Control Chart For Variables Ref: Chap 5 Besterfield

Recap Variables and Attributes Product Quality Characteristics {or} Attributes Variables Observable Conformance Measurable 1234 speed (rpm) -- discrete 9.675 weight (grammes) -- continuous 2.1 height (meters) [1, 2, 3] Wood sheen intensity

All wires connected (Y/N) Panels connected (Y/N)

Height within conformance 2.0 m < h < 2.225 m (i.e. also classified as an attribute)

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Sample Control Chart for a Variable

Introduction • No two (manufactured) objects are ever made alike. • The range of magnitude of the variations can be significant • The ability to measure variations is necessary before they can be controlled.

• Three Categories of variations – Within -Piece – Piece-To-Piece – Time -to-Time

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Causes of Variations 1. Equipment –

Tool wear, piece position slippage, vibrations, electric supply variations, maintenance schedule. Combine them all in different proportions => variation in pieces machine produces • •

EACH MACHINE HAS ITS OWN CAPABILITY/PRECISION MACHINE CAPABILITY/PRECISION CAN VARY WITH TIME, USAGE, LOCATION, MAINTENANCE, OPERATION….

2. Materials –

Variations can also be present in the raw material (someone else’s finished product!!). Includes ductile strength, thickness, porosity, moisture content etc…

3. Environment –

Temperature, lighting conditions, dust. Sometimes products are manufactured in white rooms.

4. Operator –

Method employed by operator, Physical and emotional state, Operators training. Increased automation can alleviate this problem.

5. Inspection –

The inspection process itself is subject to variations in its equipment, materials, environment and operator actions (normally considered to be 1/10 of the variation of the other factors)

Chance and Assignable Causes • Variations that fluctuate in a natural way (or expected manner) result in a stable pattern of variations are called chance causes. – These forms of variation are inevitable (and numerous) – They are difficult to detect and eliminate – However, individually they are of small importance and significance

• Those variations that are large in magnitude and are easily identified are assignable causes Process A Only has chance causes

Process B Has excessive variations

Stable and Predictable In a state of statistical control

Excessive Variation Out of control process

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Control Chart Method for Variables • Used to identify when observed variations are greater than could attributable to chance • The control chart for variations A means of visualizing the variations that occur in the central tendency and dispersion of a set of values • It’s a graphical method to record a particular characteristic.

Control Chart • This control chart records the variations in the average value of samples. • The horz. axis represents samples (each with a number of observations) called “rational subgroups” • Subgroup #1 would have been inspected first and so subgroup #5 would have been fifth, containing a set values [3.46, 3.49, 3.45, 3.44] with an average of 3.46. This average value is plotted. • The vertical axis shows the variable in question (weight here)

Averages are plotted because they indicate a change in variation much earlier Also with two or more values in a subgroup you can get a measure of dispersion.

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The Central Line 1. It can be the average of the plotted points, which means it’s the average of averages! 2. It can be a standard or reference value (based on representative prior data, economic needs or a specification) as it is here. 3. It can be the population mean, if its known.

X Xo

µ

The solid line in the center can have three interpretations depending on the available data.

The limits • The two dashed outer lines are the limits • These are established to judge the significance of the variation in the quality of the product. • Control Limits help evaluate variations in quality from subgroup to subgroup. – Don’t confuse with specification limits which apply to each individual unit.

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The limits • For the /X chart, the control limits are a function of the subgroup averages. – A frequency distribution of the subgroup averages can be determined with its corresponding average and standard deviation.

• The control limits are usually established at: ±3 standard deviations from the central line. – Covers approx 99.73% of items

When a sub group falls outside the limits the process is considered to be out of control Here we have one outof-control point.

A chart that has been posted at a machine (typically an \X chart for central tendency and an R chart for dispersion). A tool To measure hardness is a durometer. We can see; - work center - observation times - observation values in each subgroup The freq of inspection is determined by the quality of the product. Note an individual exception, meaningless for this chart For the value on the line, a cursory inspection will do.

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In subgroup 4 we see value outside the lower control limit. This will have to be signaled to the supervisor. because it means: subgroup 4 is out-of-control and corrective action is required What ever the corrective action is will be noted on the chart.

Natural and Unnatural variation • A control chart is a statistical tool that distinguishes between natural and unnatural variations. • Unnatural variation is as a result of assignable causes.

