process out of controlshari/download/qc05a control chart... · 2002-11-30 · control charts for...

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′

cchart processcapblt©smy.utm 1


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



3. For two zones 1.5σ each

(a) 2 or more points beyond 1.5σ



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.



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



3. Recurring cycles

- wavy, periodic high & low points

- seasonal effects of mtl.

- Recurring effects of temp., humidity

(morning vs evening)



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



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


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



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.



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.



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



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



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



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



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



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

cchart processcapblt©smy.utm

USL Ctr.line LSL

Visco

2 3 1 21

Batch


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)



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



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



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



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




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

process out of controlshari/download/qc05a control chart... · 2002-11-30 · control charts for...

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