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Quantitative Review III

Quantitative Review III. Chapter 16. Quality Control & Statistical Process Control (SPC). How do we ensure quality of manufacturing and service delivery processes? Using Statistical Process Control ( SPC) to help identify and eliminate unwanted causes of variation in the process

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Quantitative Review III

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  1. Quantitative Review III

  2. Chapter 16 Quality Control & Statistical Process Control (SPC)

  3. How do we ensure quality of manufacturing and service delivery processes? Using Statistical Process Control (SPC) to help identify and eliminate unwanted causes of variation in the process To monitor the degree of product conformance to a specification on continuous scale of measurement such as length, weight, and time (continuous metric), we use X-bar and R-charts. To monitor the noncontinous scale of measurement (discrete metric)such as from data that are counted (e.g. no. of compete orders, no. of good service experiences), we use p- or c-charts

  4. Control Charts for Variable Data(length, width, etc.) • Mean (x-bar) charts • Tracks the central tendency (the average or mean value observed) over time • Range (R) charts: • Tracks the spread of the distribution over time (estimates the observed variation)

  5. Control Charts for Variable Data(length, width, etc.) K: Number of samples n: Sample size The mean for each sample (sample average) R: the range for each sample The overall mean of all the sample means The average range of all the sample ranges Upper Control Limit: Lower Control Limit:

  6. standard deviation of the sample means standard deviation of the process “n” = # of observations in each sample “k”= # of samples = standard normal variable or the # of std. deviations desired to use to develop the control limits

  7. Example: X Bar-Chart Assume the standard deviation of the process is given as 1.13 ouncesManagement wants a 3-sigma chart (only 0.26% chance of alpha error)Observed values shown in the table are in ounces. Calculate the UCL and LCL.

  8. We have 3 samples here We have 4 observations in each sample

  9. Example: X Bar-Chart Inspectors want to develop process control charts to measure the weight of crates of wood. Data (in pounds) from three samples are: Sample Crate 1 Crate 2 Crate 3 Crate 4 1 18 38 22 26 2 26 30 28 20 3 26 26 26 26 What are the upper and lower control limits for this process? Z=3 = 4.27

  10. We first need to calculate x-bar for each sample = = 26 = = 26 = = 26

  11. We have 3 samples We have 4 observations in each sample

  12. Range or R Chart k = # of sample ranges Range Chart measures the dispersion or variance of the process while The X-Bar chart measures the central tendency of the process. When selecting D4 and D3 from the table, use sample size or number of observations for n.

  13. Control Chart Factors Factors for R-Chart Sample Size (n) D3 D4 2 0.00 3.27 3 0.00 2.57 4 0.00 2.28 5 0.00 2.11 6 0.00 2.00 7 0.08 1.92 8 0.14 1.86 9 0.18 1.82 10 0.22 1.78 11 0.26 1.74 12 0.28 1.72 13 0.31 1.69 14 0.33 1.67 15 0.35 1.65

  14. Example: R-Chart Sample ranges = highest observation – lowest observation

  15. R-chart Computations(Use D3 & D4 Factors: Table 6-1) D4 from control chart factors when n=4 D3 from control chart factors when n=4

  16. Example: R-Chart Ten samples of 5 observations each have been taken form a Soft drink bottling plant in order to test for volume dispersion in the bottling process. The average sample range was found To be .5 ounces. Develop control limits for the sample range. = 0.5, n = 5

  17. P- Chart Many quality characteristics assumes only two values, such as good or bad, pass or fail. The proportion of nonconforming items can be monitored using a control chart called a p-chart, where p is the proportion of nonconforming items found in a sample.

  18. (P) Fraction Defective Chart • Used for yes or no type judgments (acceptable/not acceptable, works/doesn’t work, on time/late, etc.) • p = proportion of nonconforming items = average proportion of nonconforming items

  19. (P) Fraction Defective Chart K = # of samples n = # of observations in each sample = standard normal variable or the # of std. deviations desired to use to develop the control limits

  20. P-Chart Example: A Production manager for a tire company has inspected the number of defective tires in five random samples with 20 tires in each sample. The table below shows the number of defective tires in each sample of 20 tires. Z= 3. Calculate the control limits. Solution: Note: Lower control limit can’t be negative, if LCLp is less than zero, a value of zero is used.

  21. C-Chart A p-chart monitors the proportion of nonconforming items, but a nonconforming item may have more than one conformance. E.g. a customer’s order may have several errors, such as wrong item, wrong quantity, wrong price. To monitor the number of nonconformance per unit, we use a c-chart. It is used extensively in service applications.

  22. Number-of-Defectives or C-Chart Used when looking at # of nonconformances c = # of nonconformances = average # of nonconformances per sample

  23. Number-of-Defectives or C Chart = standard normal variable or the # of std. deviations desired to use to develop the control limits

  24. C-Chart Example: The number of weekly customer complaints are monitored in a large hotel using a c-chart. Develop three sigma control limits using the data table below. Z=3. • Solution:

  25. Process Capability The natural variation in a process that results from common causes Process Capability Index A measure that quantify the relationship between the natural variation and specifications

  26. Process Capability • Product Specifications • Preset product or service dimensions, tolerances • e.g. bottle fill might be 16 oz. ±.2 oz. (15.8oz.-16.2oz.) • Based on how product is to be used or what the customer expects • Process Capability – Cp and Cpk • Assessing capability involves evaluating process variability relative to preset product or service specifications • Cp assumes that the process is centered in the specification range • Cpkhelps to address a possible lack of centering of the process

  27. Process Capability LSL USL Spec Width -3δ -2δ -1δμ 1δ 2δ 3δ If Cp is ≥ 1; process capable of meeting design specs If Cpk is ≥ 1; process capable of meeting design specs

  28. Process Capability Index • Can a process or system meet it’s requirements? Cp < 1: process not capable of meeting design specs Cp ≥ 1: process capable of meeting design specs • One shortcoming, Cpassumes that the process is centered on the specification range

  29. min = minimum of the two = mu or mean of the process If is less than 1, then the process is not capable or is not centered.

