2011 Pearson Education Inc Statistics for Business and
© 2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 12 Methods for Quality Improvement © 2011 Pearson Education, Inc
Content 12. 1 12. 2 12. 3 12. 4 Quality, Processes, and Systems Statistical Control The Logic of Control Charts A Control Chart for Monitoring the Mean of a Process: The -Chart 12. 5 A Control Chart for Monitoring the Variation of a Process: The R-Chart © 2011 Pearson Education, Inc
Content 12. 6 A Control Chart for Monitoring the Proportion of Defectives Generated by a Process: The p-Chart 12. 7 Diagnosing the Causes of Variation 12. 8 Capability Analysis © 2011 Pearson Education, Inc
Learning Objectives • Return to an examination of processes (i. e. , actions/operations that transform inputs to outputs) • Describe methods for improving processes and the quality of the output they produce • Present control charts for monitoring a process mean, variance, or proportion © 2011 Pearson Education, Inc
12. 1 Quality, Processes, and Systems © 2011 Pearson Education, Inc
Quality The quality of a good or service is indicated by the extent to which it satisfies the needs and preferences of its users. © 2011 Pearson Education, Inc
The Eight Dimensions of Quality 1. Performance • Primary operating characteristics of the product 2. Features • "bells and whistles” that supplement the product’s basic functions 3. Reliability • Probability product will function for a specified amount of time © 2011 Pearson Education, Inc
The Eight Dimensions of Quality 4. Conformance • Extent to which product meets preestablished standards 5. Durability • The life of the product 6. Serviceability • Ease of repair, speed of repair and competence and courtesy of repair staff © 2011 Pearson Education, Inc
The Eight Dimensions of Quality 7. Aesthetics • How product looks, feels, etc. 8. Other Perceptions • e. g. Company reputation © 2011 Pearson Education, Inc
Process A process is a series of actions or operations that transforms inputs to outputs. A process produces output over time. © 2011 Pearson Education, Inc
Process Inputs Information Methods Energy Materials Machines People Process Operations Outputs Finished Products Variability is present in the output of all processes © 2011 Pearson Education, Inc
System A system is a collection or arrangement of interacting processes that has an ongoing purpose or mission. A system receives inputs from its environment, transforms those inputs to outputs, and delivers them to its environment. In order to survive, a system uses feedback (i. e. , information) from its environment to understand adapt to changes in its environment. © 2011 Pearson Education, Inc
System Input Supplier Output Processes Feedback © 2011 Pearson Education, Inc Customer
The Deming’s 14 Points: Guidelines for Quality Improvement 1. Create constancy of purpose toward improvement of product and service, with the aim to become competitive, to stay in business, and to provide jobs. 2. Adopt the new philosophy. 3. Cease dependence on inspection to achieve quality. © 2011 Pearson Education, Inc
The Deming’s 14 Points: Guidelines for Quality Improvement 4. End the practice of awarding business on the basis of price tag. 5. Improve constantly and forever the system of production and service, to improve quality and productivity, and thus constantly decrease costs. 6. Institute training. 7. Institute leadership. © 2011 Pearson Education, Inc
The Deming’s 14 Points: Guidelines for Quality Improvement 8. Drive out fear, so that everyone may work effectively for the company. 9. Break down barriers between departments. 10. Eliminate slogans, exhortations, and arbitrary numerical goals and targets for the workforce that urge workers to achieve new levels of productivity and quality. 11. Eliminate numerical quotas. © 2011 Pearson Education, Inc
The Deming’s 14 Points: Guidelines for Quality Improvement 12. Remove barriers that rob employees of their pride of workmanship. 13. Institute a vigorous program of education and self-improvement. 14. Take action to accomplish the transformation. © 2011 Pearson Education, Inc
