Operations Management

1. Operations Management Supplement 6 – Statistical Process Control PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e © 2006 Prentice Hall, Inc.

2. Statistical Process Control (SPC) • Variability is inherent in every process • Natural or common causes • Special or assignable causes • Provides a statistical signal when assignable causes are present • Detect and eliminate assignable causes of variation

3. Natural Variations • Natural variations in the production process • These are to be expected • Output measures follow a probability distribution • For any distribution there is a measure of central tendency and dispersion

4. Assignable Variations • Variations that can be traced to a specific reason (machine wear, misadjusted equipment, fatigued or untrained workers) • The objective is to discover when assignable causes are present and eliminate them

5. Each of these represents one sample of five boxes of cereal # # # # # Frequency # # # # # # # # # # # # # # # # # # # # # Weight Samples To measure the process, we take samples and analyze the sample statistics following these steps (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Figure S6.1

6. The solid line represents the distribution Frequency Weight Samples (b) After enough samples are taken from a stable process, they form a pattern called a distribution Figure S6.1

7. Central tendency Variation Shape Frequency Weight Weight Weight Samples (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Figure S6.1

8. Prediction Frequency Time Weight Samples (d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Figure S6.1

9. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Prediction Frequency Time Weight Samples (e) If assignable causes are present, the process output is not stable over time and is not predicable Figure S6.1

10. Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

11. Types of Data Variables Attributes • Characteristics that can take any real value • May be in whole or in fractional numbers • Continuous random variables • Defect-related characteristics • Classify products as either good or bad or count defects • Categorical or discrete random variables

12. For variables that have continuous dimensions • Weight, speed, length, strength, etc. • x-charts are to control the central tendency of the process • R-charts are to control the dispersion of the process Control Charts for Variables

13. For x-Charts when we know s Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where x = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means = s/ n s = population standard deviation n = sample size Setting Chart Limits

14. Hour 1 Sample Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s = 1 Hour Mean Hour Mean 1 16.1 7 15.2 2 16.8 8 16.4 3 15.5 9 16.3 4 16.5 10 14.8 5 16.5 11 14.2 6 16.4 12 17.3 n = 9 UCLx = x + zsx= 16 + 3(1/3) = 17 ozs LCLx = x - zsx = 16 - 3(1/3) = 15 ozs Setting Control Limits For 99.73% control limits, z = 3

15. Variation due to assignable causes Out of control 17 = UCL Variation due to natural causes 16 = Mean 15 = LCL Variation due to assignable causes | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 Out of control Sample number Setting Control Limits Control Chart for sample of 9 boxes

16. For x-Charts when we don’t know s Upper control limit (UCL) = x + A2R Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor found in Table S6.1 x = mean of the sample means Setting Chart Limits

17. Sample Size Mean Factor Upper Range Lower Range n A2D4D3 2 1.880 3.268 0 3 1.023 2.574 0 4 .729 2.282 0 5 .577 2.115 0 6 .483 2.004 0 7 .419 1.924 0.076 8 .373 1.864 0.136 9 .337 1.816 0.184 10 .308 1.777 0.223 12 .266 1.716 0.284 Control Chart Factors Table S6.1

18. Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 Setting Control Limits

19. Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces From Table S6.1 Setting Control Limits

20. Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces UCL = 16.154 Mean = 16.01 LCLx = x - A2R = 16.01 - .144 = 15.866 ounces LCL = 15.866 Setting Control Limits

21. R – Chart • Type of variables control chart • Shows sample ranges over time • Difference between smallest and largest values in sample • Monitors process variability • Independent from process mean

22. Upper control limit (UCLR) = D4R Lower control limit (LCLR) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table S6.1 Setting Chart Limits For R-Charts

23. Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4= 2.115, D3 = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds UCL = 11.2 Mean = 5.3 LCLR = D3R = (0)(5.3) = 0 pounds LCL = 0 Setting Control Limits

24. (a) These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) UCL (x-chart detects shift in central tendency) x-chart LCL UCL (R-chart does not detect change in mean) R-chart LCL Mean and Range Charts Figure S6.5

25. (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL (x-chart does not detect the increase in dispersion) x-chart LCL UCL (R-chart detects increase in dispersion) R-chart LCL Mean and Range Charts Figure S6.5

26. Automated Control Charts

27. Control Charts for Attributes • For variables that are categorical • Good/bad, yes/no, acceptable/unacceptable • Measurement is typically counting defectives • Charts may measure • Percent defective (p-chart) • Number of defects (c-chart)

28. p(1 - p) n sp = UCLp = p + zsp ^ ^ LCLp = p - zsp ^ where p = mean fraction defective in the sample z = number of standard deviations sp = standard deviation of the sampling distribution n = sample size ^ Control Limits for p-Charts Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics

29. Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective 1 6 .06 11 6 .06 2 5 .05 12 1 .01 3 0 .00 13 8 .08 4 1 .01 14 7 .07 5 4 .04 15 5 .05 6 2 .02 16 4 .04 7 5 .05 17 11 .11 8 3 .03 18 3 .03 9 3 .03 19 0 .00 10 2 .02 20 4 .04 Total = 80 (.04)(1 - .04) 100 80 (100)(20) sp = = .02 p = = .04 ^ p-Chart for Data Entry

30. UCLp= 0.10 .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – LCLp= 0.00 UCLp = p + zsp= .04 + 3(.02) = .10 ^ Fraction defective p = 0.04 LCLp = p - zsp = .04 - 3(.02) = 0 ^ | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 Sample number p-Chart for Data Entry

31. UCLp= 0.10 .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – LCLp= 0.00 UCLp = p + zsp= .04 + 3(.02) = .10 ^ Fraction defective p = 0.04 LCLp = p - zsp = .04 - 3(.02) = 0 ^ | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 Sample number p-Chart for Data Entry Possible assignable causes present

32. UCLc = c + 3 c LCLc = c - 3 c where c = mean number defective in the sample Control Limits for c-Charts Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics

33. c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c = 6 + 3 6 = 13.35 UCLc= 13.35 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – Number defective LCLc = c - 3 c = 3 - 3 6 = 0 c = 6 LCLc= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 Day c-Chart for Cab Company

34. Upper control limit Target Lower control limit Patterns in Control Charts Normal behavior. Process is “in control.” Figure S6.7

35. Upper control limit Target Lower control limit Patterns in Control Charts One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

36. Upper control limit Target Lower control limit Patterns in Control Charts Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

37. Upper control limit Target Lower control limit Patterns in Control Charts Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

38. Upper control limit Target Lower control limit Patterns in Control Charts Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

39. Upper control limit Target Lower control limit Patterns in Control Charts Erratic behavior. Investigate. Figure S6.7

40. Using an x-chart and R-chart: • Observations are variables • Collect 20 - 25 samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart • Track samples of n observations each Which Control Chart to Use Variables Data

41. Which Control Chart to Use Attribute Data • Using the p-chart: • Observations are attributes that can be categorized in two states • We deal with fraction, proportion, or percent defectives • Have several samples, each with many observations

42. Which Control Chart to Use Attribute Data • Using a c-Chart: • Observations are attributes whose defects per unit of output can be counted • The number counted is often a small part of the possible occurrences • Defects such as number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth