Quality Management 0 9 . lecture

1. Quality Management09. lecture Statistical process control

2. Sampling methods • Less expensive • Take less time • Less intrusive • 100% sampling – during acceptance sampling or work-in-process inspection • Random sample: equal chance to be inspected, independence among observations • Systematic samples: according to time or sequence • Rational subgroup: logically homogeneous, if we not separate these groups, non-random variation can biased results

3. Variability of process • Random variation – uncontrollable, caused by chance, centered around a mean with a consistent amount of dispersion • Non-random variation – has a systematic cause, shift in process mean • Process stability – only random variation exist

4. Process control chart • Tools for monitoring process variation • Continuous variable • Attribute – either or situation Example: • Weight will be variable, while number of defective items will be attributes

5. Steps • Identify critical operations • Identify critical product characteristics • Determine whether variables or attributes • Select the proper control charts • Determine control limits and improve the process • Update the limits

6. Control limits • UCL – Upper Control limit • CL – Central line • LCL – lower Control limit • Control limits comes from the process and are very different from specification limits.

7. Distribution • Central limit theorem: • If the samples number is high (above 30) than the mean of the samples will follow normal distribution

8. Hypothesis test • H0: μ=11 cm • H1: μ≠11 cm • 95% (z=1,96) rejection limit • If σ=0,001 (n=10), than the rejection limits: • 11+1,96*0,001 and 11-1,96*0,001 • (11,00196;10,99804) • The sample mean μ=10,998 falls between the rejection limits, we accept the null hypothesis • Then we accept that a process is in control

9. Errors during hypothesis test

11. X mean and R control chart

12. Mean chart monitor the average of the process • Range chart monitor the dispersion of the process • K>25, n=4 or 5

13. 1. way σx- standard deviation of distribution of sample means σ – process standard deviation n – sample size z – standard normal deviation – average of sample means

14. 2. Way mean range n is the sample size (number of observation) Average of samples’ means Average of ranges k is the number of samples A2 is constant and depends on the sample size

15. Counting of control limits A2, D3, D4 comes from factor for control limits table

16. Exercise

17. Median chart • If counting average takes too much time or effort • Number of observations (n) is better to be odd number, (3,5,7) • 20<k<25 • In sum the number of observations must reach 100

18. Example • The table below contains observations of a process. Use median chart and determine, whether the process is in control.

19. Solution • CLx=1,4 • LCLx=1,4-0,691*0,425=1,1063 • UCLx=1,4+0,691*0,425=1,693

20. Charts for attributes

21. P-chart • p – proportion of defective items • Both p and  can be estimated from the sample • k>25 • 50<n<100 • It can be used when the sample size is different • Average sample size (easy to use) • control limits must be calculated for all sample size (more precise) • If the lower limit is under 0, then we use simply zero for LCL.

22. Exercise • A controller’s task to inspect the bills of a telecommunication company. The table below consist the number of defectives in the samples. (all sample were n=100) Create p chart which cover 99,74% of the cases if the process is in control.

23. 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

24. Process capability

25. If the process is in control, than there is only non-random variation in the process. But it doesn’t mean that the products produced by the process meet the specifications or defect-free. • Process capability refers to the ability of a process to produce a product that meet the specifications.

26. Specification limit • USL – Upper specification limit • LSL – lower specification limit • Specification limit comes from outside, determined by engineers or administration, and not calculated from the process.

27. Population capability • If there are no subgroups, calculate population capability,where •  - population mean • - population process std.dev

28. Capability index • 1. select critical operation • 2. select k sample of size n • 19<k<26 • n>50 (if n binomial) • 1<n<11 (measurement) • Use control chart whether it is stable • Compare process natural tolerance limit with specification limits • Compute capability indexes: Cpl, Cpu, Cpk •  - computed population process mean • - estimated process std.dev

29. USL LSL Cp=1 Cpk=1 6σ

30. Exercise • For an overhead projector, the thickness of component is specified to be between 30 and 40 millimeters. Thirty samples of components yield a grand mean ( ) of 34 millimeters with a standard deviation ( ) of 3,5. Calculate process capability index. If the process is not capable, what proportion of a product will not conform?

31. Solution • Cpu=(40-34)/3*3,5=0,57 • Cpl=(34-30)/3*3,5=0,38 • Cpk=0,38 • The process is not capable. • To determine the proportion of product that not conform, we need to use normal distribution table. • Z=(LSL-)/ =(30-34)/3,5=-1,14 • Z=(USL- )/ =(40-34)/3,5=1,71 • 0,1271+0,0436=0,1707 17,07% will not conform

32. Thank you for your attention!