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Statistical Process Control
Production and Process Management
Where to Inspect in the Process
• Raw materials and purchased parts – supplier certification programs can eliminate the need for inspection
• Finished goods – for customer satisfaction, quality at the source can eliminate the need for inspection
• Before a costly operation – not to waste costly labor or machine time on items that are already defective
• Before an irreversible process – in many cases items can be reworked up to a certain point, beyond that point
• Before a covering process – painting can mask deffects
Process stability and process capability
• Statistical process control (SPC) is used to evaluate process output to decide if a process is „in control” or if corrective action is needed.
• Quality Conformance: does the output of a process conform to specifications
• These are independent
Variation of the process
• Random variation (or chance) – natural variation in the output of a process, created by countless minor factors, we can not affect these factors
• Assignable variation – in process output a variation whose cause can be identified.
• In control processes – contains random variations
• Out of control processes – contains assigneable variations
Sampling distribution vs. Process distribution
• Both distribution have the same mean
• The variability of the sampling distribution is less than the variability of the process
• The sampling distribution is normal even if the profess distribution is not normal
• Central limit theorem: states thet the sample size increase the distribution of the sample averages approaches a normal distribution regardless of the shape of the sampled distribution
• In the case of normal distribution– 99,74% of the datas fall
into m± 3 σ– 95,44% of the datas fall
into m± 2 σ– 68,26% of the datas fall
into m± 1 σ – If all of the measured datas
fall into the m± 3 σ intervall, that means, the process is in control.
Sampling
• Random sampling– Each itemhas the same probability to be selected– Most common– Hard to realise
• Systhematic sampling– According to time or pieces
• Rational subgoup– Logically homogeneous– If variation among different subgroups is not accounted fo, then
an unawanted source of nonrandom variation is being introduced
– Morning and evening measurement in hospitals (body temperature)
• Variables – generate data that are measured (continuus scale, for example length of a part)
• Attributes – generate data that are counted (number of defective parts, number of calls per day)
Control limits
• The dividing lines between random and nonrandom deviation from the mean of the distribution
• UCL – Upper Control limit
• CL – Central line
• LCL – lower Control limit
• This is counted from the process itself. It is not the same as specification limits!
Specification limits
• USL – Upper specification limit
• LCL – lower specification limit
• These reflect external specifications, and determined in advance, it is not counted from the process.
Control chart
Hypothesis test
• H0 = the process is stable
Decision
Stable not stable
Reality Stable OK Type I error (risk of the producer)
not stable Type II error risk of the costumer)
OK
• Type I error – concluding a process is not in control when it is actually is – producers risk – it takes unnecessary burden on the producer who must searh fot something is not there
• Type II error – concluding a process is in control when it is actually not – customers risk – because the producer didn’t realise something is wrong and passes it on to the costumer
Control charts
and R – mean and range chart
• Sample size – n=4 or n=5 can be handled well, with short itervals,
• Sampling freuency – to reflec every affects as chenges of shifts, operators etc.
• Number of samples – 25 or more
x
• mean
• range
• n is the sample size
• Means of samples’ means
• Means of ranges
• m is the number of samples
n
xxxx n
......21
minmax xxR
m
xxxx
m
....21
m
RRRR 321 ......
Control limits
RDUCLR 4
RDLCLR 3 RAxLCLx 2
RAxUCLx 2
A2, D3, D4 are constants and depends on the sample size
Exercise
day1 6 6 5 7
day2 8 6 6 7
day3 7 6 6 6
day4 6 7 5 4
x-bar chart
024
68
1 2 3 4
day
centim
eter
Means Cl x-bar LCL x-bar UCL x-bar
Rchart
0
2
4
6
1 2 3 4
Day
Cen
tim
eter
Sample Range R-bar UCL R
Control charts for attributes
• When the process charasterictic is counted rather than measured
• p-chart – fraction of defective items in a sample
• c-chart – number of defects per unit
p-chart
• p-average fraction defective in the population
• P and σ can be counted from the samples
• min 25 samples – m• Big samlpe size is
needed (50-200 pieces) – n
• Number of defective item –np
• If the LCL is negativ, lower limit will be 0.
