lec 5 variables control chart

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• Chance causes - natural variability/background noise; in statistical control

• Assignable causes - unacceptable: out of control- sources: improperly adjusted or controlled machines,

operator errors or defective raw materials

Major objective of SPC is to quickly detect the occurrence of assignable causes of process shifts so that investigation of the process and corrective actions may be undertaken before many nonconforming units are manufactured.

CONTROL CHARTS

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Statistical Basis A control chart contains

A center line An upper control limit A lower control limit

A point that plots within the control limits indicates the process is in control No action is necessary

A point that plots outside the control limits is evidence that the process is out of control Investigation and corrective

action are required to find and eliminate assignable cause(s)

There is a close connection between control charts and hypothesis testing

CONTROL CHARTS


A. Most important use of CC is to improve the process1. Most processes do not

operate in a state of statistical control

2. Routine and attentive use of control charts will identify assignable causes

3. The chart will only detect assignable causes. > Management, operator and engineering action

B. Control Charts can be used as an estimating device-- Determine the capability of the process to produce acceptable products

C. Two general types1. Variables Control Charts

• Continuous scale of measurement• Quality characteristic described by central tendency and a measure of variability

2. Attributes Control Charts • Conforming/nonconforming• Counts

D. Design of Control Charts1. Sample size2. Control limits3. Frequency of sampling

Chapter 5 5

Reasons for Popularity of Control Charts

1. Control charts are a proven technique for improving productivity.

2. Control charts are effective in defect prevention.

3. Control charts prevent unnecessary process adjustment.

4. Control charts provide diagnostic information.5. Control charts provide information about

process capability.

Chapter 5 6

3-Sigma Control Limits Probability of type I error is 0.0027

Probability Limits Type I error probability is chosen directly For example, 0.001 gives 3.09-sigma control limits

Warning Limits Typically selected as 2-sigma limits

Choice of Control Limits

Chapter 5 7

Size of the shift that is to be detected In general, larger samples will make it easier to detect small

shifts in the process

Frequency of sampling Most desirable (from the point of view of detecting shifts) :

take large samples very frequently Small samples at short intervals or large samples at longer

interval

Sample Size and Sampling Frequency

Chapter 5 8

Average Run Length (ARL) The average number of points to be plotted before a

point indicates an out of control condition ARL = 1/p ( p is the probability that any point exceeds

the control limits)

Average Time to Signal (ATS) ATS = ARL h (h = time interval between samples)

Sample Size and Sampling Frequency

STATISTICAL CONTROL


A phenomenon is said to be in statistical control when, through the use of past experience, we can predict how the phenomenon will vary in the future.

- Walter Shewhart

Time element- help predict error level and plan

resources

Chapter 5 10

Chapter 5 11

Sensitizing Rules

Chapter 5 12

Phase I and Phase II of Control Chart Application

Phase I is a retrospective analysis of process data to construct trial control limits Charts are effective at detecting large,

sustained shifts in process parameters, outliers, measurement errors, data entry errors, etc.

Facilitates identification and removal of assignable causes

In phase II, the control chart is used to monitor the process Process is assumed to be reasonably stable Emphasis is on process monitoring, not on

bringing an unruly process into control

Chapter 5 13

Control Chart Selection Process

Characteristic Selected

Variable Data?

Homogeneous or NOT able to subgroup data?

Individuals Chart (I-MR) N>10

YES NO

Average-Range Chart (xbar-R)

NO

YESAverage Standard

Deviation Chart (xbar-s)

VARIABLE CONTROL CHARTS ATTRIBUTE CONTROL CHARTS

YES NO

Proportion

Constant sample size

P-chart

NO

YES

NOCount

P-chart or np chart

YES

Constant sample size

u-chart

NO

C chart

YES

14

Learning Objectives

VARIABLE CONTROL CHARTS

1. Understand the statistical basis of Shewhart control charts for variables2. Know how to design variables control charts3. Know how to set up and use Xbar and R control charts4. Know how to estimate process capability from the control chart

information5. Know how to interpret patterns on the Xbar and R control charts6. Know how to set up and use xbar and s or s2 control charts7. Know how to set up and use control charts for individual measurements8. Understand the importance of the normality assumption for individuals

control charts and know how to check this assumption9. Understand the rational subgroup concept for variables control charts10. Determine the average run length for variables control charts

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VARIABLE CONTROL CHARTS ROADMAP

N>10

VARIABLE

Average-Range Chart (xbar-R)

NO

YESAverage Standard

Deviation Chart (xbar-s)

N = 1YES

I-MR Chart

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Phase I Application of and R Charts

Eqns 6.4 and 6.5 are trial control limits Determined from m initial samples

• Typically 20-25 subgroups of size n between 3 and 5

Any out-of-control points should be examined for assignable causes• If assignable causes are found, discard points from

calculations and revise the trial control limits• Continue examination until all points plot in control• Adopt resulting trial control limits for use• If no assignable cause is found, there are two options

1. Eliminate point as if an assignable cause were found and revise limits

2. Retain point and consider limits appropriate for control

If there are many out-of-control points they should be examined for patterns that may identify underlying process problems

x

18

Example 6.1 The Hard Bake Process

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Example 6.1 The Hard Bake Process

Chapter 5 20

• Pattern is very nonrandom in appearance• 19 of 25 points plot below the center line, while only 6 plot above• Following 4th point, 5 points in a row increase in magnitude, a run up• There is also an unusually long run down beginning with 18th point• Run - a sequence of observations of the same type

Patterns on Control Charts

Chapter 5 21Introduction to Statistical Quality Control, 6th Edition by Douglas C.

