JF608: Quality Control - Unit 3

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UNIT 3 :UNIT 3 :CONTROL CHART FOR VARIABLES

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LOGOOUTLINEOUTLINE

Introduction

Variation

Purpose of Control Charts

Patterns in Control Charts

X – R Charts

X – S Charts

Process Capability

LOGOINTRODUCTIONINTRODUCTION

CONTROL CHART

“It is a statistical tool used to distinguish between variation in a process resulting from common cause

& special cause.

Its presents graphical display of process stability or instability over a time.”

OR

Control charts are particularly useful for monitoring quality and giving early warnings that a process may be going “Out of Control” and on its way to

producing defective parts.

LOGOVARIATION

“The variation concept is a law of nature in that no two natural items

in any category are the same”

LOGOVARIATION

The variation may be quite large and easily noticeable

The variation may be very small. It may appear that items are identical; however, precision instruments will show difference

The ability to measure variation is necessary before it can be controlled

LOGOVARIATION

There are three categories of variation in piece part production:

1. Within-piece variation: Surface

2. Piece-to-piece variation: Among pieces produced at the same time

3. Time-to-time variation: Difference in product produced at different times of the day

LOGOVARIATION

Materials

Tools

Operators Methods Measurement Instruments

HumanInspectionPerformance

EnvironmentMachines

Sources of Variation in production processes:

INPUTSINPUTS PROCESSPROCESS OUTPUTSOUTPUTS

LOGOPURPOSE OF CONTROL CHARTMonitor process variation over a timeDifferentiate between special cause and common cause

variationsAssess the effectiveness of changes to improve a

processCommunicate how a process performed during specific

period Improve productivity of process Prevent defects in process Prevent unnecessary process adjustmentsCollect diagnostic informationCollect information about process capability

LOGOTYPES OF CONTROL CHARTS

There are two main categories of Control Charts, those that display attribute data,and those that display variables data.

Attribute Data: This category of Control Chart displays data that result from counting the number of occurrences or items in a single category of similar items or occurrences. These “count” data may be expressed as pass/fail, yes/no, or presence/absence of a defect.

Variables Data: This category of Control Chart displays values resulting from the measurement of a continuous variable. Examples of variables data are elapsed time, temperature, and radiation dose.


Variable datax-bar and R-charts

x-bar and s-charts

Charts for individuals (x-charts)

Attribute dataFor “defectives” (p-chart, np-chart)

For “defects” (c-chart, u-chart)


Continuous Numerical Data

Categorical or Discrete Numerical Data

LOGOCONTROL CHARTS ELEMENTS1. Title. The title

briefly describes the information which is displayed.

2. Legend. This is information on how and when the data were collected.

3. Data Collection Section. The counts or measurements are recorded in the data collection section.

LOGOCONTROL CHARTS ELEMENTS4a. The Upper Graph4b. The Lower Graph5. Y-Axis. This axis

reflects the magnitude of the data collected.

6. X-Axis. This axis displays the chronological order in which the data were collected.

7. Control Limits (CL). Control limits are set at a distance of 3 sigma above and 3 sigma below the centerline.

8. Centerline. This line is drawn at the average or mean value of all the plotted data.

LOGOCONSTRUCTING AN X-BAR AND R CHART

The X-Bar and R Control Chart is used with variables data when subgroup or sample size is between 2 and 15. The steps for constructing this type of Control Chart are:

Step 1 - Determine the data to be collected. Decide what questions about the process you plan to answer. Refer to the Data Collection module for information on how this is done.


Step 2•Collect and enter the data by subgroup.•A subgroup is made up of variables data that represent a characteristic of a product produced by a process.•The sample size relates to how large the subgroups are.


Step 3•Calculate and enter the average for each subgroup.•Use the formula below to calculate the average (mean) for each subgroup

nxxxx n.....21


Step 4•Calculate and enter the range for each subgroup. •Use the following formula to calculate the range (R) for each subgroup.

valuevalueR minmax


Step 5 - Calculate the grand mean of the subgroup’s average. The grand mean of the subgroup’s average (X-Bar) becomes the centerline for the upper plot.

Where:

subgroupsofnumberthekaveragessubgroupeachforaveragethex

averagessubgroupindividualallofmeangrandthex

kxxxxx n.......321


Step 6 - Calculate the average of the subgroup ranges. The average of all subgroups becomes the centerline for the lower plotting area.

Where:

subgroupsofnumbertheksubgroupeachforrangeindividualtheRsubgroupsallforrangestheofaveragetheR

i

kRRRR n.........21

LOGOCONSTRUCTING AN X-BAR AND R CHARTStep 7 - Calculate the upper control limit (UCL) and lower

control limit (LCL) for the averages of the subgroups. To find the X-Bar control limits, use the following formula:

NOTE: Constants, based on the subgroup size (n), are used in determining control limits for variables charts.

RAXLCL

RAXUCL

x

x

2

2

n A2 n A2 n A2

2 1.880 7 0.419 12 0.266

3 1.023 8 0.373 13 0.249

4 0.729 9 0.337 14 0.235

5 0.577 10 0.308 15 0.223

6 0.483 11 0.285

LOGOCONSTRUCTING AN X-BAR AND R CHARTStep 8 - Calculate the upper control limit for the ranges. When

the subgroup or sample size (n) is less than 7, there is no lower control limit. To find the upper control limit for the ranges, use the formula:

NOTE: Constants, based on the subgroup size (n), are used in determining control limits for variables charts.

n D4 n D4 n D4

2 3.267 7 1.924 12 1.717

3 2.574 8 1.864 13 1.693

4 2.282 9 1.816 14 1.672

5 2.114 10 1.777 15 1.653

6 2.004 11 1.744

RDLCL

RDUCL

R

R

3

4


Step 9•Select the scales and plot the control limits, centerline, and data points, in each plotting area. •The scales must be determined before the data points and centerline can be plotted. •The scales for both the upper and lower plotting areas should allow for future high or low out-ofcontrol data points.


Step 10 - Provide the appropriate documentation. Each Control Chart should be labeled with who, what, when, where, why, and how information to describe where the data originated, when it was collected, who collected it, any identifiable equipment or work groups, sample size, and all the other things necessary for understanding and interpreting it. It is important that the legend include all of the information that clarifies what the data describe.


Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3

2 1.880 3.268 0

3 1.023 2.574 0

4 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 0

7 0.419 1.924 0.076

8 0.373 1.864 0.136

9 0.337 1.816 0.184

10 0.308 1.777 0.223

12 0.266 1.716 0.284

CONTROL CHART FACTORS


EXERCISE 1: A team collected the variables data recorded in the table below.

Use these data to answer the following questions and plot a Control Chart:1. What type of Control Chart would you use with these data?2. Why?3. What are the values of X-Bar for each subgroup?4. What are the values of the ranges for each subgroup?5. What is the grand mean for the X-Bar data?6. What is the average of the range values?7. Compute the values for the upper and lower control limits for both the upper and lower plotting areas.8. Plot the Control Chart.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X1 6 2 5 3 2 5 4 7 2 5 3 6 4 5 3 6

X2 5 7 6 6 8 4 6 4 3 5 1 4 3 4 4 2

X3 2 9 4 6 3 8 3 4 7 2 6 2 6 6 7 4

X4 7 3 2 7 5 4 6 5 1 6 5 2 6 2 3 4


EXERCISE 1 ANSWER KEY:1. X-Bar and R.2. There is more than one measurement within each subgroup.5. Grand Mean of X = 4.526. Average of R = 4.387. UCLx = 7.71 LCLx = 1.33 UCLR = 10.0 LCLR = 0

LOGOX-BAR AND R CHART FORMULA

1 1

g g

i i

i i

i

i

X RX and R

g g

where

X average of subgroup averages

X average of the ith subgroup

g number of subgroups

R average of subgroup ranges

R range of the ith subgroup

Central Lines are obtained using:


3 3

3 3

R RX X

R RX X

X

R

UCL X UCL R

LCL X LCL R

where

UCL=upper control limit

LCL=lower control limit

population standard deviation of the subgroup averages

population standard deviation of the range

If σ are known, control limits are obtained using :


If σ are unknown, control limits are obtained using :

2 4

2 3

RX

RX

UCL X A R UCL D R

LCL X A R LCL D R

NOTE :A2,D3 and D4 are factors that vary with the subgroup size and are found in Table.