The Six Major Sources of Process Variation 1. People 2. Machines 3. Materials 4. Methods 5. Measurement 6. Environment © 2011 Pearson Education, Inc
12. 2 Statistical Control © 2011 Pearson Education, Inc
Control Chart A control chart is a graphical devices used for monitoring process variation, identifying when to take action to improve the process, and assisting in diagnosing the causes of process variation. © 2011 Pearson Education, Inc
Time Series Plot (Run Chart) • Graphically shows trends and changes in the data over time • Time recorded on the horizontal axis • Measurements recorded on the vertical axis • Points connected by straight lines © 2011 Pearson Education, Inc
Time Series Plot (Run Chart) © 2011 Pearson Education, Inc
Oscillating Sequence Center line © 2011 Pearson Education, Inc
Patterns of Process Variation © 2011 Pearson Education, Inc
Patterns of Process Variation © 2011 Pearson Education, Inc
Patterns of Process Variation © 2011 Pearson Education, Inc
Patterns of Process Variation © 2011 Pearson Education, Inc
Process Variation 1. At any point in time, the output variable of interest can be described by a particular probability distribution. 2. The particular value of the output variable that is realized at a given time can be thought of as being generated or produced according to the distribution described in point 1. 3. The distribution that describes the output variable may change over time. © 2011 Pearson Education, Inc
Process Variation © 2011 Pearson Education, Inc
Stability A process whose output distribution does not change over time is said to be in a state of statistical control, or simply in control. If it does change, it is said to be out of statistical control, or simply out of control. The figure on the next slide illustrates a sequence of output distributions for both an in-control and an outof-control process. © 2011 Pearson Education, Inc
Stability © 2011 Pearson Education, Inc
Statistical Process Control The process of monitoring and eliminating variation in order to keep a process in a state of statistical control or to bring a process into statistical control is called statistical process control (SPC). © 2011 Pearson Education, Inc
Stability © 2011 Pearson Education, Inc
Common Causes of Variation Common causes of variation are the methods, materials, machines, personnel, and environment that make up a process and the inputs required by the process. Common causes are thus attributable to the design of the process. Common causes affect all output of the process and may affect everyone who participates in the process. © 2011 Pearson Education, Inc
Special Causes of Variation Special causes of variation (sometimes called assignable causes) are events or actions that are not part of the process design. Typically, they are transient, fleeting events that affect only local areas or operations within the process (e. g. , a single worker, machine, or batch of materials) for a brief period of time. Occasionally, however, such events may have a persistent or recurrent effect on the process. © 2011 Pearson Education, Inc
12. 3 The Logic of Control Charts © 2011 Pearson Education, Inc
Control Chart Uses • Monitor process variation • Differentiate between variation due to common causes v. special causes • Evaluate past performance • Monitor current performance © 2011 Pearson Education, Inc
Specification Limits Specification limits are boundary points that define the acceptable values for an output variable (i. e. , for a quality characteristic) of a particular product or service. They are determined by customers, management, and product designers. Specification limits may be two sided, with upper and lower limits, or one sided, with either an upper or a lower limit. © 2011 Pearson Education, Inc
Specification Limits © 2011 Pearson Education, Inc
4 Possible Outcomes • H 0: Process is in control • Ha: Process is out of control Conclusion Conclude process is in control H 0 True Ha True Reality H 0 True Ha True Correct decision Type I Error Type II Error Correct decision Conclude process is out of control© 2011 Pearson Education, Inc
