Ch5 Control Charts for Variables

- Variablenumerical measurement
- A single measurable quality

characteristic, such as a - dimension, weight, or volume, is called a

variable.

- Process mean?Process variability????control.
- (a) Mean and standard deviation at nomial levels.

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Statistical Basis of the Charts

?X ??Normal,?n ???,???C.L.?????

relative range.

App. VI)

?n?????

??exhibit in control.

C.C. for process mean

C.C. for process mean

R chart???

App. VI)

Example 1

25?samples,??n5(???)

(???)

(???)

Estimating Process Capability

?????????Capability(??)

- ??????specification limits?74 0.05mm,
- ??????????????????

- PCR(process capability ratio)?????capability???

????????natural tolerance limits????? USL?LSL???

- PCR gt 1

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Revision of Control Limits and Center Lines

(?????C.L.?Center Lines)

?????????25?50?100????

- ?R chart??in control?process mean?????variable??,?

??center line??target value,??process

mean??????variables???,????????? - ?R chart??out of control,?eliminate????????center

line?control limit?

- ???????

- ?????tolerance chart(??tier diagram)?

- ?n?7 or 8?,????Box plot.

- CL ?SL?NTL CL ?SL??????????????

- CL??process?natural variability(?NTL)????

- SP???management?engineer?customers??
- (??????????)?

- Control chart??CL?

- Tolerance chart??specification limits?

2. ?R chart?? ? ??process variability within

sample

??variability within samples???chance ?random

causes

??C.C.???

- CC?????

- ?????cost of sampling, costs of investigating

- and possibly correcting the process in

response to out - of control signals, costs associated with

producing a - product that does not meet specifications.

- small n ???shift?????sample???

- ????small, frequent or larger samples less

frequently, - ?????????small, frequent?????

- ??????rate of production???

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Example 2??????

??process exhibits good control

(???)

(???)

(?????Type I level????control limitUK?????)

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- ??????,???out of control???!

Case 1 Cycles

Case 2 Mixture pattern

???? ?????overlapping ??????????

??,overlap??,? ????????????overcontrol????

Case 3 Shift in process level

???? ?????????? ??????????

???????,?? skill????????motivations?

Case 4 Trend

???? tool?????,operator

fatigue?presence of supervisor(????

critical process component)?????????????

??????????(??????)?

Case 5 Stratification (?????????? ???,lack of

natural variability!)

- ???????underlying????????????
- ????,???????probability limit,????
- CC,????????,underlying???????,
- ??????CC?normality???robustness???

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The Operating-Characteristic Function

(???)

(??k?sample(after the shift)?detect?shift???)

(???????????)

R chart?O.C. curve

(???)

??O.C. curve????????,?????2025 subgroups???C.C. ?

(???)

Average Time to Signal ATSARL.k

Expected number of individual units

sampled InARL (???)

?

2. The sample size n is variable.

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Example 3

???????(???)

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Example 4

??????n3n5(???)

????sample,n5?C.L.

(??????)

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(recommended by some practitioners)

Control Limits Base on a Small Number of

Samples (Short Production Runs)

Control Charts for Individual Measurements

- ?n1???
- 1. ??????????
- 2. ??????,???ngt1?
- 3. ?????????????????????,
- (e.g. ????),???????????
- 4. ????????????,???????,
- ?????????,????std. dev.?????

- ??detect?shift???,????Ch 7?cumulative
- sum?exponential weighted moving-average

C.C. ?

Example 5

- ????????
- ????????
- ????????
- ??batch???,
- ????????
- ???batch?

Control chart for the moving range

In control

Control chart for individual observations

- ???,???????1630 batches?

??Moving range C.C.????moving ranges???,

??,???Moving range C.C.??pattern?,?

???????????moving range chart,?

???????center line??,?????out of

control,???x chart?,??????? out of

control !

- ???individual?moving range C.C.?3-sigma limit
- ???narrow???detect small process shift???,
- ?????????cumulative sum C.C.?
- exponentially weighted moving-average C.C. ?

- ?individual C.C.??,?????normality???,?
- 1. ??percentiles of underlying

distribution???control - limits?
- 2. ?transfer variable??????Normal???,??
- ?C.C.?

Applications of Variable Control Charts

Example 7

Supplier A

Center??,?variability??(??SPC??)?

Supplier B

Center??,?variability??(????,?? ????NC?????center?

?????)?

Example 8

- Job-shop???????????short production runs?

- ???? process characteristic.

- ?????????part?,?????????,
- ???????????,????????????
- ???????correct diameter,????????
- deviation of the actual hole diameter from

the nominal - diameter, ??????????????

Example 9

- ??????,??????specification limits,?
- process capability?????!

Example 10

- ????????????????,???????
- ????????????(???)?

- ??????

1. ??????????5?????(vane), ??????(i.e. n5)?

4. ??????,????part??5????,? ??????rational

subgroup?

- ?????five vanes???????individual
- measurement,??moving-range chart?monitor
- between sample?variability?

- ??S chart Moving range chart tolerance

chart - ????????

- ???quality characteristic?casting??opening

diameter? - ?????????,???opening???24?(??
- ??????)(i.e. n24)

- S chart??????????part?roundness,??out
- of control????,??????????(???) ?

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Inside diameter measurements(mm) on forged piston

rings

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R chart for Example 1

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Continuation of the R chart for Example 1

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Tolerance chart for Example 1

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Recalculated control limits for the piston- ring

data to reflect changing the sample size form

n5 to n3.

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O.C. curve for the R chart with 3-sigma limits

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Computation of control limits for Example 4

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Control chart for individual observations

Control chart for the moving range

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Actions taken to improve a process

Is the process capable?

Yes

No

Yes

Is the process in control?

No

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Control chart for S and tolerance chart for vane

height data, Example 10

?S chart???within part ?variation??????

?

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S chart for diameter, Example 10. This is

essentially a control chart for roundness

??wax pattern??,????wooden cooling

fixture???pattern?????

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An aerospace casting

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