1. Production and Operations Management: Manufacturing and Services PowerPoint Presentation for Chapter 7 Supplement Statistical Quality Control © The McGraw-Hill Companies, Inc., 1998 and (c) Stephen A. DeLurgio, 2000 Irwin/McGraw-Hill

2. 2 Chapter 7 Supplement - 1 Statistical Quality Control Process Control Procedures - 1 –Variable data –Attribute data Process Capability - 2 Acceptance Sampling - 3 –Operating Characteristic Curve

3. 3 Basic Forms of Statistical Sampling for Quality Control Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). Trying to Inspect Quality Into Product! Sampling to determine if the process is within acceptable limits (Statistical Process Control). Building Quality Into Product and Process!

5. IMPORTANT UNDERLYING PRINCIPLE IT IS POSSIBLE TO DESIGN A PROCESS SO THAT EVEN WHEN WE DETECT IT AS BEING OUT OF CONTROL, NO DEFECTS ARE PRODUCED. OUR GOAL REDUCE PROCESS VARIATION SO MUCH THAT DEFECTS ARE NOT PRODUCED. WE DO THAT BY CREATING CONTROL DEVICES, ELIMINATING THE CAUSES OF LARGE, ASSIGNABLE PROCESS VARIATIONS, AND COORDINATING PRODUCT DESIGN AND PROCESS CAPABILITY.

6. PRODUCTIVITY/QUALITY GAINS FROM SPC ARE TRULY EXTRAORDINARY ! WE STUDY SCIENTIFIC METHODS OF SPC TO Eliminate Causes of Defects Identify Assignable Variations Adjust the Process Reduce Risks of Defective Products ACHIEVE VALUE FOR EVERYONE!

7. UNDERSTANDING VARIABILITY To understand variability, we need to understand some basic statistics and random behavior. These concepts apply to industrial processes, how we perform at sports, how physical and biological systems behave, and many other occurrences. Well designed processes yield output that is Normally Distributed. Your understanding of the Normal Distribution(ND) is Essential -WHAT IS AND WHAT CAUSES NORMALLY DISTRIBUED VALUES? WHY IS THIS IMPORTANT?

8. NORMALLY DISTRIBUTED MEAN +/- ONE STANDARD DEVIATION 68% MEAN +/- 1.96 STANDAR DEVIATRIONS 95% MEAN +/- 3.00 STANDARD DEVIATIONS 99.73% MEAN +/- 4.00 STANDARD DEVIATIONS 99.994%

9. ND CHARACTERISTICS SYMMETRICAL - BELL SHAPED DISCOVERED BY K. F. GAUSS DEFINED COMPLETELY BY MEAN AND STANDARD DEVIATION GENERATED BY IN CONTROL RANDOM PROCESS CONTINUOUS DISTRIBUTION FROM -INFINITY TO + INFINITY

10. WHAT GENERATES ND OUTPUT? “IF AN EVENT IS THE RESULT OF A RELATIVELY LARGE NUMBER OF SMALL, CHANCE, INDEPENDENT INFLUENCES, THEN ITS OUTPUT WILL BE ND.” MANY PROCESSES ARE ND BECAUSE: WE HAVE WORKED HARD TO ELIMINATE THE VERY LARGE INFLUENCES, THUS ONLY A RELATIVELY LARGE NUMBER OF SMALL, INDEPENDENT INFLUENCES REMAIN.

11. FOR EXAMPLE: THINK ABOUT THE PROCESS OF PRODUCING GOLD COINS, IT IS IMPORTANT THAT EACH WEIGHS 1.0 OZ. TO ACHIEVE A 1 OZ. WEIGHT WE CONTROL: THE SIZE OF GOLD STRIPS GOING INTO THE PRESS. THE ADJUSTMENTS ON THE MACHINE. THE TEMPERATURE OF THE MACHINE. THE HUMIDITY OF THE ROOM. THE CLEANLINESS OF THE SET UP. THE CONDITION OF THE TOOLS (DIES) USED. ALL OTHER FACTORS THAT INFLUENCE WEIGHT. 1 OZ.

12. COINING OUTPUT FOR n = 600 NOTE SYMMETRY AND BELL SHAPE

13. HISTOGRAM OF COINING OUTPUT, n=600 NOTE SYMMETRY AND BELL SHAPE

14. IN CONTROL PROCESS VARIATION BY ELIMINATING ALL OF THE LARGE INFLUENCES WE ARE LEFT WITH MANY SMALL INFLUENCES ACTING SEPARETLY. THIS YIELDS A PROCESS WITH: MEAN = 1 OZ. STD. DEV. =.001 OZS. AND IMPORTANTLY, THE OUTPUT IS NORMALLY DISTRIBUTED CONSIDER THE INTERVALS:

15. MEAN = 1 OZ., STD DEV=.001 1 +/-.001 68% 6,800 OF 10,000 IN THIS RANGE 1 +/-.00196 95% 9,500 OF 10,000 IN THIS RANGE 1 +/-.003 99.73% 9,973 OF 10,000 IN THIS RANGE 1 +/-.004 99.994% 9,999.4 OF 10,000 IN THIS RANGE

16. DESCRIPTIVE STATISTICS MEAN = CENTER OF DEVIATIONS POPULATION MEAN,  =  X / N MEDIAN VALUE HAVING 50% ABOVE, 50% BELOW MODE MOST FREQUENT VALUE FOR SYMMETRICAL DISTRIBUTION MEAN = MEDIAN = MODE

17. COINING OUTPUT FOR n = 600 NOTE SYMMETRY AND BELL SHAPE

18. STANDARD DEVIATION MEASURES VARIATION OR SCATTER SQUARE ROOT OF THE MEAN SQUARED ERROR   x =   ( X -  ) 2 /N Population std. deviation of X with census.  S x =   ( X -  X ) 2 /(n-1) Sample standard deviation of X. Formulas may not yield much information, not as meaningful unless for known distribution.

19. MEAN = 1 OZ., STD DEV=.001 1 +/-.001 68% 6,800 OF 10,000 IN THIS RANGE 1 +/-.00196 95% 9,500 OF 10,000 IN THIS RANGE 1 +/-.003 99.73% 9,973 OF 10,000 IN THIS RANGE 1 +/-.004 99.994% 9,999.4 OF 10,000 IN THIS RANGE

20. MEAN +/- ONE STANDARD DEVIATION 68% MEAN +/- 1.96 STANDAR DEVIATRIONS 95% MEAN +/- 3.00 STANDARD DEVIATIONS 99.73% MEAN +/- 4.00 STANDARD DEVIATIONS 99.994% MEAN +/- 5.00 STANDARD DEVIATIONS 99.99994% MEAN +/- 6.00 STANDARD DEVIATIONS 99.99999% OTHER ND INTERVALS

21. THE CENTRAL LIMIT THEOREM NOTE THAT SAMPLE MEANS ARE ND!

22. THE CENTRAL LIMIT THEOREM DISTRIBUTION OF SAMPLE MEANS IS ND FOR LARGE SAMPLES FROM ANY GENERAL POPULATION! MEAN OF MEANS ARE ND __ __ X =   Z  /  n MEAN OF MEAN = POP MEAN STD. DEV. OF MEANS = POP STD.DEV /n^.5 __ ___ X = 1.0  Z.001/  100

23. 15 Control Limits Let’s establish control limits at +/- 3 standard deviations, then We expect 99.7% of observations to fall within these limits x LCLUCL

24. CONTROL CHARTS BASED ON ND

25. TIME TO THE CONTROL CHART ADDS POWERFUL INFERENCES!

26. ALL POINTS IN CONTROL

27. A B “A” IS OUT OF CONTROL, TWO PTS. IN B ARE OUT OF CONTROL, TREND OF 7 = OUT OF CONTROL 7=TREND

29. X-BAR CHART FORMULAS When using known mean  and standard deviation  : _ __ X =   Z  /  n When  and  are unknown, they are estimated: _ = _ __ X = X  Z S/  n When using measured Ranges: _ = _ X = X  A 2 R

30. THE RELATIONSHIP BETWEEN COOKBOOK FORMULAS AND THEORY  A 2  R = 3   n

31. S-Charts and R-Charts The S-chart uses the following formula: S =   Z  /  2n The R-Chart uses the following formulas: D 4  R (UCL) R = { D 3  R (LCL) The results of both will be the same in use, however, numerical values using S and R will be different, the plots will look nearly identical.

32. A LITTLE MORE THEORY When small samples (n<30) are used, the assumption is that the sample comes from a ND. When this is not true, then the above formulas MAY NOT BE valid. If the process is NOT ND, then large samples are necessary, or other statistical tests called Nonparametric methods must be used.

33. 23 Example: x-Bar and R Charts

34. 24 Calculate sample means, sample ranges, mean of means, and mean of ranges.

35. 25 Control Limit Formulas

36. 26 x-Bar Chart UCL LCL

37. 27 R-Chart UCL LCL

38. 6 Statistical Sampling--Data Attribute (Go no-go information) –Defectives--refers to the acceptability of product across a range of characteristics. –Defects--refers to the number of defects per unit-- may be higher than the number of defectives. Variable (Continuous) –Usually measured by the mean and the standard deviation.

39. DISTRIBUTION OF SAMPLE PROPORTIONS POP IS NOT ND  =.98 SAMPLE LOOKS LIKE POP, P =.99 DIST. OF SAMPLE P’S ARE ND

40. P-CHARTS Require large samples n  30. When population proportion is known: ————— P =   Z   (1 -  )/n When population proportion is unknown: _ _ _ P = P  Z  P (1 - P )/n Where P-Bar is an estimate of 

41. 17 Constructing a p-Chart

42. 18 Statistical Process Control--Attribute Measurements (P-Charts)

43. 1. Calculate the sample proportion, p, for each sample. 19 © The McGraw-Hill Companies, Inc., 1998 Irwin/McGraw-Hill

44. 2. Calculate the average of the sample proportions. 3. Calculate the standard deviation of the sample proportion 20 © The McGraw-Hill Companies, Inc., 1998 Irwin/McGraw-Hill

45. 4. Calculate the control limits. UCL = 0.093 LCL = -0.0197 (or 0) 21 © The McGraw-Hill Companies, Inc., 1998 Irwin/McGraw-Hill

46. 22 p-Chart (Continued) 5. Plot the individual sample proportions, the average of the proportions, and the control limits

47. 1. Calculate the sample proportion, p, for each sample. 19 © The McGraw-Hill Companies, Inc., 1998 Irwin/McGraw-Hill