12. 4 A Control Chart for Monitoring the Mean of a Process: The -Chart © 2011 Pearson Education, Inc
–Chart • • Monitors changes in the mean of samples Horizontal axis: Sample number Vertical axis: Mean of sample Control limits based on sampling distribution of x – Standard deviation of x: © 2011 Pearson Education, Inc
Sample –Chart © 2011 Pearson Education, Inc
Sampling Distribution of © 2011 Pearson Education, Inc
Determining the Centerline k = number of samples of size n = sample mean of the ith sample is an estimator of © 2011 Pearson Education, Inc
Control Limits Upper control limit: Lower control limit: © 2011 Pearson Education, Inc
Estimating σ 1. Determine the Range of each sample Range = Maximum – Minimum 2. Determine the Average Range of the k samples 3. Divide R by the constant d 2 (based on sample size and found in Table XI of Appendix B) © 2011 Pearson Education, Inc
Determining the Control Limits where © 2011 Pearson Education, Inc
The Two Most Important Decisions in Constructing an -Chart 1. The sample size, n, must be determined. 2. The frequency with which samples are to be drawn from the process must be determined (e. g. , once an hour, once each shift, or once a day). © 2011 Pearson Education, Inc
Rational Subgroups Samples whose size and frequency have been designed to make it likely that process changes will occur between, rather than within, the samples are referred to as rational subgroups. © 2011 Pearson Education, Inc
Rational Subgrouping Strategy The samples (rational subgroups) should be chosen in a manner that 1. Gives the maximum chance for the measurements in each sample to be similar (i. e. , to be affected by the same sources of variation) 2. Gives the maximum chance for the samples to differ (i. e. , be affected by at least one different source of variation) © 2011 Pearson Education, Inc
–Chart Summary 1. Collect at least 20 samples of size n ≥ 2 2. Calculate the mean and range of each sample 3. Calculate where k = number of samples (i. e. , subgroups) = sample mean for the ith sample Ri = range ©of the ith sample 2011 Pearson Education, Inc
–Chart Summary 4. Plot the centerline and control limits where A 2 is a constant that depends on n. Its values are given in Table XI in Appendix B, for samples of size n = 2 to n = 25. © 2011 Pearson Education, Inc
–Chart Summary 5. Plot the k sample means on the control chart in the order that the samples were produced by the process. Note: Most quality control analysts use available statistical software to perform the calculations and generate the -chart. © 2011 Pearson Education, Inc
–Chart Example Samples from a machine filling 12 oz soda cans © 2011 Pearson Education, Inc
–Chart Centerline Solution © 2011 Pearson Education, Inc
–Chart Control Limits Solution © 2011 Pearson Education, Inc
–Chart Control Limits Solution © 2011 Pearson Education, Inc
–Chart Solution UCL = 12. 61 LCL = 11. 38 © 2011 Pearson Education, Inc
Interpreting Control Charts • Six zones – Each zone is one standard deviation wide Upper A–B boundary Upper B–C boundary Lower A–B boundary Zone A Zone B Zone C Zone B Zone A © 2011 Pearson Inc Order of. Education, production UCL centerline LCL
Zone Boundaries 3–sigma control limit zone boundaries: Upper A–B Boundary: Lower A–B Boundary: Upper B–C Boundary: Lower B–C Boundary: © 2011 Pearson Education, Inc
Zone Boundaries Example Samples from a machine filling 12 oz soda cans © 2011 Pearson Education, Inc
Zone A–B Boundaries Solution Recall Upper A–B: Lower A–B: © 2011 Pearson Education, Inc
Zone B–C Boundaries Solution Recall Upper B–C: Lower B–C: © 2011 Pearson Education, Inc
–Chart Solution UCL = 12. 61 A 12. 4 B 12. 2 C C B 11. 8 11. 6 A LCL = 11. 38 © 2011 Pearson Education, Inc
Pattern Analysis Rules • Rule 1: One point beyond Zone A – Either lower or upper half of control chart © 2011 Pearson Education, Inc
Pattern Analysis Rules • Rule 2: Nine points in a row in Zone C or beyond – Either lower or upper half of control chart © 2011 Pearson Education, Inc
Pattern Analysis Rules • Rule 3: Six points in a row steadily increasing or decreasing © 2011 Pearson Education, Inc
