MBA 8452 Systems and Operations Management

1. MBA 8452 Systems and Operations Management Statistical Quality Control

2. Objective: Quality Analysis • Process Variation • Be able to explain Taguchi’s View of the cost of variation. • Statistical Process Control Charts and Process Capability • Be able to develop and interpret SPC charts. • Be able to calculate and interpret Cp and Cpk • Be able to explain the difference between process control and process capability • Sample Size • Be able to explain the importance of sample size

3. Statistical Quality Control Approaches • Statistical Process Control (SPC) • Sampling to determine if the process is within acceptable limits (under control) • Acceptance Sampling • Inspects a random sample of a product to determine if the lot is acceptable

4. 10 9 1 2 3 4 5 6 7 8 Line graph shows plot of dataand variation from the average Process target or average Sample number

5. Why Statistical Quality Control? • Variations in Manufacturing/Service Processes • Any process has some variations: common and/or special • Variations are causes for quality problems • If a process is stable (no special variation), it is able to produce product/service consistently • As variation is reduced, quality is improved • Statistics is the only science that is dedicated to dealing with variations. • Measure, monitor, and reduce variations in the process

6. Assignable (special) Natural (common) Types of Variation • Exogenous to process • Not random • Controllable • Preventable • Examples • tool wear • human factors (fatigue) • poor maintenance • Inherent to process • Random • Cannot be controlled • Cannot be prevented • Examples • weather • accuracy of measurements • capability of machine

7. High High Incremental Cost of Variability Incremental Cost of Variability Zero Zero Lower Spec Target Spec Upper Spec Lower Spec Target Spec Upper Spec Traditional View Taguchi’s View Cost of Variation: Traditional vs. Taguchi’s View

8. Statistical Process Control • On-line quality control tool used when the product/service is being produced • Purpose: prevent systematic quality problems • Procedure • Take periodic random samples from a process • Plot the sample statistics on control chart(s) • Determine if the process is under control • If the process is under control, do nothing • If the process is out of control, investigate and fix the cause

9. Statistical Process ControlTypes Of Data • Attribute data (discrete values) • Quality characteristic evaluated about whether it meets the required specifications • Good/bad, yes/no • Variable data (continuous values) • Quality characteristic that can be measured • Length, size, weight, height, time, velocity

10. Statistical Process ControlControl Charts • Charts for attributes • p-chart (for proportions) • c-chart (for counts) • Charts for variables • R-chart (for ranges) • -chart (for means)

11. 10 9 1 2 3 4 5 6 7 8 Control ChartGeneral Structure Upper control limit (UCL) Process target or average Lower control limit (LCL) Sample number

12. A Process Is In Control If ... • No sample points outside control limits • Most points near the process average • About an equal # points above & below the centerline • Points appear randomly distributed

13. One observation outside the limits Sample observations consistently below or above the average Sample observations consistently decrease or increase Common Out-of-control Signs

14. Issues In Building Control Charts • Number of samples: around 25 • Size of each sample: large (100) for attributes and small (25) for variables • Frequency of sampling: depends • Control limits: typically 3-sigma away from the process mean

15. 95 % 99.74 % m+1s m+2s m+3s m m-3s m-2s m-1s Control Limits:The Normal Distribution X If we establish control limits at +/- 3 standard deviations (s), then we would expect 99.74% of observations (X) to fall within these limits.

16. Control Limits: General Formulas UCL = mean + z (stand dev) LCL= mean – z (stand dev) • z is the # of standard deviations • z = 3.00 is the most commonly used value with 99.7% confidence level • Other z values can be used (e.g. z=2 for 95% confidence and z=2.58 for 99% confidence)

17. Control Charts for Attributesp-charts

18. total defectives total sample observations 200 20 (100) Proportion SampleDefect Defective 1 6 .06 2 12 .12 3 4 .04 . . . . . . . . . 20 18.18 Total 200 p = = 0.10 = n=100 jeans in each sample p-Chart Example 20 Samples of 100 pairs of jeans each were randomly selected from the Western Jean Company’s production line.