pp zpUCL
pp zpLCL
n
ppp
)1(
mn
npp
Exercise
• z=3,00• p=220/(20*100)=0,11• σ=(0,11(1-0,11)/
100)1/2=0,03• UCL=0,11+3*0,03=0,2• LCL=0,11-3*0,3=0,02
c-chart• To control the occurrences (defects) per unit• c1, c2 a number of defects per unit, k is the number of units
ccUCLc 3
ccLCLc 3
Exercise
Solution
5,218
45c
024,25,235,2 cLCL
24,75,235,2 cUCL
Run and trend tests • Determine
– Runs up and down (u/d)– Above and below median (med)
• Count the number of runs and compared with the number of runs that would be expected in a completely random series.
– N number of observations or data points, – E(r) expected number of runs
• Determine the standard deviation• Too few or too maní runs can be an indication of nonrandomness• Determine z score using the following formula:
• counted z must be fall into the interval of (-2;2) to accept nonrandomness (this means that the 95,5% of the time random process will produce an observed number of runs within 2σ of the expected number)
90
2916/
NDU
3
12)( /
NrE DU1
2)(
NrE med
4
1
Nmed
)(rEobs
z
It can be (-1,96;1.96) 95% of time
Or (-2,33;2,33) 98% of time
Example
Solution
• E(r)med=N/2+1=20/2+1=11
• E(r)u/d=(2N-1)/3=(2*20-1)/3=13
• σmed=[(N-1)/4]1/2=[(20-1)/4]1/2=2,18
• σu/d= =[(16N-29)/90]1/2 =[(16*20-29)/90]1/2=1,80
• zmed=(10-11)/2,18=-0,46
• Zu/d=(17-13)/1,8=2,22
• Although the median test doesn’t reveal any pattern, the up down test does.
Index of process capability
• CP (capability process) – it refers to the inherent variability of process output relative to the variation allowed by designed specifications
• the higher CP the best capablity• I the case of CP>1 the process can fulfill the requirements • It make sense when the process is centered
6LSLUSL
Cp
Process capability when process is not centered
- estimated process average (using grand mean of the samples)
• - estimated standard deviation
);min{ CplCpuCpk
2
ˆd
R
ˆ3
)ˆ( USL
Cpu
ˆ3
)ˆ( LSLCpl
x
Process capability when process is not centered II
• When sampling is not achievable, than for the total population
};min{ PplPpuPpk
3
)( USL
Ppu
3
)( LSLPpl
)1(
)( 2
n
xxi
6)( LSLUSL
Pp
USL
6σ
LSL
Cp=1
Cpk=1
• When the process is not centered the is the fault of operator but when standard deviation is higher than the tolerance limit, managers must interfer in a new machine is needed ,
Cp>1 Cp<1
Cpk>1 process capacity is proper
It can’t occure
Cpk<1 Process capacity is not proper it is the workers fault
Managers responsible for
Exercise
• For an overheat projector, the thickness of a component is specified to be between 30 and 40 millimeters. Thirty samples of components yielded a grand mean ( ) 34 mm, with a standard deviation ( ) 3,5 mm. Calculate the process capability index by following the steps previously outlined. If the process is not highly capable, what proportion of product will not conform?
x
Solution
• Process is out of control• To determine number of products use table of normal
distribution
• 0,1271+0,0436=0,1707 17,07% of products doesn’t meet specification
71,15,3
3440ˆ
xUSL
zu
14,15,3
3430ˆ
xLSL
zl
57,05,33
3440ˆ3
)ˆ(
USLCpu
38,05,33
3034ˆ3
)ˆ(
LSLCpl
5,33
3040
ˆ6
LSLUSL
Cp
Thank you for your attention