Pattern Analysis Rules • Rule 4: Fourteen points in a row alternating up and down © 2011 Pearson Education, Inc
Pattern Analysis Rules • Rule 5: Two out of three points in Zone A or beyond – Either lower or upper half of control chart © 2011 Pearson Education, Inc
Pattern Analysis Rules • Rule 6: Four out of five points in a row in Zone B or beyond – Either lower or upper half of control chart © 2011 Pearson Education, Inc
Interpreting an –Chart • Process is considered out of control if any of the pattern analysis rules are detected • Process is considered in control if none of the pattern analysis rules are detected Assumption: The variation process is stable. © 2011 Pearson Education, Inc
Interpreting –Chart Example What does the chart suggest about the stability of the process? UCL = 12. 61 12. 4 B 12. 2 A C C B 11. 8 11. 6 A LCL = 11. 38 © 2011 Pearson Education, Inc
Interpreting –Chart Solution Since none of the six pattern analysis rules are observed, the process is considered in control © 2011 Pearson Education, Inc
Interpreting –Chart Thinking Challenge Ten additional samples of size 5 are taken. What does the chart suggest about the stability of the process? UCL = 12. 61 A 12. 4 B 12. 2 C C B 11. 8 11. 6 A LCL = 11. 38 © 2011 Pearson Education, Inc
Interpreting – Chart Solution Rule 5 and Rule 6 are violated. Process is out of control UCL = 12. 61 A 12. 4 B 12. 2 C C B A © 2011 Pearson Education, Inc 11. 8 11. 6 LCL = 11. 38
12. 5 A Control Chart for Monitoring the Variation of a Process: The R-Chart © 2011 Pearson Education, Inc
R–Chart • • Monitors changes in process variation Horizontal axis: Sample number Vertical axis: Sample ranges Control limits based on sampling distribution of R – Mean of sampling distribution of R: μR – Standard deviation of sampling distribution of R: σR © 2011 Pearson Education, Inc
Estimating μR and σR Estimate of μR: k = number of samples of size n ≥ 2 Ri = sample range of the ith sample Estimate of σR: © 2011 Pearson Education, Inc
Determining the Control Limits Note: If n ≤ 6, the LCL will be negative. Since the range can’t be negative the LCL is meaningless. © 2011 Pearson Education, Inc
R–Chart Summary 1. Collect at least 20 samples of size n ≥ 2 2. Calculate the range of each sample 3. Calculate the mean of the sample ranges where k = the number of sample (i. e. , subgroups) Ri = the range of the ith sample © 2011 Pearson Education, Inc
R–Chart Summary 4. Plot the centerline and control limits where D 3 and D 4 are constants that depend on n. Their values can be found in Table XI in Appendix B. When n ≤ 6, D 3 = 0, indicating that the control chart does not have a lower control limit. © 2011 Pearson Education, Inc
R–Chart Summary 5. Plot the k sample ranges on the control chart in the order that the samples were produced by the process. © 2011 Pearson Education, Inc
Zone Boundaries Upper A–B Boundary: Lower A–B Boundary: Upper B–C Boundary: Lower B–C Boundary: © 2011 Pearson Education, Inc
Interpreting an R–Chart • Process is considered out of control if any of the pattern analysis rules 1 – 4 are detected: – One point beyond Zone A – Nine points in a row in Zone C or beyond – Six points in a row steadily increasing or decreasing – Fourteen points in a row alternating up and down • Process is considered in control if none of the pattern analysis rules are detected © 2011 Pearson Education, Inc
R–Chart Example Samples from a machine filling 12 oz soda cans © 2011 Pearson Education, Inc
R–Chart Solution Calculate the mean of the ranges: © 2011 Pearson Education, Inc
R–Chart Solution Calculate the control limits. n=5 D 4 = 2. 114 D 3 = 0 (LCL will be zero) © 2011 Pearson Education, Inc
R–Chart Solution Determine the A–B zone boundaries Upper A–B Boundary: Lower A–B Boundary: © 2011 Pearson Education, Inc
R–Chart Solution Determine the B–C zone boundaries Upper B–C Boundary: Lower B–C Boundary: © 2011 Pearson Education, Inc
R–Chart Solution A B UCL = 2. 3 1. 9 1. 5 C C B . 7 . 3 A LCL = 0 The variation of the process is in control © 2011 Pearson Education, Inc
