Business Finance - Operations Management OPMT 620: Operations Management - Case Study McDonalds Assignment
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Stevenson7ce_PPT_Ch101.pptxStatistical Quality Control
Chapter
Sam Lampropoulos George Brown College
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Trek Bicycle
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Trek Bicycle has a well earned reputation for building some of the most technologically advanced bikes in the world. The company has gotten to where it is by relentless innovation and continuous improvement of its production processes
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List and explain various elements of the statistical process control planning process.
Explain how control charts are designed and the concepts that underlie their use, and solve typical problems.
Assess and solve problems involving process capability.
Describe Six Sigma quality and design of experiments.
Learning Objectives
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Introduction to Statistical Process Control
Control Charts
Process Capability
Six Sigma Quality and Design of Experiments
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Chapter Outline
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Introduction
Statistical Quality Control uses statistical techniques and sampling to monitor and test the quality of goods and services.
Acceptance sampling relies on inspection, determines to accept or reject a product
Statistical process control determines if process is operating within acceptable limits during production
Inspection is the appraisal of goods/services against standards.
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The best companies design quality into the process, thereby reducing the need for inspection/tests.
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p385
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Phases of Quality Control
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Phases of statistical quality control/ improvement in a company.
Figure 10-1
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Figure 10-1 Page #345
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Statistical Process Control Planning Process
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1. Define important quality characteristics, and how to measure
2. For each characteristic,
a. Determine a quality control point
b. Plan
c. Plan the corrective action process.
i. How to inspect
ii. How much to inspect
iii. Where centralized or on-site
Inspection
How Much/How Often
Where/When
Centralized vs. On-site
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Location of use of an acceptance sampling and statistical process control within production
Figure 10-2
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Figure 10-2 Page #345
In manufacturing, some of the typical quality control points are:
1. At the beginning of process. There is little sense in paying for goods that do not meet quality standards and in spending time and effort on material that is bad to begin with.
2. At the end of process. Customer satisfaction and company’s image are at stake here, and repairing or replacing products in the field is usually much more costly than doing it at the factory.
3. At the operation where a characteristic of interest to customers is first determined. In particular, before a costly, irreversible, or covering (e.g., painting) operation.
The HACCP system described in the previous chapter also provides some guidelines for determining the quality control points.
In the service sector, inspection points include where personnel and customer interfaces (e.g., service counter) and the facility.
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Where to Inspect in the Process: Quality Control Point
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1. At the beginning of the process:
Raw materials and purchased parts
3. At the operation where a characteristic of interest to customers is first determined:
Before a costly operation
Before an irreversible process Before a covering process
2. At the end of the process:
Finished products
Examples of Inspection Points
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Similar to Table 10-1
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Similar to Table 10-1 Page # 347
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Inspection Costs
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The amount of inspection is optimal when the sum of the costs of inspection and passing defectives is minimized.
Figure 10-3
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Figure 10-3 Page # 348
“As illustrated in Figure 10–3, if inspection activities increase, inspection costs increase, but the costs of undetected defects decrease. The goal is to minimize the sum of these two costs. In other words, it may not pay to attempt to catch every defect, particularly if the cost of inspection exceeds the penalties associated with letting some defects get through.”
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Centralized vs. On-Site Inspection
Immovable product (e.g. ship)
Simple or handheld measuring equipment
Automated inspection
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On-Site
specialized equipment,
Skilled quality control inspectors,
More favourable test environment
Are the advantages of specialized lab tests worth the time and interruption?
In Lab
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P388-9
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(SPC) Statistical Process Control
Statistical Process Control: Statistical evaluation of the product during production
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Types of Variations
Sampling and Sampling Distributions
Control Charts & Their Design
Sample Mean and Range Control Charts
Individual Unit, Moving Range Control Charts
Control Charts for Attributes.