19. p-ChartExample 0.2 UCL 0.18 0.16 0.14 0.12 p 0.1 Proportion defective 0.08 0.06 0.04 0.02 LCL 0 2 6 0 4 8 12 16 18 10 14 20 . . Sample number

20. Control Charts For VariablesX-bar charts and R-charts Where X = average of sample means = Xi / m R = average of sample ranges = Ri / m Xi = mean of sample i, i = 1,2,…,m Ri = range of sample i, i = 1,2,…,m m = total number of samples A2, D3, and D4 are constants from Exhibit TN7.7

21. Example • If a company makes jeans, there are a specifications that must be met. • The back pockets of the jeans can’t be too small or too large. • The control chart can be established to monitor the measurements of the back pocket • Given 15 samples with 5 observations each, we can determine the Upper and Lower control limits for the range and x-bar charts.

22. R X (Xi) X-bar and R ChartsExample (Ri)

23. X-bar and R Charts Example Exhibit TN7.7 Since n=5, from Exhibit TN7.7 (also right table), we find A2=0.58 D3=0 D4=2.11

24. UCL LCL X-bar and R Charts: Example R chart R

25. UCL LCL X-bar and R ChartsExample X-bar chart X

26. Process Capability • The ability of a process to meet product design/technical specifications • Design specifications for products (Tolerances) • upper and lower specification limits (USL, LSL) • Process variability in production process • natural variation in process (3 from the mean) • Process may not be capable of meeting specifications if natural variation in a process exceeds allowable variation (tolerances)

27. (a) (b) specification specification natural variation natural variation (c) (d) specification specification natural variation natural variation Process CapabilityIllustrations

28. Process CapabilityFurther Illustrations Target Target LSL USL LSL Capable process USL Process variation Tolerance variation Highly capable process Process not capable Process not capable

29. Specification LimitsControl Limits • Specification limits are pre-established for products before production • Control limits are used to monitor the actual production process performance • It is possible that a process is under control, but not capable to meet specifications • It is also possible that a process that is within specifications is out-of-control

30. UCL USL UCL USL LSL LCL LCL LSL (1) In control and within specifications (2) In control but exceeds specifications USL UCL LCL LSL (3) Out-of-control and within specifications Control Limits Vs. Specification LimitsIllustrations

31. USL LSL Cp > 1 Cp = 1 Cp < 1 Process Capability Index:Cp -- Measure of Potential Capability

32. is the standard deviation of the production process Process Capability Index:Cpk -- Measure of Actual Capability Cpk considers both process variation () and process location (X)

33. Process Capability IndexExample A manufacturing process produces a certain part with a mean diameter of 2 inches and a standard deviation of 0.03 inches. The lower and upper engineering specification limits are 1.90 inches and 2.05 inches. Therefore, the process is not capable (the variation is too much and the process mean is not on the target)

34. Impact of Process Location on Process Capability Cp = 2.0 Cpk = 2.0 Cp = 2.0 Cpk = 1.5 Cp = 2.0 Cpk = 1.0 Cp = 2.0 Cpk = 0

35. Acceptance Sampling • Determines whether to accept or rejectan entire lot of goods based on sample results • Measures quality in percent defective • Usually applied to incomingraw materials or outgoingfinished goods

36. Sampling Plan • Guidelines for accepting or rejecting a lot • Single sampling plan • N = lot size • n = sample size • c = max acceptance number of defects • d = number of defective items in sample • If d <= c, accept lot; else reject Sampling plan is developed based on the tradeoff between producer’s risk and consumer’s risk

37. Producer’s & Consumer’s Risk • Producer’s Risks • reject a good lot (TYPE I ERROR) • a = producer’s risk = P(reject good lot) • 5% is common • Consumer’s Risks • accept a bad lot (TYPE II ERROR) • b = consumer’s risk = P(accept bad lot) • 10% is typical

38. Quality Definitions • Acceptable quality level (AQL) • Acceptable proportion of defects on average • “good lot” = the proportion of defects of the lot is less than or equal to AQL • Lot tolerance percent defective (LTPD) • Maximum proportion of defects in a lot • “bad lot” = the proportion of defects of the lot is greater than LTPD

39. a = .05 (producer’s risk) 1 0.9 0.8 0.7 n = 99 0.6 c = 4 Probability of acceptance 0.5 0.4 0.3  =.10 0.2 (consumer’s risk) 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 AQL LTPD Operating Characteristic Curve Percent defective in a lot