  30. Capability Indexes • Centered Process (Cp): • Any Process (Cpk): • Cp=Cpkwhen process is centered

  31. Example • Design specifications call for a target value of 16.0 +/-0.2 ounces (USL = 16.2 & LSL = 15.8) • Observed process output has a mean of 15.9 and a standard deviation of 0.1 ounces

  32. Computations • Cp: • Cp < 1: process not capable of meeting design specs • Cpk: C pk <1, then this measure also indicates that process is not capable

  33. Chapter 18 Project Management

  34. Project Planning, Scheduling and Controlling Calculate Path Durations • The longest path (ABDEGIJK) limits the project’s duration (project cannot finish in less time than its longest path) • ABDEGIJK is the project’s critical path

  35. ES=16 EF=30 LS=16 LF=30 ES=32 EF=34 LS=33 LF=35 ES=10 EF=16 LS=10 LF=16 ES=4 EF=10 LS=4 LF=10 Project Network E Buffer ES=39 EF=41 LS=39 LF=41 ES=35 EF=39 LS=35 LF=39 D(6) E(14) H(2) B(6) A(4) K(2) G(2) J(4) ES=0 EF=4 LS=0 LF=4 C(3) ES=30 EF=32 LS=30 LF=32 F(5) I(3) ES=4 EF=7 LS=22 LF=25 ES=7 EF=12 LS=25 LF=30 ES=32 EF=35 LS=32 LF=35 Critical Path: the sequence of activities that takes the longest time & defines the total project completion time

  36. Calculate Early Starts & Finishes ES=32 EF=34 ES=16 EF=30 ES=10 EF=16 ES=4 EF=10 D(6) E(14) H(2) B(6) Latest EF = Next ES ES=35 EF=39 ES=39 EF=41 A(4) K(2) G(2) J(4) ES=0 EF=4 LS=0 LF=4 ES=30 EF=32 LS=30 LF=32 C(3) F(5) I(3) ES=4 EF=7F=25 ES=7 EF=12 LS=25 LF=30 ES=32 EF=35 LS=32 LF=35

  37. Calculate Late Starts & Finishes ES=32 EF=34 LS=33 LF=35 ES=10 EF=16 LS=10 LF=16 ES=16 EF=30 LS=16 LF=30 ES=4 EF=10 LS=4 LF=10 ES=39 EF=41 LS=39 LF=41 ES=35 EF=39 LS=35 LF=39 D(6) E(14) H(2) B(6) A(4) K(2) G(2) J(4) ES=0 EF=4 LS=0 LF=4 ES=30 EF=32 LS=30 LF=32 C(3) F(5) I(3) ES=4 EF=7 LS=22 LF=25 ES=7 EF=12 LS=25 LF=30 ES=32 EF=35 LS=32 LF=35

  38. What is the critical path? E(5) B(3) G(4) A(4) H(3) C(5) D(2) F(6) • ABEGH • ACEGH • ACFGH • ADFGH

  39. What is ES, EF for G? E(5) B(3) G(4) A(4) H(3) C(5) D(2) F(6) • 14, 18 • 15, 19 • 16, 20 • 17, 21 Latest EF = Next ES

  40. What is LS, LF for C? E(5) B(3) G(4) A(4) H(3) C(5) D(2) F(6) • 4, 9 • 5, 10 • 3, 8 • 6, 11

  41. E(5) B(3) H(3) G(4) C(5) A4 F(6) D2 ES 9 LS 10 EF 14 LF 15 ES 4 LS 7 EF 7 LF 10 ES 19 LS 19 EF 22 LF 22 ES 15 LS 15 EF 19 LF 19 ES 4 LS 4 EF 9 LF 9 ES 0 EF 4 ES 4 LS 7 EF 6 LF 9 ES 9 LS 9 EF 15 LF 15 Latest EF = Next ES Earliest LS = Previous LF

  42. Activity Slack Time TES = earliest start time for activity TLS = latest start time for activity TEF = earliest finish time for activity TLF = latest finish time for activity Activity Slack = TLS - TES = TLF - TEF

  43. Calculate Activity Slack

  44. Path Slack Duration of Critical Path (41) - Path Duration = Path Slack Path Slack 1 0 19 18

  45. A (3) B (5) C (2) E (12) F (3) H(5) D (5) G (6) What is ES and EF for E? ES for E: 3+5+5=13 EF for E: 13+12 =25 What is LS and LF for C? LF for C: = ES for E =13 LS for C: 13-2=11 What is the activity slack for C? ES for C: 3+5 =8 Activity slack for C = LS-ES=11-8=3 EF for C: 8+2=10 If as a project manager you see that Activity F in the project above is going to take 3 extra weeks, what should you do? Watchful waiting

  46. Waiting Line Models

  47. How long you wait in line depends on a number of Factors: • the number of people served before you • the number of servers working • the amount of time it takes to serve each individual customer Wait time is affected by the design of the waiting line system. A waiting line system (or queuing system) is defined by two elements: • The customer population (people or objects to be processed) • The process or service system

  48. Waiting Line System Performance Measures • The average number of customers waiting in queue • The average number of customers in the system • The average waiting time in queue • The average time in the system • The system utilization rate (% of time servers are busy)

  49. Infinite Population, Single-Server, Single Line, Single Phase Formulae

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