Control ChartsControl ChartsChapter 5Chapter 5

IntroductionIntroduction In industrial processes, goals are to:In industrial processes, goals are to:

- .- .

We need to We need to ______________________ an engineering an engineering process to ensure these goals are met.process to ensure these goals are met.

Examples:Examples:- monitor process for filling 16 oz cans- monitor process for filling 16 oz cans

of pepsiof pepsi- maintain correct outside diameter of- maintain correct outside diameter of

ball point pens ball point pens

ExampleExample

A 10-oz packaging line for a popular A 10-oz packaging line for a popular cereal has a target net weight of cereal has a target net weight of 10.5oz. Historically, process 10.5oz. Historically, process standard deviation is 0.1oz. The standard deviation is 0.1oz. The company monitors net weights by a company monitors net weights by a random sample of 5 boxes of cereal random sample of 5 boxes of cereal each hour. The mean net weights for each hour. The mean net weights for 30 such subgroups is:30 such subgroups is:

DataData10.56, 10.56, 10.46,10.46, 10.49, 10.52, 10.56, 10.56, 10.46,10.46, 10.49, 10.52, 10.53,10.47, 10.44, 10.54, 10.49, 10.52, 10.53,10.47, 10.44, 10.54, 10.49, 10.52, 10.53,10.50, 10.50, 10.48, 10.55, 10.68, 10.53,10.50, 10.50, 10.48, 10.55, 10.68, 10.67,10.62, 10.45, 10.47, 10.40, 10.43, 10.67,10.62, 10.45, 10.47, 10.40, 10.43, 10.47,10.50, 10.38, 10.43, 10.50, 10.5110.47,10.50, 10.38, 10.43, 10.50, 10.51

The target net mean weight is The target net mean weight is μμ00=10.5oz.=10.5oz.

Let yLet yii = the weight of the ith box of cereal = the weight of the ith box of cereal (i=1,…,5) at hour t. (t=1,…,30).(i=1,…,5) at hour t. (t=1,…,30).

Let Let μμ(t) be the process mean (the true (t) be the process mean (the true mean weight at this time).mean weight at this time).

A model we can use is:A model we can use is:

εεii is the random error associated with this is the random error associated with this box of cereal.box of cereal.

We assume here that the process variance We assume here that the process variance σσ22 is constant. is constant.

Factors that may influence processFactors that may influence process

.. .. .. ..

These factors change over time, whichThese factors change over time, whichwill influence the true mean weight ofwill influence the true mean weight ofthe cereal boxes.the cereal boxes.

Typical Drift in a Process Mean Typical Drift in a Process Mean over timeover time

μ(t)

μ0

t

Dealing with Process ChangesDealing with Process Changes Do:Do:

Inspect ________ item and remove the Inspect ________ item and remove the unacceptable items. Very expensive, not unacceptable items. Very expensive, not always 100% effective.always 100% effective.

Produce batches of products, use statistics Produce batches of products, use statistics to determine if entire batch unacceptable. to determine if entire batch unacceptable.

Constantly monitor true mean weight, and Constantly monitor true mean weight, and adjust process whenever it begins to drift. adjust process whenever it begins to drift. Can use control charts here. Can use control charts here.

Create a process that cannot produce an Create a process that cannot produce an unacceptable product. What we strive for. unacceptable product. What we strive for.

Control ChartsControl Charts

Are used to:Are used to: A random sample (a subgroup) is A random sample (a subgroup) is

taken from process on a:taken from process on a:

As soon as data collected:As soon as data collected:

A control chart for a process meanA control chart for a process mean

Is equivalent to carrying out a hypothesis Is equivalent to carrying out a hypothesis test at each time point. In cereal example:test at each time point. In cereal example:

HH00::

HH11::

If we reject the null hypothesis at a If we reject the null hypothesis at a specified level of significance (e.g. 5%), we specified level of significance (e.g. 5%), we conclude the true mean weight of cereal conclude the true mean weight of cereal boxes has moved away from the target boxes has moved away from the target value of 10.5 oz at that time point. value of 10.5 oz at that time point.