Natural Variations

– Generally requires corrective action

•

Natural variation is the result of chance causes. • It requires management intervention to achieve quality improvement. • In this regard, between 80% and 85% of the quality problems are due to management or the system, • and 15% to 20% are due to operations.

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Objectives of Variable Control Charts Provide information on: 1. For Quality improvement 2. To determine process capability 3. For decisions in regard to product specification – Obtaining true process capability means effective specs can be determined (i.e. if process capability is ±0.003 then specs of ±0.004 are attainable) 4. For decisions in regard to the production process – Does control exist? Use a control chart to find out. – Control exists! use a control chart to maintain control. 5. For decisions in regard to recently produced items – Should items be released to the next phase of sequence?

Control Chart Techniques -Step 1 •

The following six steps help build a pair of control charts for the Average X and Range (R) (or Standard deviation).

1. Select the quality characteristic • Must be measurable and expressible as a number such as time, weight, height, velocity etc • Give priority to those characteristics that are given problems wrt production and/or cost .. possibly identify with a Pareto diagra m

- Luminosity - Max. input Voltage - Time to produce

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Step 2 - Choose the rational subgroup – Control charts plot points derived from a group of items (i.e. average of five weights is one point) – Data in a subgroup cannot be collected randomly – Rational implies variation in a group is only due to chance causes – Remember we are examining within-subgroup variations!

– Schemes for collecting data – Generally speaking the lots should be homogenous (same machine same operator etc)

1

2

3

4

Most recent four Instant-Time Most Common

2

1 3 Last hour

4

Period-of-Time Most sensitive

Step 2 Continued – Sample size in sub- group guidelines 1. As the subgroup size increases, the control limits become closer to the central value: makes the control chart more sensitive to small variations in the process average. 2. As the subgroup size increases, the inspection cost per subgroup increases. (Is the increased cost of larger subgroups justified?) 3. When destructive testing is used and the item is expensive, a small subgroup size of 2 or 3 is necessary (Why?) 4. Because of the ease of computation a sample size of 5 is quite common in industry; (Does this hold for computerized checks?) 5. Four or more samples are statistically significant. 6. When the subgroup size exceeds 10, the s chart should be used instead of the R chart for the control of the dispersion. (Why?)

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Step 2- Frequency of taking sub-groups

Produce 5,000 pieces per day? Then 75 inspections are suggested.

Step 3 -Collect the data Quality characteristic is shaft keyway depth of 6.35mm Rational-subgroup of four. Obtain 5 sub-groups per day for 5 days Data coded from 6mm Data recorded on a run-chart prior to proper statistical control (collected by a technician)

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(Run-time) Chart for this data

Recap The Central Line 1. It can be the average of the plotted points, which means it’s the average of averages! 2. It can be a standard or reference value (based on representative prior data, economic needs or a specification) as it is here. 3. It can be the population mean, if its known.

X Xo

µ

The solid line in the center can have three interpretations depending on the available data.

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Step 4 Trial-Central Line 1. Determine the trial central line g

X=

∑

g

Xi

i =1

R=

g

∑R

i

i =1

g

X = average of the subgroup averages X i = average of the i th subgroup g = number of subgroups R = average of the subgroup ranges R i = Range of i th subgroup Earlier given

∑ X = 160 .25 ∑ R = 2.19

160 .25 25 = 6.41mm

X=

and g = 25

2. 19 25 = 0.0876 mm

R=

Step 4 - Trial Control Limits for the chart • Trial control limits at ±3 standard deviations from the central line are established (Note this is the simplified form): UCL X = X + A 2 R UCLR = D 4 R LCL X = X − A 2 R

LCL R = D 3 R

Note you will need Table B to solve A2 , D3 and D4 UCL X = X + A 2 R = 6.41 + (0.729)( 0.0876) = 6.47 mm LCL X = X − A 2 R

UCL R = D4 R = (2 .282)(0 .0876) = 0.20mm LCL R = D3 R

= 6.41 − ( 0.729)(0 .0876)

= (0)(0.0876 )

= 6.35 mm

= 0mm

Table B Values for the factors for sub-group size of (n) 4 are A2 =0.729, D3 = 0 and D4 = 2.282. This gives Trial Control Limits

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/X and R chart for Prelim data with trial control limits Post the preliminary data to the chart along with the control limits and central lines

Control Limits

Central Lines

Are the Central Lines good enough to be a standard? – Having posted the data and the trial central lines and trial control limits the next job is to ask Are the central lines good enough - will they be my standard? – You will have to estimate if the values are good enough to be a standard – Does an analysis of the preliminary data show good control? – Yes: Then we set we set the standard values to these values.

