not forget that it is not necessary for us to have points outside the control limits for us to have to act under the process. Remember, for example, that a process that follows a trend toward the lower or upper limits already indicates that it needs intervention. In other words, when we see that there is a car stopped ahead, we do not wait for the collision to take action. We brake or steer away to avoid the impact; that is, corrective action was taken before the event effectively became a problem.Exercises1) In a manufacturing process for toilet paper, each roll must have 30 meters. The cutting machine operates with an average of 30 meters, but from time to time it leaves the roll a little larger or a little smaller. The standard deviation for samples of 50 units calculated is 30 centimeters. Calculate the upper and lower limits. For which of the following values should you stop production to check the machine: 30.63 meters; 29.85 meters; 29.06 meters; 30.89 meters.2) There is a 5% probability of a product being defective. You take samples of 100 units to check the condition of the machinery. What are the upper and lower limits?
Sign in and join the conversation
— Comments 0
, Reactions 1
Be the first to comment