Montgomery.Copyright (c) 2009 John Wiley &

Sons, Inc.

Patterns on Control Charts

Cycles may be due to systematic environmental changes: temperature, fatigue, regular rotation of workers or machines or fluctuations in pressure or voltage

Stratification -- tendency for the points to cluster around the center line and may be due to wrong calculation of control limits or samples are taken from different groups

Chapter 5 22

Patterns on Control Charts• Shift in process levels -- may result from the introduction of new workers, methods, raw materials or machines; change in the inspection method or standards; change in either the skill, attentiveness or motivation of the operators

• Trend -- usually due to a gradual wearing out or deterioration of a tool or some other critical process component; result from seasonal influences such as temperature; human fatigue or presence of supervision

• Mixtures -- result from overcontrol, where the operators make process adjustments too often, responding to random variation in the output rather than systematic causes

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Revision of Control Limitsand Center Lines

Effective use of control charts requires periodic review and revision of control limits and center lines

Sometimes users replace the center line on the chart with a target value

When R chart is out of control, out-of-control points are often eliminated to recompute a revised value of which is used to determine new limits and center line on R chart and new limits on chart

x

R

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Phase II Operation of Charts

Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage (Figure 6.4)

A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram (Figure 6.5), may reveal patterns or unusual observations in the data

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Control vs. Specification Limits

Control limits are derived from natural process variability, or the natural tolerance limits of a process

Specification limits are determined externally, for example by customers or designers

There is no mathematical or statistical relationship between the control limits and the specification limits

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Rational Subgroups

charts monitor between-sample variability R charts measure within-sample variability Standard deviation estimate of used to

construct control limits is calculated from within-sample variability

It is not correct to estimate using

x

34

Control Charts for Xbar and s

Preferable than the xbar-R chart• sample size n is moderately large – n > 10• the sample size is variable

37

Xbar and s Charts with Variable sample Size

Summary of Xbar and R/S Control Charts

Sample Size

Parameters X-bar Chart R Chart Distribution

Small,

Constant

, known or given

UCL = + A

CL =

LCL = - A

UCL = D2

CL = d2

LCL = D1

= R/d2

R = d3

Small,

Constant

, unknown

UCL = + A2Ř

CL = x

LCL = - A2Ř

UCL = D4Ř

CL = Ř

LCL = D3Ř

= R/d2

R = d3

x

xx

Summary of Xbar and R/S Control Charts

Sample Size

Parameters X-bar Chart S Chart Distribution

Bigger,

Constant

, known or given

UCL = + A

CL =

LCL = - A

UCL = B6

CL = c4

LCL = B5

= s/c4

s = 1-c42

Bigger,

Constant

, unknown

UCL = + A3š

CL =

LCL = - A3š

UCL = B4 š

CL = š

LCL = B3 š

= s/c4

s = 1-c42x

x

x

42

Shewhart Control Chart for Individual Measurements

Situations where n = 1• automated inspection and measurement technology is used, and every unit manufactured is analyzed so there is no basis for rational subgrouping

• data comes available relatively slowly, and it is inconvenient to allow sample sizes of n>1 to accumulate before analysis. The long interval between observations will cause problems with rational subgrouping.

• repeat measurements on the process differ only because of laboratory or analysis error, as in many chemical processes

• multiple measurements are taken over the same unit of a product, such as measuring oxide thickness at several different locations on a wafer in semicon manufacturing

• in process plants, such as papermaking, measurements on some parameters such as coating thickness across the roll will differ very little and produce a standard deviation that is too much too small if the objective is to control coating thickness along the roll

43

Shewhart Control Chart for Individual Measurements

48

Assumption of normality for I-MR Chart

• In control ARL is dramatically affected by non-normal data

• If the process shows evidence of even moderate departure from normality, the control limits may be entirely inappropriate for phase II process monitoring

• Transform the original variable to a new variable that is approximately normally distributed

49

Assumption of normality for I-MR Chart

50

Applications of Variable Control Charts

Using Control Charts to Improve Suppliers’ Processes•A large aerospace manufacturer purchased an aircraft component from two suppliers.

• Exhibited excessive variability; impossible to assemble into final product

• Resulted in expensive rework costs and occasionally caused delays in finishing the assembly of an airplane

•100% inspection of the parts• They maintained Xbar-R charts on the dimension of interest for

both suppliers• Fraction nonconforming the same for both suppliers• Supplier A could produce parts with mean dimension equal to

the required value, but the process was out of statistical control• Supplier B maintained statistical control ; less variability than

parts from Supplier A; process was centered so far off the nominal required dimension that many parts were out of specifications

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Using Control Charts to Improve Suppliers’ Processes•A large aerospace manufacturer purchased an aircraft component from two suppliers.

• Persuade Supplier A to install an SPC activity and to begin working at continuous improvement

• Assist Supplier B to find out why his process was consistently centered incorrectly

• Supplier B’s problem was ultimately tracked to some incorrect code in an NC machine

• Use of SPC at Supplier A resulted in considerable variability over a 6-month period

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lec 5 variables control chart

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