X = X o and R = R o

Standard Values !!

Xo and R are the standard values o

– Good control means: no out of control points, no long runs on either side of the central line and no unusual patterns of variation.

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Revised Central Lines Are the central lines good enough - will they be my standard? – Does an analysis of the preliminary data show good control? Commonly its No! – Most processes are not in control when they are first analyzed.

Note out of control points on both graphs. There is also a lot points below the central line. We need to examine each of these out of control points

Examining the charts • The R chart is analyzed first to determine its stability. • For out-of-control point at subgroup #18 has an assignable cause (oil line damaged) it can be discarded. • Remaining plot indicates stable process.

Discarded “Range values” are for subgroup #18 (0.3) Note however that its “average value” stays!

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Examining the charts • The /X chart now analyzed. • Subgroups #4 and #20 have an assignable cause and are discarded from the data. • Subgroup #16 has no assignable cause => we assume it is due to chance cause and is part of a natural variation and stays • As a rule if either the /X or R value of a subgroup is out-ofcontrol with an assignable cause then discard it. Discarded “average values” are for subgroup #4 (6.65) and subgroup #20 (6.51) Note however that their Range values stay!

New Central Line values required! • After assessing the out-of-control points we need to revisit out central lines. g

X new =

∑X − X i =1

g − gd

g

d

R new =

∑R−R

d

i =1

g − gd

X d = discarded subgroup averages g = number of discarded subgroups d R d = discarded subgroup ranges

160.25 − (6.65 + 6.51) 25 − 2 = 6.40mm

X new =

2.19 − (0.30) 25 − 1 = 0.079mm

R new =

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New (Revised) Standard Values Required • We now need to re-compute our standard values X o = X new , R o = R new and σ o =

Ro d2

New Standard Values !!

• Where d2 is a factor from Table B. • The standard (or reference values) can be considered to be the best estimate with the available data. As more data becomes available better estimates (or more confidence in these values) will be possible. X o = X new = 6.40mm R o = R new = 0. 079(0. 08 for chart) σo =

R o 0.079 = = 0.038 d2 2.059

Now redo the Control Limits • Using the standard values we can redo the 3σ control limits for actual operation using the formulas UCL X = Xo + Aσo

UCL R = D 2σ o

LCL X = X o − Aσ o

LCL R = D1σ o

UCL X = X o + Aσ o

UCL R = D 2 σ o

= 6.40 + (1.500 )( 0.038 )

= (4.698 )(0.038 )

= 6.46 mm

= 0.18mm

LCL X = X o − Aσ o = 6.40 − (1.500 )(0.038 ) = 6.34 mm

LCL R = D1σ o = (0)(0.038) = 0 mm

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Trial control limits and Revised Control Limits The revised control limits are for plotting further subgroups

Achieve the objectives • When control charts are first introduced at a work center an improvement in the process performance usually occurs. (Is this a case of Big Brother watching?)

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Sample Standard Deviation Control Chart • Some companies prefer the standard deviation, s , as a measure of the subgroup dispersion. • In comparing with an R chart, the R chart is easier to compute and easier to explain. • However the subgroup sample standard deviation for the deviation for the s chart is calculated using all the data (unlike the max and min in the range, R) • The s chart is common in larger subgroup sizes (> 10 typ) • Over we have the same data values but we have a standard deviation column now.

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Trial Control Limits • The formulae for computation of trial control limits are; g

g

∑ Si

∑ Xi

S = i =1 X = i =1 g g s i = sample s tan dard deviation of the subgroup values s = average of the subgroup sample s tan dard deviations UCL X = X + A3 s

UCL s = B 4 s

LCL X = X − A 3s

LCL s = B3 s

• A3 , B3 , B4 = factors found in Table B

Formula for revised control Limits • Using the standard values

X o and σ o So = Snew =

∑S −S g − gd

d

X o = X new =

∑X −X g − gd

d

σo =

So C4

s d = s tan dard deviation for the discarded group UCL X = X o + Aσ o

UCL s = B 6 σ o

LCL X = X o − Aσ o

LCL s = B 5σ o

• c4 , A, B5 , B6 factors found in Table B

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