– and R–Charts In practice, the -chart and the R-chart are not used in isolation. Rather, they are used together to monitor the mean (i. e. , the location) of the process and the variation of the process simultaneously. In fact, many practitioners plot them on the same piece of paper. The appropriate procedure is to first construct and then interpret the R-chart. If it indicates that the process variation is in control, then it makes sense to construct© 2011 and interpret -chart Pearson Education, Inc the
12. 6 A Control Chart for Monitoring the Proportion of Defectives Generated by a Process: The p-Chart © 2011 Pearson Education, Inc
p–Chart • • • Used for qualitative data Monitors variation in the process proportion Horizontal axis: Sample number Vertical axis: Sample proportion Control limits based on sampling distribution of p^ ^ μ^p – Mean of sampling distribution of p: ^ σ – Standard deviation of sampling distribution of p: p ^ © 2011 Pearson Education, Inc
Estimating μp^ and σp^ Estimate of μp^: p = Total number of defective units in all k samples Total number of units sampled Estimate of σp^: © 2011 Pearson Education, Inc
Determining the Control Limits Note: If the LCL is negative do not plot it on the control chart. © 2011 Pearson Education, Inc
p–Chart Summary 1. Collect at least 20 samples of size p 0 is an estimate of p 2. Calculate the proportion of defective units in each sample p^ = Number of defective units in the sample Number of units in the sample © 2011 Pearson Education, Inc
p–Chart Summary 3. Plot the centerline and control limits where k is the number of samples of size n and is the overall proportion of defective units in the nk units sampled. is an estimate of the © 2011 proportion Pearson Education, Inc p. unknown process
p–Chart Summary 4. Plot the k sample proportions on the control chart in the order that the samples were produced by the process. © 2011 Pearson Education, Inc
Zone Boundaries Upper A–B Boundary: Lower A–B Boundary: Upper B–C Boundary: Lower B–C Boundary: © 2011 Pearson Education, Inc
Interpreting a p–Chart • Process is considered out of control if any of the pattern analysis rules 1 – 4 are detected: – One point beyond Zone A – Nine points in a row in Zone C or beyond – Six points in a row steadily increasing or decreasing – Fourteen points in a row alternating up and down • Process is considered in control if none of the pattern analysis rules are detected © 2011 Pearson Education, Inc
p–Chart Example A manufacturer of pencils knows about 4% of pencils produced fail to meet specifications. How many pencils should be sampled for monitoring the process proportion? Solution: Samples of size 216 or more should be selected. © 2011 Pearson Education, Inc
p–Chart Example The pencil manufacturer has decided to select samples of size n = 225. The table shows the results for the past 20 samples. Construct a p– chart. © 2011 Pearson Education, Inc
p–Chart Solution Calculate the centerline: © 2011 Pearson Education, Inc
p–Chart Solution Calculate the control limits: *Since LCL is negative, do not plot it on the control chart © 2011 Pearson Education, Inc
p–Chart Solution Determine the A–B zone boundaries Upper A–B Boundary: . 06407 Lower A–B Boundary: . 01281 © 2011 Pearson Education, Inc
p–Chart Solution Determine the B–C zone boundaries Upper B–C Boundary: . 05126 Lower B–C Boundary: . 02562 © 2011 Pearson Education, Inc
p–Chart Solution UCL =. 07689. 06407 B. 05126 A C C . 02562 B. 01281 A The process is in control © 2011 Pearson Education, Inc
12. 7 Diagnosing the Causes of Variation © 2011 Pearson Education, Inc
Statistical Process Control 1. Monitoring process variation 2. Diagnosing causes of variation 3. Eliminating those causes © 2011 Pearson Education, Inc
Methods of Diagnosing Variation 1. 2. 3. 4. Flowcharting Pareto analysis Cause-and-effect diagram Experimental design © 2011 Pearson Education, Inc
Cause-and-Effect Diagram © 2011 Pearson Education, Inc
Example Pizza Deliveries © 2011 Pearson Education, Inc
Example Pizza Deliveries © 2011 Pearson Education, Inc
12. 8 Capability Analysis © 2011 Pearson Education, Inc