Revised Central LinesRevised Central Lines

d dnew new

d d

d

d

d

X X R RX and R

g g g g

where

X discarded subgroup averages

g number of discarded subgroups

R discarded subgroup ranges


Standard ValuesStandard Values

00 0 0

2

new newR

X X R R andd

0 0 2 0

0 0 1 0

RX

RX

UCL X A UCL D

LCL X A LCL D


Sample Std. Deviation Chart (x - s control chart ) Both R and s measure dispersion of

data R chart - simple, only use XH

(highest) and XL(lowest) s chart - more calculation - use ALL

xi’s more accurate, need calculate sub-group sample standard deviation

When n < 10 R chart s chart n 10 - s chart better , R not

accurate any more

sAXUCL 3x

sAXLCL 3x

sBUCL 4s sBLCL 3s

44

oo C

sCs

LOGOX-BAR AND S CHART FORMULA

For subgroup sizes >=10, an s chart is more accurate than an R Chart. Control limits are given by:

1 1

3 4

3 3

g gi ii i

sX

sX

s Xs X

g g

UCL X A s UCL B s

LCL X A s LCL B s

LOGOX-BAR AND S CHART FORMULA

0

00 0

4

0 0 6 0

0 0 5 0

4 5 6, , ,

dnew

d

dnew

d

sX

sX

d

X XX X

g g

s s ss s

g g c

UCL X A UCL B

LCL X A LCL B

where

s discarded subgroup averages

c A B B factors found in Table B

Revised Limits for s chartRevised Limits for s chart

LOGOPATTERNS IN CONTROL CHARTS

Upper control limit

Target

Lower control limit

Normal behavior. Process is “in control.”


One plot out above (or below). Investigate for cause. Process is “out of control.”

Upper control limit

Target

Lower control limit


Upper control limit

Target

Lower control limit

Trends in either direction, 5 plots. Investigate for cause of progressive change.


Upper control limit

Target

Lower control limit

Two plots very near lower (or upper) control. Investigate for cause.


Upper control limit

Target

Lower control limit

Run of 5 above (or below) central line. Investigate for cause.


Upper control limit

Target

Lower control limit

Erratic behavior. Investigate.

LOGOProcess is OUT of control if :

One or multiple points outside the control limits

Eight points in a row above the average value

Multiple points in a row near the control limits

LOGOProcess is IN of control if :

The sample points fall between the control limits

There are no major trends forming, i.e.. The points vary, both above and below the average value.

LOGOSTATE OF CONTROL

When the process is in control:

1. Individual units of the product or service will be more uniform

2. Since the product is more uniform, fewer samples are needed to judge the quality

3. The process capability or spread of the process is easily attained from 6ơ

4. Trouble can be anticipated before it occurs

LOGOSTATE OF CONTROL

When the process is in control:

5. The % of product that falls within any pair of values is more predictable

6. It allows the consumer to use the producer’s data

7. It is an indication that the operator is performing satisfactorily

LOGOPROCESS CAPABILITY (CP)

The range over which the natural variation of a process occurs as determined by the system of common causes

Measured by the proportion of output that can be produced within design specifications


Cp = Upper Specification - Lower Specification

6

A capable process must have a Cp of at least 1.0

Does not look at how well the process is centered in the specification range

Often a target value of Cp = 1.33 is used to allow for off-center processes

Six Sigma quality requires a Cp = 2.0


Example

Cp = Upper Specification - Lower Specification6

Insurance claims process

Process mean x = 210.0 minutesProcess standard deviation = .516 minutesDesign specification = 210 ± 3 minutes

LOGOPROCESS CAPABILITY (CP)Calculation

Cp = Upper Specification - Lower Specification6

Insurance claims process

Process mean x = 210.0 minutesProcess standard deviation = .516 minutesDesign specification = 210 ± 3 minutes

= = 1.938213 - 2076(.516)

Process is capable

LOGOPROCESS CAPABILITY INDEX (Cpk)

A capable process must have a Cpk of at least 1.0

A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

Cpk = minimum of ,

UpperSpecification - xLimit

Lowerx - Specification

Limit

LOGO

Example

New Cutting MachineNew process mean x = .250 inchesProcess standard deviation = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches

PROCESS CAPABILITY INDEX (Cpk)

LOGO

Calculation

New Cutting MachineNew process mean x = .250 inchesProcess standard deviation = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches

Cpk = = 0.67.001

.0015

New machine is NOT capable

Cpk = minimum of ,(.251) - .250

(3).0005.250 - (.249)

(3).0005

Both calculations result in


LOGO

Interpreting Cpk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1


LOGOEXERCISE

Dalam suatu proses pengeluaran sebuah mentol lampu, didapati purata jangka hayat µ=900jam dan σ=77jam. Nilai norminal bagi julat kelegaan adalah 1000 jam dengan spesifikasi atas = 1200jam dan spesifikasi bawah = 800jam. Jika anda sebagai seorang pengurus operasi, tentukan adakah proses berupaya untuk menghasilkan spesifikasi mentol lampu. Huraikan dapatan anda.

LOGO

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JF608: Quality Control - Unit 3

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