Statistical Applications in Quality and Productivity Management Sections 1 – 8. Skip 5.
Introductory Notes • Quality is an important concept for effective competition in our global economy. • Quality refers to both goods and services (good examples on page 752). • Some tools: control charts, Pareto diagrams, and histograms (and understanding of random variables).
18.1: Total Quality Management (TQM) • Global Marketplace • USA companies have been interested in quality since the middle to late 1950s. • The systematic approach to management that emphasizes quality and continuous improvement of products and services is called Total Quality Management. See page 753.
Deming’s 14 Points: A Theory of Management • Japan as a model. • Shewhart-Deming Cycle: Figure 18.1
18.2 Six Sigma Management • Quality improvement from Motorola 1980s. • Create processes that result in no more than 3.4 defects per million. • D—define • M—measure (the CTQ characteristics) • A—analyze (why defects occur) • I—improve (often with experiments) • C—Control (maintain benefits)
18.3: The Theory of Control Charts • Control chart shows the plot of data over time. • The data being plotted is related to quality. • The control chart gives insight into variability. • Use the control chart to improve the process. • Different types of data have different types of control chart.
Variation • To separate the special (assignable) causes of variation from chance (common causes). • Special causes of variation are correctable without changing the system. • Reducing variation from common causes requires that the system be changed (by management).
Control Limits • Typically calculate +/- 3 standard deviations of the measure of interest (mu, proportion, range, etc.). • Upper Control Limit (UCL) = process average plus 3 standard deviations. • Lower Control Limit (LCL) = process average minus 3 standard deviations. • A process that produces data outside of the control lines is said to be “out of control.”
Out of Control • First thing: identify sources of variation. • Hopefully we can find the assignable cause(s). • Figure 18.3, page 757: • Panel A: stable—in control • Panel B: special cause detected in 2 places • Panel C: not in control, in a trend • Define Trend, page 757.
In Control--Stable • Only common cause variation. • Is the common cause variation small enough to satisfy the buyers of the product? • Yes: monitor process. • No: change process.
18-4: Control Chart for the Proportion of Nonconforming Items—The p Chart • p chart is an attribute chart: shows the results of classifying sample observations as conforming or nonconforming. • The p chart shows the results, i.e. the proportion of nonconforming items in a sample.
Theory • p chart is based on binomial distribution. • Use formulae 18.2 and 18.3 to calculate the upper and lower control limits
Mechanics • Negative values of LCL mean that there is no LCL. • “Subgroup” means number of days that samples were taken. • The formulas work out nicer if the sample size is the same for each subgroup as is demonstrated in Table 18.1. • PHStat and Minitab will produce control charts.
Example 18.1 • Making sponges. • data: k=32 (the proportion was calculated 32 times). • The sample sizes were different for each sample. • Figure 18.6 and 18.7 • One instance of special cause variation. • Minitab calculates new control limits for each day.
18.5: The Red Bead Experiment: Understanding Process Variability • This experiment demonstrates some aspects of production and quality control. • In the experiment the “workers” must produce white beads. A red bead is nonconforming. • The “inspectors” count the red beads. • Production input is 80/20 white to red ratio. • Production tools are limited.
Lessons learned from the Experiment • Processes have variation. • Worker performance is primarily determined by the system. • Only managers can change the system. • Some workers will be above average and some will be below average.
18.6: Control Chart for an Area of Opportunity—The c Chart • Instead of looking at the proportion of nonconforming items, look at the number of nonconforming items per unit. • The “area of opportunity” is called a “unit.” • In considering the number of typos on a page, the page is the unit. • In considering the number of flaws in a square foot of carpet, the unit is the square foot of carpet.
Theory of the c Chart • The probability distribution is Poisson. • If the size of the unit remains constant, the normal probability distribution can be used to derive the formulae in 18.3. • In other words, the “3” in Formulae 18.3 is theoretically derived.
Example using Table 18.4 • Number of complaints per week for 50 weeks. • “Unit” = week. • k = 50. • c-bar = sum of complaints / 50. • LCL does not exist. • Figure 18.9: out of control due to trend. • Obvious managerial question is ___.
18.7: Control Charts for the Range and the Mean • Instead of measuring qualities such as “defective” or “Nonconforming,” we often want to measure quantities. • Variables control chartsare used with quantities. • Variables control charts are used in pairs: • one for variability: the range chart. • One for the process average: the x-bar chart.
The R Chart • Examine the Range chart first. • In control? Use it to develop mean chart. • Out of control? Use it to achieve control. Mean chart is not useful until in control. • Formulae 18.4 and 18.5 define the control limits. The constants are found in Table E.13. What do we need to know?
Example from Table 18.5 • How much time is required to move luggage from lobby to guest room? • Examine 5 deliveries per day for 28 days. • Calculate the average time per day and the daily range. • Examine R chart. In control?
The x-bar Chart • Formulae 18.6 and 18.7 show that you need x-bar-bar and the average range. • The “A” factors are found in Table E.13.
Figure 18.11 • The x-bar control chart for luggage delivery times. • In control? • Since the R-chart and X-bar-chart show processes that are “in control” if a change is desired, management must change the process.
18.8: Process Capability • Answers the question: can our process satisfy the quality requirements of our customers? • To answer this question, we must know: • what the customers expect. • that our process is in control.
3 Approaches to Capability • Process Capability is the ability of a process to consistently meet requirements. • Use the “specification limits” to calculate the probability of falling within specifications. • Calculate the Cp index. • Calculate the Cpk index.
Probability of Falling within Spec. • Process must be in control. • You will need x-bar-bar, R-bar, n, d2, and the customer specification limits. • Customer specification limits are called LSL and USL. • Need to assume that the population of measured values—the x values—is approximately normally distributed. • Use formulae 18.8.
The Cp Index • An “overall” measure of ability to meet specification. • Cp is the most common index used. • Formula 18.9. • Ratio of (1) distance between specification limits and (2) actual process spread. • Bigger is better; less than 1 is not good.
Problems with Cp • This index shows potential capability. Since it does not consider x-bar, the actual capability is in question. • Most optimistic assumption is that the process is operating near the center of the control area, i.e. near x-bar-bar. • Many companies require values near 2.0—very strict control!
The Cpk Index • Cpk = min [CPL, CPU] • CPL and CPU are capability indices that show the process capability relative to the actual operation of the process. • Formulae 18.10. • Bigger is better. • The minimum CP is the conservative value of process capability.