Run Tests and Using Control Charts
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Statistical Process Control (SPC) Steps
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Take periodic samples from process
If outside limits, stop process and take corrective action
Compare to predetermined limits
If inside limits, continue process
Types of Variations
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Main task of SPC is to distinguish assignable from random variation
Random variation: Natural variations in the output of process, created by countless minor factors.
Assignable variation: A variation whose source can be identified.
Figure 10-4
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Figure 10-4 Page #351
The variability of a sample statistic is described by its sampling distribution, which is the theoretical distribution of the values of the statistic for all possible samples of a given size from the process.
The sampling distribution of sample mean exhibits less variability than the process distribution because of the averaging that occurs in computing the sample means. The mean of the sampling distribution is exactly equal to the mean of the process. Most process distributions and sampling distributions are approximately Normal. Furthermore, the central limit theorem implies that sampling distributions will be approximately Normal, even if the population (i.e., the process) is not.
If the process has only random variability, then the sample mean should most likely fall between 2 (with 95.5 percent probability for Normal distribution) or 3 (with 99.7 percent probability for Normal distribution) standard deviations of the process mean (see Figure 10–5). If it doesn’t, then we can conclude that the process mean most likely has changed, and hence there is most likely an assignable cause.
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Normal Distribution
Only a small percentage of sample means fall more than 2 or 3 standard deviations from the process mean.
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Figure 10-6
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Figure 10-6 Page # 352
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Control Chart
Purpose: to monitor process output to distinguish between random and assignable variation
A time ordered plot of sample statistics (e.g. means) obtained from an ongoing process
Upper and lower control limits define the range of acceptable variation
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Control Chart
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Figure 10-7
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Figure 10-7 Page #353
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Control Limits
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Sampling distribution
Process distribution
Process Mean
Control limits are set at 2 or 3 standard deviations of the process mean.
Lower control limit
Upper control limit
The dividing lines between random and assignable deviations from the process mean.
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Figure 10-8 p392
Control limits are calculated for 2 (with 95.5 percent probability for Normal distribution) or 3 (with 99.7 percent probability for Normal distribution) standard deviations of the process mean.
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Type I and Type II Error
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Type II error: concluding a process is in control when it is actually not (assignable variation is present).
Type I error: concluding that a process has changed (assignable variation) when it has not.
Figure 10-9
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Figure 10-9 Page #354
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Type I and Type II Errors
| In control | Out of control | |
| In control | No Error | Type I error (producers risk) |
| Out of control | Type II Error (consumers risk) | No error |
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Table 10-2
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Table 10-2 Page #354
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Observations from Sample Distribution
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Each sample mean (a red dot) is compared to the control limits.
Figure 10-10
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Figure 10-10 Page # 355
each sample mean is compared to the extremes of the sampling distribution (i.e., the control limits) to judge if it is within the acceptable (random) range.
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Designing Control Charts
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If the process is out of control it is due to assignable variation. Assignable variation can be identified and removed from the process.
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1. Determine a sample size
2. Obtain 20 to 25 samples
3. Establish and graph preliminary control limits
4. Plot sample statistic values on control chart
5. Are any points outside control limits (CL)?
a. NO Assume no assignable cause
Process is in control
b. YES Investigate and correct
Process is out of control
Control Charts for Variables
Sample Mean control charts
Used to monitor the mean (centre) of a process.
X-bar charts
Sample Range control charts
Used to monitor the process dispersion (variation)
R charts
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Variables generate data that are measured.
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Upper and Lower Control Limits for Sample Mean Chart
where
x = Standard deviation of sampling distribution of sample means =
= Process standard deviation
n = Sample size
z = Standard Normal deviate (usually z = 3)
= Average of sample means = grand mean
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Formula 10-1 Page #355
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Upper and Lower Control Limits for Sample Mean Chart
where
A2 can be obtained from Table 10–3
= Average of sample ranges
Sample range = maximum value – minimum value in the sample
= Average of sample means = grand mean
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Alternate Method
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Example: Control Chart
Twenty samples of n = 8 have been taken of the weight of a part. The average of sample ranges for the 20 samples is .016kg, and the average of sample means is 3kg. Determine three sigma control limits for sample mean of this process.
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Solution
Example 10-3
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Example 10-3 Page # 357
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Upper and Lower Control Limits for Sample Range Control Chart
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Sample range (R) control chart: the control chart for sample range, used to monitor process dispersion or spread.
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Example: Control Chart
Twenty-five samples of n=10 observations have been taken from a milling process. The average of sample ranges is .01 centimetre. Determine upper and lower control limits for sample range.
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Solution
UCLR = 1. 78(.01) = 0.0178cm
LCLR = 0.22(.01) = 0.0022cm
Example 10-4
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Example 10-4 Page # 358
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Example: Sample Mean and Range Charts
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Data from 15 samples each with 5 observations.
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Chart helps explain how sample mean and range are determined. Not in text.
Calculate sample means, sample ranges, grand mean, and average of sample ranges.
Example: Sample Mean and Range Charts
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Chart helps explain how sample mean and range are determined. Not in text. Sample mean =10.73, sample range = 0.22
From Table 10-3
Choose factor for sample size
Determine Control Limits
Example: Sample Mean and Range Charts
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Create x-bar Chart and Plot Values
Example: Sample Mean and Range Charts
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UCL
LCL
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The sample means, as plotted on the x-bar chart, shows a process in control. Random variation.
UCL
LCL
Create R-chart and Plot Values
Example: Sample Mean and Range Charts
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The range chart sample plots show sample #8 above the upper control limit. The process is out of control based on amount of variation. This is due to assignable cause.
Sample Mean and Range Charts
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Figure 10-11A.
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Figure 10-11A. Page # 359
x-bar and R-charts used together.
E.g. Thanksgiving turkey in the oven. What can go wrong with the temp. of oven if set at 350 F? The avg. temp during cooking could be 250 F instead. Or, could avg 350 F, but actually fluctuate during cooking time between 200 & 500. Either way, turkey won’t be properly cooked. x-bar would detect inaccurate avg. temp. R-chart detect changes in temp.
Use R-chart first. If out of control, then process variation is out of control. Next investigate cause. No need to interpret x-bar chart if R-chart out of control. If R-chart in control, then interpret x-bar chart. If out of control, then process average is out of control.
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Sample Mean and Range Charts
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Figure 10-11B
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Figure 10-11B Page # 359
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Individual Unit Control Charts
where
= Process standard deviation
z = Standard Normal deviate (usually z = 3)
= Average of individual observations (estimate of process mean)
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Individual unit (X) control chart
Used to monitor single observations ( n = 1)
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Moving Range Control Charts
where
= Average of moving ranges (absolute value of the difference between two consecutive observations)
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Moving range (MR) control chart
MR is difference between consecutive observations
used to monitor dispersion or spread when n = 1
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Example 10-5 Page # 359
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Control Charts for Attributes
p-Chart - Control chart used to monitor the proportion of defectives in a process.
c-Chart - Control chart used to monitor the number of defects per unit.
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Control charts for attributes are used when the process characteristic is counted rather than measured.
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Control Charts for Attributes
p-Chart – for sample proportion of defectives in a process
c-Chart – for the number of defects per unit
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Note: Because formula for control limits is an approximation, sometimes LCLp will be negative. In this case, zero should be used as the lower control limit.
If the value of c is unknown, as is generally the case, the sample estimate, , is used in place of , where = number of defects number of samples.
When the lower control limit is negative, it is set to zero.
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Use of p-Charts
When observations can be placed into two categories.
Good or bad
Pass or fail
Operate or don’t operate
When the data consists of multiple samples of several observations each.
Sample proportion of defectives.
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Use of c-Charts
Use only when number of occurrences per unit of measure can be counted; non-occurrences cannot be counted.
Scratches, chips, dents, or errors per item
Cracks or faults per unit of distance
Breaks or tears per unit of area
Bacteria or pollutants per unit of volume
Calls, complaints, failures per unit of time
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At what points in the process to use control charts?
