Ch5 Control Charts for Variables

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Ch5 Control Charts for Variables

• Variablenumerical measurement
• A single measurable quality
  characteristic, such as a
• dimension, weight, or volume, is called a
  variable.

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• 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.?????
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relative range.
App. VI)
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?n?????
??exhibit in control.
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C.C. for process mean
C.C. for process mean
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R chart???
App. VI)
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Example 1
25?samples,??n5(???)
(???)
(???)
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Estimating Process Capability
?????????Capability(??)

• ??????specification limits?74 0.05mm,
• ??????????????????

• PCR(process capability ratio)?????capability???

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????????natural tolerance limits????? USL?LSL???
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• PCR gt 1

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Revision of Control Limits and Center Lines
(?????C.L.?Center Lines)
?????????25?50?100????

1. ?R chart??in control?process mean?????variable??,?
   ??center line??target value,??process
   mean??????variables???,?????????
2. ?R chart??out of control,?eliminate????????center
   line?control limit?

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• ???????

• ?????tolerance chart(??tier diagram)?

• ?n?7 or 8?,????Box plot.

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• CL ?SL?NTL CL ?SL??????????????

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

• SP???management?engineer?customers??
• (??????????)?

• Control chart??CL?

• Tolerance chart??specification limits?

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2. ?R chart?? ? ??process variability within
sample
??variability within samples???chance ?random
causes
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??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.

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• small n ???shift?????sample???

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• ????small, frequent or larger samples less
  frequently,
• ?????????small, frequent?????

• ??????rate of production???

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Example 2??????
??process exhibits good control
(???)
(???)
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(?????Type I level????control limitUK?????)
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• ??????,???out of control???!

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Case 1 Cycles
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Case 2 Mixture pattern
???? ?????overlapping ??????????
??,overlap??,? ????????????overcontrol????
Case 3 Shift in process level
???? ?????????? ??????????
???????,?? skill????????motivations?
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Case 4 Trend
???? tool?????,operator
fatigue?presence of supervisor(????
critical process component)?????????????
??????????(??????)?
Case 5 Stratification (?????????? ???,lack of
natural variability!)
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• ???????underlying????????????
• ????,???????probability limit,????
• CC,????????,underlying???????,
• ??????CC?normality???robustness???

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The Operating-Characteristic Function
(???)
(??k?sample(after the shift)?detect?shift???)
(???????????)
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R chart?O.C. curve
(???)
??O.C. curve????????,?????2025 subgroups???C.C. ?
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(???)
Average Time to Signal ATSARL.k
Expected number of individual units
sampled InARL (???)
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?
2. The sample size n is variable.
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Example 3
???????(???)
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Example 4
??????n3n5(???)
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????sample,n5?C.L.
(??????)
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(recommended by some practitioners)
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Control Limits Base on a Small Number of
Samples (Short Production Runs)
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Control Charts for Individual Measurements

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

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

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Example 5

• ????????
• ????????
• ????????
• ??batch???,
• ????????
• ???batch?

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Control chart for the moving range
In control
Control chart for individual observations
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• ???,???????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 !
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• ???individual?moving range C.C.?3-sigma limit
• ???narrow???detect small process shift???,
• ?????????cumulative sum C.C.?
• exponentially weighted moving-average C.C. ?

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• ?individual C.C.??,?????normality???,?
• 1. ??percentiles of underlying
  distribution???control
• limits?
• 2. ?transfer variable??????Normal???,??
• ?C.C.?

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Applications of Variable Control Charts
Example 7
Supplier A
Center??,?variability??(??SPC??)?
Supplier B
Center??,?variability??(????,?? ????NC?????center?
?????)?
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Example 8

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

• ???? process characteristic.

• ?????????part?,?????????,
• ???????????,????????????
• ???????correct diameter,????????
• deviation of the actual hole diameter from
  the nominal
• diameter, ??????????????

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Example 9

• ??????,??????specification limits,?
• process capability?????!

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Example 10

• ????????????????,???????
• ????????????(???)?

• ??????

1. ??????????5?????(vane), ??????(i.e. n5)?
4. ??????,????part??5????,? ??????rational
subgroup?
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• ?????five vanes???????individual
• measurement,??moving-range chart?monitor
• between sample?variability?

• ??S chart Moving range chart tolerance
  chart
• ????????

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• ???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
Back
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Recalculated control limits for the piston- ring
data to reflect changing the sample size form
n5 to n3.
Back
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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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