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Control Charts for Variables & Process Capability
PROCESS OUT OF CONTROL 1. A point falls outside control limits
- assignable cause present - process producing subgroup avg. not
from stable process - must be investigated, corrected - frequency distribution of sx′
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Control Charts for Variables & Process Capability
2. Unnatural runs of variation even within ±3σ limits (a) 7 or more points above or below center
line (in a row) (b) 10 out of 11 points on one side (c) 12 out of 14 points on one side (d) 6 points increasing/decreasing (e) Z out of 3 in Zone A (WL) (f) 4 out of 5 in Zone B
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Control Charts for Variables & Process Capability
3. For two zones 1.5σ each
(a) 2 or more points beyond 1.5σ
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Control Charts for Variables & Process Capability
ANALYSIS FOR OUT-OF-CONTROL Patterns
1. Change/Jump in level
- shift in mean
- causes :
• process parameters change
• diff / new operator
• change in raw mtl.
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Control Charts for Variables & Process Capability
2. Trend or steady change in level
- drifting mean
- very common
- upward or downward direction
- tool wear, gradual change in temp.
viscosity of chemical used
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Control Charts for Variables & Process Capability
3. Recurring cycles
- wavy, periodic high & low points
- seasonal effects of mtl.
- Recurring effects of temp., humidity
(morning vs evening)
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Control Charts for Variables & Process Capability
4. Two populations (mixture)
- many points near or outside limits
- due to
• large difference in material quality
• 2 or more machines
• different test method
• mtls from different supplier
Other reasons for out-of-control
Mistakes by quality personnel
• Measuring equipment out of calibration
• Calculation error
• Error in method of using test equipment
• Mistake of samples from different populations
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Control Charts for Variables & Process Capability
SPECIFICATION LIMIT & PROCESS LIMIT Look at indv. values and avg. values of x’s Indv x’s n = 84 - can be considered as
population Avg’s s'x n = 21 - sample taken
xXandX = same (in this case)
Normally distributed individual x’s and avg. values s'x having same mean, only the spread
is different σ > xσ
Relationship nx
σ=σ xσ = popu. Std. dev.
of avg’s If n = 5 xσ = 0.45σ σ = popu. std. dev.
of indv. x’s SPREAD OF AVGS IS HALF OF SPREAD FOR INDV. VALUES cchart processcapblt©smy.utm 8
Control Charts for Variables & Process Capability
Assume Normal Dist. ‘Estimate’ popu. std. dev.
4c
sˆ =σ c4 ≈ 3n4)1n(4
−− ; n = 84
∴ 99699.0
16.4ˆ =σ c4 = 0.99699
= 4.17
417.4
nx =σ
=σ = 2.09
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Control Charts for Variables & Process Capability
CENTRAL LIMIT THEOREM ‘If the population from which samples are taken is NOT normal, the distribution of SAMPLE AVERAGES will tend toward normality provided that sample size, n, is at least 4.’ Tendency gets better as n↑ And standardized normal for distribution of
avg’s Z = n
xσ
µ−
- Central Limit Theorem is one reason control
chart works - - No need to worry about distribution of x’s is
not normal, i.e. indv. values.
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Control Charts for Variables & Process Capability
CONTROL LIMITS & SPECIFICATIONS • Control limit - limits for avg’s - established as a func. of
avg’s
• Specifications - allowable variation in size
∴ for individual values - est. by design engrs
• Control limits, Process spread, Dist of avg’s
& dist. of indv. values are interdependent. – determined by the process
• C. Charts CANNOT determine process meets spec.
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Control Charts for Variables & Process Capability
PROCESS CAPABILITY & TOLERANCE
• When spec. established without knowing process capable of meeting it or not serions situations result
• Process capable or not – actually looking at process spread, which is called process
capability (6σ)
• Define specification limit as tolerance (T) i.e. T = USL -LSL
3 types of situation can result
(I) 6σ < USL-LSL
(II) 6σ = USL - LSL
(III) 6σ > USL - LSL
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Control Charts for Variables & Process Capability
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Control Charts for Variables & Process Capability
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Control Charts for Variables & Process Capability
PROCESS CAPABILITY Procedure (s – method) 1. Take subgroup size 4 for 20 subgroups
