Quality Control and Improvement

Here we will use some statistical techniques.

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Process Quality Control

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Process quality control is the inspecting or testing of the product or service while it is being produced.Implementation:1) Take a sample of output (more than 1 unit),2) Do some figuring (we see more later),3) If the figures are good, keep producing as you have been.If figures are bad, stop process and search for an assignable cause (the reason for the bad figures),4) If figures are bad and an assignable cause is found – fix it and start the production process again.(It is beyond the scope of the course to not be able to find the source of the problem.)

Random Variability

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Random variability, or common causes, is (are) going to happen in most every production process. As an example (that is not exactly like production, but has the flavor we want here), consider that I put on a pot of coffee most every day at the office. Do I put in the same amount of coffee particles and water with each pot? No! But, I do a close enough job so that from day to day the java is acceptable.

The aim of process control is to find the natural random variation of the process and to make sure the production stays in this range.

It is natural that people I will not make coffee exactly the same every day. But I am within limits of acceptability.

Special Causes

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Unnecessary variation, or special causes, is (are) the real problem (and are the assignable errors mentioned before). Lax procedures, untrained workers, or improper machine maintenance are examples that can be lead to too much variation in production.

New faculty have to be trained to make coffee because our coffee maker here is a little more sophisticated than an at home model. New faculty may not know how many scoops to put in or not know to turn on the warmer. We straiten them out so the coffee is awesome!

Quality Control Chart

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UCL

Average or CL

LCL

Time or successive sample observation

Quality characteristic being observed

Quality Control Chart

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On the previous slide I have a graph. On the “x” axis you see time, or more importantly at successive time periods we take a sample and study some characteristic. On the “y” we have various concepts related to the measurement.

The CL is the center line, or average will be our average of all observations up to the current sample. UCL is the upper control limit, or the maximum acceptable sample value. LCL is the lower control limit, or the minimum acceptable sample value.

What I do not have in the graph, yet, are actual sample values. We will get to that.

Quality Control Chart

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If we can assume the production process, when under control, has a normal distribution (and we will assume that here) then 99.74% of all sample means or proportions will be within 3 standard deviations from the mean. So, we make the UCL 3 standard deviations above the mean and the LCL 3 standard deviations below the mean.If the sample value obtained in a sample has a value outside the range of ± 3 standard deviations, then the process will be stopped and a search for assignable cause will be conducted.

Attribute Control

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Sometimes our quality characteristic is measured on what is called a discrete scale. In a sample we will have n observations. On each observation we will note if a defect is present. A defect is a bad thing. In a batch or sample of n we could have anywhere from 0 to n defects. This is the idea of discrete scale. Then we will call p the percent of defects in one sample and will be calculated as (the number of defects/n). In each sample we will calculated p. If we did this t times we would have p1, p2, …, pt. Each of these p’s would be put into the control chart.

The CL will be called “p bar” (really in the book this is the letter p with a line over it – it is a pain to type so I have you think p bar) and is calculated as (p1 + p2 + … + pt)/t

Attribute Control

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In this case the standard deviation is calculated as the square root of [(p bar)(1 – p bar)/n].Example from page 166Each sample has n = 200. t = 11 samples are taken.Note .005 is .5%. I prefer to leave the data as a relative frequency and thus p bar = [.005 + .01 + .015 + .02 + .015 + .01 + .015 + .005 + .01 + .015 + .02]/11 = .14/11 = .0127 when rounded to 4 decimals.

A standard deviation here is sqrt[.0127(.9873)/200] = .0079

Thus UCL = .0127 + [3(.0079)] = .0364 and LCL = .0127 - [3(.0079)] = - .011. Since LCL is negative and any p can not be negative we make LCL = 0

Example in book on page 166

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UCL = .0364

CL = .0127

LCL = 0

Time or successive sample observation

Percent of defects

The data points again are .005, .01, .015, .02, .015, .01, .015, .005, .01, .015, .02 and all fall with the limits from 0 to .0364 So, we have no special causes, or assignable cause, here! (note I did not plot each value – maybe you should – yea, yea, you do it!)

Variables Control

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Let’s think about an example. Have you ever purchased Captain Crunch cereal? (I especially like it the day after I have burned the roof of my mouth with hot pizza!) Well, each box probably does not have exactly the amount of ounces as printed on the box. Some have more, some have less. Over time the thinking is your purchases will average out. The variable ounces in a box is an example of a continuous variable. With really precise measurement we can get any value, even fractional amounts, for the actual ounces in the box.

