Statistical Quality ControlSimple applications of Statistics in TQM

IntroductionThe production processes are not perfect!Which means that the output of these processes will not be perfect correct and deterministic.Successive runs of the same production process will produce non-identical parts.Alternately, seemingly similar runs of the production process will vary, by some degree, and impart the variation into the some product characteristics.Because of these variations in the products, we need probabilistic models and robust statistical techniques to analyze quality of such products.

IntroductionNo matter how carefully a production process is controlled, these quality measurements will vary from item to item, and there will be a probability distribution associated with the population of such measurements.If all important sources of variations are under control in a production process, then the slight variations among the quality measurements usually cause no serious problems.Such a process should produce the same distribution of quality measurements no matter when it is sampled, thus this is a stable system.

IntroductionObjective of quality control is to develop a scheme for sampling a process, making a quality measurement of interest on sample items, and then making a decision as to whether or not the process is in the stable state, or in control.If the sample data suggests that the process is out of control, a cause is for the abnormality is sought.A common method for making these decisions involves the use of control charts.These are very important and widely used techniques in industry, and everyone in the industry, even if not directly related to quality control, should be aware of these.

SPCStatistical process controlMethodology for monitoring a process to identify special causes of variation and signaling the need to take corrective action.When special causes are present, the system said to be statistically out of control.If the variations are due to common causes alone, the process is said to be in statistical control.SPC relies heavily on control charts.

Quality control measurementsAttributes A performance characteristics that is either present or absent in the product or service under consideration.Examples: Order is either complete or incomplete; an invoice can have one, two, or more errors. Attributes data are discrete and tell whether the characteristics conforms to specifications.Attributes measurements typically represented as proportions or rates. e.g. rate of errors per opportunity.Typically measured by Go-No Go gauges.

Quality control measurementsVariable Continuous data that is concerned with degree of conformance to specifications.Generally expressed with statistical measures such as averages and standard deviations.Sophisticated instruments (caliper) used.

In statistical sense, attributes inspection less efficient than variable inspection. Attribute data requires larger sample than variable inspection to obtain same amount of statistical information. Most quality characteristics in service industry are attributes.

The chartA typical quality control plan requires sampling one or more items from a production process periodically, and making the appropriate quality measurements. Usually more than one items are measured each time to increase accuracy and measure variability. The chart helps the quality control person decide whether the center (or average, or the location of central tendency) of the measurement has shifted.

The chartSuppose that n observations are to be made at each time in which the process is checked. Let Xi be the ith observation (i = 1,,n) at the specified time point. Let be the average of n observations at time j. If

for a process in control, then should be approximately normally distributed with

The chartNow since has approximately a normal distribution, we can find the interval that will have a probability of 1- of containing .This interval is

If the mean and variation of the population (the and ) is known then,

we can set the lower control limit at

and the upper control limit at

The chartIf a control chart is being constructed for a new process, the and (the population parameters) will not be known and hence must be estimated from the sample data.For establishing control limits, it is generally recommended that at least 30 time points be sampled before the control limits are calculated.For each of the k samples, we compute the sample mean, the sample variance, and the range .Note that the range of a sample is the difference between the largest and smallest value in the sample.

The chart

The chart

The chartComputation of the mean standard deviation can be avoided by using the range data instead of calculating the adjusted standard deviation.

The chart

The r-ChartDeciding whether the center of the distribution of quality measurements has shifted up or down may not be enough. It is frequently of interest to decide if the variability of the process measurements has significantly increased or decreased.A process that suddenly starts turning out highly variable products could cause severe problems in the operations.Since we already have established the fact that there is a relationship between the ranges (Rj) and , it would be natural to base our control chart for variability on the Rjs.A control chart for variability could be based on the adjusted std dev also (Sj), but use of ranges provides nearly as much accuracy for much less computation.

The r-ChartUsing the same argument as control chart for central tendency (x-bar), one could argue that almost all the Rjs should be within three standard deviations of the mean of Rj. Now

The r-ChartAs in the previous chart, if is not specified, it must be estimated from the data. The best estimator of based on the range is , and the estimator of the control then becomes:

The r-Chart

Control charts for variable dataTwo basic types: x-bar-chart and R-chart one used to monitor centering of the process, the other for variation.Construction and establishing statistical control:

Interpreting patterns in control chartsGeneral rules to determine whether a process is in control:

No points outside the control limits.The number of points above and below the center line are about the same.Points seem to fall randomly above and below the center line.Most points are near center line, and only a few are close to control limits.

Basic assumption: Central Limit Theorem.