Capability Analysis If a process were in statistical control, but the level of variation was unacceptably high, common causes of variation should be identified and eliminated. © 2011 Pearson Education, Inc
Six In-Control Processes The processes produce a high percentage of items that are outside the specification limits. None of these processes is capable of satisfying its customers. a: the process is centered on the target value, but the variation due to common causes is too high; b: the variation is low relative to the width of the specification limits, but the process is off-center; c: both problems exist. © 2011 Pearson Education, Inc
Six In-Control Processes All three processes are capable: the process distribution fits comfortably between the specification limits. However, any significant tightening of the specification limits would result in the production of unacceptable output and necessitate the initiation of process improvement activities to restore the process’ capability. © 2011 Pearson Education, Inc
Capability Analysis Diagram © 2011 Pearson Education, Inc
Capability Index The capability index for a process centered on the desired mean is where is estimated by s, the standard deviation of the sample of measurements used to construct the capability analysis diagram. © 2011 Pearson Education, Inc
Interpretation of Capability Index, Cp Cp summarizes the performance of a stable, centered process relative to the specification limits. It indicates the extent to which the output of the process falls within the specification limits. 1. If Cp = 1 (specification spread = process spread), process is capable. 2. If Cp > 1 (specification spread > process spread), process is capable. 3. If Cp < 1 (specification spread < process © 2011 Pearson Education, Inc spread), process is not capable.
Interpretation of Capability Index, Cp If the process follows a normal distribution, Cp = 1. 00 means about 2. 7 units per 1, 000 will be unacceptable. Cp = 1. 33 means about 63 units per million will be unacceptable. Cp = 1. 67 means about. 6 units per million will be unacceptable. Cp = 2. 00 means about 2 units per billion will be unacceptable. © 2011 Pearson Education, Inc
Key Ideas Total Quality Management (TQM) Involves the management of quality in all phases of a business Statistical Process Control (SPC) The process of monitoring and eliminating variation to keep a process in control © 2011 Pearson Education, Inc
Key Ideas In Control Process Has an output distribution that does not change over time Out-of-Control Process Has an output distribution that changes over time © 2011 Pearson Education, Inc
Key Ideas Dimensions of Quality 1. 2. 3. 4. 5. 6. 7. 8. Performance Features Reliability Conformance Durability Serviceability Aesthetics Reputation © 2011 Pearson Education, Inc
Key Ideas Major Sources of Process Variation 1. 2. 3. 4. 5. 6. People Machines Materials Methods Measurement Environment © 2011 Pearson Education, Inc
Key Ideas Causes of Variation 1. Common causes 2. Special (assignable) causes Types of Control Charts 1. -chart: monitor the process mean 2. R-chart: monitor the process variation 3. p-chart: monitor the proportion of nonconforming items © 2011 Pearson Education, Inc
Key Ideas Specification Limits Define acceptable values for an output variable LSL = lower specification limit USL = upper specification limit (USL – LSL) = specification spread © 2011 Pearson Education, Inc
Key Ideas Capability Analysis Determines if process is capable of satisfying its customers Capability Index (Cp) Summarizes performance of a process relative to the specification limits Cp = (USL – LSL)/6 © 2011 Pearson Education, Inc
Key Ideas Pattern-Analysis Rules Determine whether a process is in or out of control Rational Subgroups Samples designed to make it more likely that process changes will occur between (rather than within) subgroups© 2011 Pearson Education, Inc
Key Ideas Sample Size for p-Chart n > 9(1 – p 0)/ p 0 where p 0 estimates true proportion defective Cause-and-Effect Diagram Facilitates process diagnosis and documents causal factors in a process © 2011 Pearson Education, Inc
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