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 H 0 = the process is stable Decision Stablenot stable RealityStable OKType 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

mean range n is the sample size Means of samples’ means Means of ranges m is the number of samples

Control limits A 2, D 3, D 4 are constants and depends on the sample size

Exercise day16657 day28667 day37666 day46754

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 ( pieces) – n Number of defective item –np If the LCL is negativ, lower limit will be 0.

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 c 1, c 2 a number of defects per unit, k is the number of units

Exercise

Solution

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)

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 σ med =[(N-1)/4] 1/2 =[(20-1)/4] 1/2 =2,18 σ u/d = =[(16N-29)/90] 1/2 =[(16*20-29)/90] 1/2 =1,80 z med =(10-11)/2,18=-0,46 Z u/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

Process capability when process is not centered  - estimated process average (using grand mean of the samples) - estimated standard deviation

Process capability when process is not centered II When sampling is not achievable, than for the total population

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>1Cp<1 Cpk>1process capacity is proper It can’t occure Cpk<1Process 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?

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

Thank you for your attention