Interpreting patterns in control chartsOne point outside control limits:

measurement or calculation error, power surge, a broken tool, incomplete operation.Sudden shift in process average:

new operator or inspector, new machine setting.Cycles:

operator rotation or fatigue at the end of shift, different gauges, seasonal effects such as temperature and humidity.Trends:

x-bar-chart learning effect, dirt or chip buildup, tool wear, aging of equipment; R-chart (increasing trend) gradual decline in material quality; R-chart (decreasing trend) improved skills, better materials.

Interpreting patterns in control chartsHugging the center line:

sample taken over various machines canceling out the variation within the sample.Hugging the control limits:

sample taken over various machines not canceling out the variation within the sample.Instability:

difficult to identify causes. Typically, over-adjustment of machine.

Always, R-chart analysis before the x-bar-chart analysis.

Interpreting patterns in control chartsDownward trend in R-chart

Interpreting patterns in control charts.causes smaller variation in x-bar-chart.

Process capabilityParticularly process characteristics study: Performed over a period of time under actual operating conditions to capture the variations in material and operators.Here we differentiate between the control limits and specified tolerance limits (also called specification limits).Control limits are obtained from the data itself. Probability of finding similar data within the calculated control limits can be found.Specification limits are specified by product/process designers.Process capability study checks the probability of getting product dimension within the specified limits.

Process capabilityNote that we are interested in finding individual product dimensions within limits (and not the sample means).We need to use the estimate for population variance and not variance of sample mean.From the previous analyses, we have:

Process capabilityLet the specified tolerance limits be (a, b).To determine that the process of capable of producing parts within these limits, we calculate the probability of finding parts outside these limits. Hence, calculate

We calculate the probability area to the right of z1 and left of z2 from the standard normal table.Addition of these two probabilities gives us the probability that a part will not meet the specifications.

Process capability

Process capability

Process capabilityPCI Process Capability Index

The p-ChartThe control charts we looked at so far were applicable to quality measurements that possessed continuous probability distribution. This sampling is commonly referred to as sampling by variables.In many cases we merely want to assess whether or not a certain item is defective.We can observe a number of defective items from a particular sample of series of samples. This is referred as sampling by attributes.

The p-ChartSuppose a series of k independent samples, each of size n, is selected for a particular process.Let p denote the proportion of defective items in the population (total production for a certain time period) for a process in control.Let Xi denote the number of defective items in the ith sample.Then Xi is a binomial variable, assuming random sampling from a large lot, with mean E(Xi) = np, and Var(Xi) = np(1-p).We typically consider the fraction of defective items in a sample (Xi/n) rather than the observed number of defectives.

The p-Chart

The p-ChartThus the estimated control limits are given by:

The c-ChartIn many quality control problems the particular items being subjected to inspection may have more than one defect.We may wish to count # of defects instead of merely classifying at item as to whether or not it is defective.If Ci denotes the # of defects observed in the ith inspected item, we can safely assume that Ci has a Poisson distribution.Let this Poisson distribution have a mean of for a process in control.For us to be 99.74% sure, almost all the Cis should fall within three standard deviations of the mean if the process is in control.

The c-Chart

Acceptance sampling by attributesInspections in which a unit of product is classified simply as defective or non-defective is called inspection by attributes.Lots of items, either raw materials or finished products, are sold by a producer to consumer with some guarantee for quality.We determine quality, here, by the proportion p of defective items in the lot.To check this characteristics of the lot, the consumer will sample some of the items, test them, and observe the number of defectives.

Acceptance sampling by attributesGenerally, some defectives are allowed because defective-free lots may be too expensive for the consumer to purchase.But if the number of defectives is too large, the consumer will reject the lot and return it to the producer.Sampling is generally used because the cost of inspecting the entire lot may be too high or the inspection process may be destructive.Before sampling inspection takes place for a lot, the consumer must have in mind a proportion p0 of defectives that will be acceptable.Thus, if the true proportion of defectives p is no greater than p0, the consumer wants to accept the lot, and reject otherwise.

Acceptance sampling by attributesThis maximum proportion of defectives satisfactory to the consumer is called the Acceptable Quality Level (AQL).So, we have a hypothesis-testing problem. We are testing null hypothesis of the true defectives proportions being less than or equal to the acceptable proportion; against an alternative hypothesis that the true proportion is actually greater than the acceptable one.

Acceptance sampling by attributesA random sample of n items is selected from the lot, and the number of defectives Y is observed.The decision concerning rejection or non-rejection of Ho (and consequently rejecting or accepting the lot) will be based on the observed value of Y.The probability of Type I error () in this problem is the probability that the lot is rejected by the consumer when, in fact, the proportion of defectives is satisfactory to the consumer.This is referred to as the producers risk.

Acceptance sampling by attributesThe value of calculated at p = p0 is the upper limit to the proportion of good lots rejected by the sampling plan being considered.The probability of Type II error calculated for some proportion of defective p1, where p1 > p0, represents the probability that an unsatisfactory lot will be accepted by the consumer.This is referred to as the consumers risk.The value of calculated at p1 = p is the upper limit to the proportion of bad lots accepted by the sampling plan being considered, for all .