2. Calculate sample s.d., s, for each subgroup
3. Calculate avg. sample s.d. s = ∑s/g
4. Calculate est. population s.d.
4o csˆ =σ
5. Calculate Process Capability = 6 σ̂
R - method
1. Same as 1. above
2. Calculate R for each subgroup
3. Calculate avg. Range, R = ∑R/g
4. Calculate 2o dRˆ =σ
5. Calculate 6σ oˆ
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Control Charts for Variables & Process Capability
PROCESS CAPABILITY (6σ) AND TOLERANCE
Cp = o6
Tσ
; Capability Index T = U-L
Cp = 1 ⇒ Case II 6σ = T
Cp = 1 ⇒ Case I 6σ < T
Cp = 1 ⇒ Case III 6σ > T USUALLY Cp = 1.33 (defect std.) MEASURE OF PROCESS PERFORMANCE Shortfall - measure not in terms of nominal or
target value
⇓ use Cpk
Cpk =
3)MIN(Z ; Z (USL) =
σ− xUSL
Z (LSL) = σ
− LSLx
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Control Charts for Variables & Process Capability
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Control Charts for Variables & Process Capability
Determine Cp = Cpk for a process with average 6.45, σ = 0.030 having USL = 6.50 , LSL = 6.30
x6.45 =
6.50 6.30
T L U T = 0.2
Cpk
Cp = σ6
T Z(U) = σ
− xUSL
= )03.0(6
2. = 1.11 = 03.0
45.65.6 −
= 1.67
∴ Cpk = 3
)MIN(Z Z(L) = σ
− LSLx
= 367.1 =
03.03.45. 66 −
= 0.56 = 5.00 Process NOT capable since not centered. Cp > 1 doesn’t mean capable. Have to check Cpk
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Control Charts for Variables & Process Capability
COMMENTS ON Cp, Cpk 1. Cp does not change when process center
(avg.) changes
2. Cp = Cpk when process is centred
3. Cpk ≤ Cp always this situation
4. Cpk = 1.00 de facto standard
5. Cpk < 1.00 → process producing rejects
6. Cp < 1.00 → process not capable
7. Cpk = 0 ⇒ process center is at one of spec.
limit (U or L)
8. Cpk < 0 ⇒ i.e. – ve value, avg outside of
limits
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Control Charts for Variables & Process Capability
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Control Charts for Variables & Process Capability
OTHER SPC TECHNIQUES Continuous processes - 24 hrs/day, 7 days a weak, only maint. stop.
eg. paper making - basically need an automated process
control. i.e. computer aided on-line SPC Batch processes - paint, adhesives, sealants - SPC in batches have 2 forms
• within batch variation • between batch variation
- minimal within batch variation get one value
→ x – R chart - greater/exhibit within batch like soup →
x - R chart
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USL Ctr.line LSL
Visco
2 3 1 21
Batch
Control Charts for Variables & Process Capability
5. Chart for Moving average & Moving Range - Combine a number of indv. values and
plot - When one reading taken at a time
Value 3-period x R Moving ∑1
35 - - - 26 - - - 28 89 29.6 9 32 89 28.6 6 6. Chart for Median & Range
- middle value
• 36, 39, 35 Md = 36 R = 4
• 35, 32, 36 Md = 35 R = 4 . - UCL = Mdmd+ AsRmd . - LCL = Mdmd+ AsRmd . Find R 7. Chart for Indv. values (x chart)
- only one measurement, expensive time consuming, etc.
gxx Σ
= gRR Σ
= (Moving Range)
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Control Charts for Variables & Process Capability
SHORT RUN SPC - Some process, prod. Completed before we
can calculate control limits eg. job shop small sample size
- Use spec. chart, pre-control chart,deviation
chart, etc. SPECIFICATION CHART
- Central line, Control limits est. from spec
eg. spec 25 ± 0.12
Centered line oX = 25.00
USL – LSL = 0.24 ∴ estimate σo If case II process Cp = 1.00
Cp = o6
Tσ
= 1.00
σo = 624.0
= 0.04
URL = ox + Aσo
LRL = ox - Aσo
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Control Charts for Variables & Process Capability
PRE-CONTROL - technique to detect shifts or upsets in the
process resulting in prod. of non-conf. units - first step to ensure capable process 6σ ≤ T i.e. Cp ≥ 1.00 - Pre-control lines established
RED Yellow GR
GREEN Yellow
RED
USL PC xo
PC Nominal LSL
PC lines located ½ between nominal & spec. limits
86%
7% 7%
3.25 3.05 3.10 3.15 3.20
PC N PC U L Cp = Cpk = 1.00
eg: 3.15±0.1 (1) Divide T/4 (2) + LSL (3) - USL
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Control Charts for Variables & Process Capability
Cp = σ6
T ox = 3.15
Cp = 1.00 ∴ T = 6σ T = 3.25 – 3.05
σ = 62. = .2
= 0.33
Z3.1 = 03.0
15.310.3 −
= - 1.5 Table, Area = 0.0668 ~ 7%
7% ≈ 141
1 out of every 14 times OR
µ ± 3σ
+ 1.56 - 1.56
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Control Charts for Variables & Process Capability
Pre-control has 2 stages : start-up & run
RUN stage
Outside RESET
Between PC & spec
lines
2 in a row
RESET
Go to FREQ. TESTING 6(six) Pairs between
adjust
Continue 5
consec.
Inside PC
lines
Start
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Control Charts for Variables & Process Capability
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EXERCISE 1. Find Cp, Cpk
x = 129.7 Length of radiator hose
σ = 2.35
Spec. 130.0 ± 3.0
What is the % defective?
2. Find Cp, Cpk
Spec. U = 58 mm
L = 42 mm
σ = 2 mm
(a) When x = 50
(b) When x = 54
3. Find Cp, Cpk
U = 56
L = 44
σ = 2
(a) When x = 50
(b) When x = 56
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