When we have a continuous variable we will make two control charts – a central tendency (the average) chart and a variability (we will the range as the variability measure, where range is largest minus smallest value) chart.

Variables Control

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You may recall from your statistics training that when the mean is calculated that extreme values pull the mean toward the extreme value. Examples: Data set A: 1, 2 and 3 has mean = 2.

Data set B: 1, 2, and 9 has mean = 4.Data set B has extreme value of 9 and thus mean is moved toward the value of 9.If we just watch the average we could get tricked. Think back to our cereal box example. Data set a: boxes have values 19, 20 and 21 ounces for an average = 20. The range here is 21 -19 = 2Data set b: boxes have 10, 20 and 30 ounces for an average of 20. The range here is 30 – 10 = 20. The mean hasn’t changed but the range sure has. We look at charts for both averages and variability as measured by the range and stop the process if either gets outside its limits.

Variables Control

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Call x bar the mean of a sample and call R the range of a sample (not to be confused with R squared in regression). The values of interest for the average chart are:CL = x double bar or the average of the averages!UCL = x double bar + [A2 times R bar], LCL = x double bar - [A2 times R bar]. The regular points in the chart will be sample averages.

R bar is just the average of the sample ranges. A2 helps us get three standard deviations from the mean. The way to get the value was not even covered in basic stats. We will pick the value to use from the table on page 167 when we know the sample size. For example, if the sample size is 24 A2 = .157.

Variables Control

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The values of interest for the range chart are CL = R bar,UCL = D4 times R bar, andLCL = D3 times R bar. The regular points in the chart will be sample ranges.D4 and D3 are similar in nature to A2. It is beyond the scope of the course to have you calculate A2, D3, and D4. But, we want you to use them.Example on page 167A company has a machine that is producing bolts. Each hour a sample of six bolts are taken and the diameter of each bolt is measure. Say in 1 sample we have the values.536, .507, .530, .525, .530 and .520. The average is .525 and the range is (.536 - .507) = .029 (the R in the book should not be R bar for this one specific example – this is a typo.)

Example on page 167

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Note in this example you only have information on 1 specific example. The sample size of the sample is n = 6. We are toldx double bar is .513 and R bar is .02.For the average chart we have values:CL = .513UCL = .513 + .483(.02) = .523LCL = .513 - .483(.02) = .503For the range chart we have values:CL = .02UCL = 2.004(.02) = .040LCL = 0(.02) = 0Again we only had one sample in this example with x bar = .525 and R = .02. Note our sample mean .525 is above the UCL = .523 so we have a problem. (we are okay on the range chart). The process is out of control and should be shut down to find an assignable cause.

http://r.ghinconnect.com/gh/iihbp81.cgi?c=0410801&g=7680758

Continuous Improvement

The aim here is to reduce variability of the product or process.

Pareto AnalysisSome guy named Vilfredo Pareto was very busy back around 1900. One incite he had is that when you look at a population a few items constitute a significant percentage of the whole group – what might be called the vital few.On page 169 in the book table 9.2 contains an example. 2347 front end loaders (a tractor of some sort, right?) were inspected. 412 had defects. Let’s reorder what is in the table with highest number of defect category first and so on. We see the updated table on the next slide with an added column, the cumulative percentage. The cumulative percentage just starts in the first row and as you move down the down you accumulate, or add all the separate percentages.

ExampleDefective item number % Cumulative%Loose Connections 193 46.8 46.8Cracked Connectors 131 31.8 78.6Fitting burrs 47 11.4 90.0Improper torque 25 6.1 96.1O-rings missing 16 3.9 100

So, in the population of defects, the vital few are loose connections and cracked connectors.Figure 9.3 page puts some of this info in a graph. Note on the “x” axis we just have the item names listed in “sorted” order. On the left we have the number column and on the right we have the cumulative %.

Cause and Effect (CE) diagramOnce the vital few have been decided on, each can be investigate more. Each vital few gets its own CE diagram. In the diagram a “fishbone” structure is built. The main spine identifies which vital few is being considered. Those involved in the process brainstorm possible causes and then each is examined more closely.

Pareto charts and CE diagrams aid in reducing problems.