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