Control Limits of an …. ChartControl Limits of an …. Chart Upper Control Limit UCL:Upper Control Limit UCL:

Lower Control Limit LCL:Lower Control Limit LCL:

X

Cereal Example X-bar ChartCereal Example X-bar Chart

UCL=10.63

μ0 =10.5

Out of control

Out of control

In control

In control

LCL=10.37

time

.. Chart of Cereal Box Weights.. Chart of Cereal Box WeightsX

Conclusions for Cereal ExampleConclusions for Cereal Example

The control chart suggests the The control chart suggests the process is:process is:

After this it is within control limits, After this it is within control limits, but appears to be a downward trend.but appears to be a downward trend.

Process:Process:

The np control chartThe np control chart

Instead of monitoring the mean of Instead of monitoring the mean of the process, for example, average the process, for example, average weight of cereal packages, we may weight of cereal packages, we may be interested in the:be interested in the:

Here, we also use a control chart, but Here, we also use a control chart, but a slightly different one.a slightly different one.

Brick ExampleBrick Example A study looked at the number of non-A study looked at the number of non-

conforming bricks from a manufacturing conforming bricks from a manufacturing process. It is known from historic data that process. It is known from historic data that the target proportion of non-conforming the target proportion of non-conforming bricks is 0.06625. Process monitored by bricks is 0.06625. Process monitored by taking a random sample of 200 bricks every taking a random sample of 200 bricks every day for 16 days.day for 16 days.

Each brick was classified as eitherEach brick was classified as either

- conforming- conforming

- non-conforming.- non-conforming.

Let pLet p00 be the target proportion of be the target proportion of nonconforming bricks.nonconforming bricks.

Let yLet yii be the number of nonconforming be the number of nonconforming bricks on day i.bricks on day i.

Expected value E[yi]=npExpected value E[yi]=np00

Variance Var[yi]=npVariance Var[yi]=np00(1-p(1-p00).).

As process stays in control, with same As process stays in control, with same sample size each day, the expected value sample size each day, the expected value and variance stay constant.and variance stay constant.

An np control chartAn np control chart

Examines the:Examines the:

Is equivalent to carrying out a Is equivalent to carrying out a

hypothesis test at each time point.hypothesis test at each time point.

nn is the number of items collected: is the number of items collected:

In Brick ExampleIn Brick Example

Our true hypotheses are:Our true hypotheses are:

HH00::

HH11::

Our control limits take the form:Our control limits take the form:

In Brick ExampleIn Brick Example

The control limits are:The control limits are:

Brick ExampleBrick Example

Check data and Figure 5.22 on page Check data and Figure 5.22 on page 330 in text.330 in text.

Class ExerciseClass Exercise

Airplanes approaching the runway for Airplanes approaching the runway for landing are required to stay within a landing are required to stay within a certain distance (left and right) of the certain distance (left and right) of the runway. Deviations are called runway. Deviations are called “exceedences”. Each day, one airline “exceedences”. Each day, one airline randomly selects 200 flights and randomly selects 200 flights and records the number in exceedence. records the number in exceedence. The following data presents the The following data presents the number of flights in exceedence on number of flights in exceedence on each of 30 days in a row.each of 30 days in a row.

16 25 23 16 15 23 24 15 23 13 16 25 23 16 15 23 24 15 23 13

21 19 23 28 30 19 18 15 18 1021 19 23 28 30 19 18 15 18 10

24 22 20 27 21 26 30 25 26 3024 22 20 27 21 26 30 25 26 30

It is known historically that theIt is known historically that the

exceedence rate is 0.0985 for thisexceedence rate is 0.0985 for this

airline. Calculate the appropriate controlairline. Calculate the appropriate control

limits and plot the control chart.limits and plot the control chart.

Here Here nn=200=200

Days

Exce

edence

28252219161310741

35

30

25

20

15

10

__NP=19.7

UCL=32.34

LCL=7.06