Acceptance sampling by attributesThe null hypothesis will be rejected if the observed value of Y is larger than some constant a. Since the lot is accepted if Y is less than or equal to a, the constant a is called the acceptance number. In this context, the significance level is given by:

Acceptance sampling by attributesSince p0 may not be known precisely, it is of interest to see how P(A) behaves as a function of true p, for a given n and a.A plot of these probabilities is called an operating characteristic curve (OC curve).

Acceptance sampling by attributesExample of OC curve plotting:Suppose that one sampling plan calls for (n = 10, a = 1) while another calls for (n = 25, a = 3). Plot the OC curve for both of them. If the plant using these plans can operate with 30% defective raw materials, considering the price, but cannot operate efficiently if the proportion gets close to 40%, which plan do you recommend?

Solution: We need to calculate the P(A) for various values of p. For each plan, P(A) is the binomial probability of finding at the most a number of defects in n trials.

OC curve plottingFor the first sampling plan, we have:

For the second plan:

p00.10.20.30.40.6P(A)10.7360.3760.150.0460.002

p00.10.20.30.4P(A)10.7640.2340.0330.002

OC curve plotting

OC curve plottingNote that OC curve for first plan (n = 10, a = 1) drops slowly and does not get appreciably small until p is in the neighborhood of 0.4. For p around 0.3 this plan still has fairly high probability of accepting the lot.The other curve for the second plan (n = 25, a = 3) drops much more rapidly and falls to a small P(A) at p = 0.3.Hence, the second plan would be better for inspection if the plant can afford to sample n = 25 items out of each lot before making a decision.

Demings kp ruleObjective is to minimize the average total cost of inspection of incoming materials and final products for processes that are stable.It calls for 0% or 100% inspection.It can be shown that, if the process is stable, the distribution of the nonconforming items in the sample is independent of the distribution of nonconforming items in the remainder of the lot.Items are submitted in lots, and a random sample is drawn from that lot to make a decision regarding the entire lot.

Demings kp ruleAssumptionsInspection process is completely reliable.All items are inspected prior to moving forward to the next customer in the process.The vendor provides the buyer with an extra supply of items in order to replace any nonconforming items that are found. The cost is assumed to included in the contract.

Demings kp ruleNotationsp: average proportion of nonconforming items in the lotsk1: cost of initial inspection of an itemk2: cost of repair or reassembly due to the usage of a nonconforming itemk: average cost to find a conforming item from the additional supply to replace a detected nonconforming item = k1/(1-p)xi =1 if item i is nonconforming; 0 otherwise.

Demings kp ruleThe unit cost of initial inspection and replacement, if an item is nonconforming, is given by:

Unit cost to repair and replace a nonconforming item:

These costs components are mutually exclusive (if one is present, other is zero.)

Demings kp ruleIf the item is inspected, the total cost is (k1+kxi). If the item is not inspected, the total cost is (k2+k)xi.It can be extended to the whole lot, where the proportion of nonconforming items is p.So, the average total cost per item if items are inspected is (k1+kp). And the average total cost per item if items are not inspected is (k2+k)p.The break-even point for p is obtained by equating the average total costs. We get: p = k1/k2.

Demings kp ruleIf k1/k2 is greater than p, conduct no inspection. This situation occurs if the proportion of nonconforming items is low, cost of inspection is high, and cost of repairing a nonconformity is low.If k1/k2 is less than p, conduct 100% inspection. In this situation, it is quite expensive for a nonconforming item to be allowed into production.If k1/k2 is equal to p, either no or 100% inspection may be conducted. Usually, if the estimated value of p is not very reliable, 100% inspection is conducted.

Critique of the kp ruleEstimating proportion p requires sampling. Contradiction: for sampling, initially some inspection would have to be carried out.Proportion p may not be stationary over a long period of time. Once again, sampling may be needed to verify.Estimating the costs is not easy. Linearity of costs may not hold.What 100% inspection policy, if it involves destructive testing?

Acceptance sampling by variablesAt times the characteristics under study involves a measurement on each sampled item.In these cases, one could base the decision to accept or reject on the measurements themselves, rather than simply on the number of defective items.Defective items would mean those items having measurements that do not meet the standards.When looking at actual measurements obtained from variables method, the experimenter may gain some insights into degree of nonconformance and may be able to quickly suggest methods of improvements.

Acceptance sampling by variablesFor variables, arriving at the correct sample size and correct acceptance factor is slightly tedious and depends on the non-standard statistical tables.US Military standards (MIL-STD-414) is one such standard for calculating the same.