What size samples to take?
What type of control chart to use (i.e., variables or attribute)?
How often samples should be taken?
Managerial Considerations Concerning Control Charts
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Nonrandom Patterns in Control Charts: Run Tests
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Fig 10-12
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Fig 10-12 Page #364
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Ideally both control charts and run tests should be used to analyze process output, along with a plot of the data.
The procedure involves the following steps:
Compute control limits for the process output
Conduct median and up/down run tests
Note: If there is no indication that the process output is nonrandom. Plot the sample data and check for patterns (e.g., cycling). If you see a pattern, the output is probably not random. Otherwise, conclude the output is random and that the process is in control
Using Control Charts and Run Tests Together
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ideally both control charts and run tests should be used to analyze process output, along with a plot of the data. The procedure involves the following three steps:
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Process Capability
Design specifications
Range of acceptable values established by engineering design or customer requirements
Process variability
Natural variability in a process
Process capability
Ability of a process to meet the design specification
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Three terms relate to the process capability:
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Process Capability Indices
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Process capability ratio=
specification width
process width
If the process is centered use Cp
If the process is not centered use Cpk
Upper specification – Process mean
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Smaller of:
Process mean – Lower specification
3
and
Upper specification – lower specification
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Cp =
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Process centered means the process mean is in the centre of the limits. Sometimes the limits are smaller, or even zero, on one side of the process mean.
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Capability Example
| Machine | Standard Deviation | Machine Variability 6 | Machine Capability Cp = spec/6 |
| A | 0.13 | 0.78 | 0.80/0.78 = 1.03 |
| B | 0.08 | 0.48 | 0.80/0.48 = 1.67 |
| C | 0.16 | 0.96 | 0.80/0.96 = 0.83 |
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Cp > 1.33 is desirable
Cp = 1.00 process is barely capable
Cp < 1.00 process is not capable
The design specification for the width of a part is between 101 mm and 101.8 mm (= .8 mm). Which of these machines are capable?
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Similar to Example 10-9, different values
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Process Capability Analysis
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Figure 10-16
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Similar to Figure 10-16 Page #368
Note: For this looking at PROCESS distribution, individual items, not samples
If less than 1 then not capable
For six sigma, must be 2.
If not capable might:
1) redesign process
2) use alternate process
3) use more inspection
4) see if can relax specs
If > 1 then better than needed. Is it costing more for that, or can we charge more for better quality?
Capability Analysis
If incapable:
Redesign process or reduce variability
Use alternative process
Use 100-percent inspection
Examine/relax design specification
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Capable = process output falls within specifications.
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Process Capability Example: Cookie Packages
A company creates small packages of cookies in a 16 gram package. Government standards state that weights must be within ± 5 percent of the weight advertised on the package.
The design specifications are:
Upper design specification = 16 + .05(16) = 16.8 grams
Lower design specification = 16 – .05(16) = 15.2 grams
Inspectors test 1,000 packages of cookies and find an average weight of 15.875 grams with a standard deviation of .529 grams.
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Is the process capable?
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New example
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Process Capability Example: Cookie Packages
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Specification Limits
Upper Spec = 16.8 g
Lower Spec = 15.2 g
Observed Weight
Mean = 15.875 g
Std Dev = .529 g
What is the Cp index for
this process?
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Not capable.
Use Cpk b/c the process is not centered on 16 (spec = 16, but mean = 15.88). The Cp = 0.504 also excessive variation in the process.
What does a Cpk of .4253 mean?
An index that shows how well the units being produced fit within the specification limits.
Process considered capable if Cpk 1.
This process will produce a relatively high number of defects.
Many companies look for a Cpk of 1.3 or better… Six-Sigma companies want 2.0!
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Bigger is Better!
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Six Sigma Quality
Goal: achieving process variability so small that the half-width of design specification equals six standard deviations of the process.
Cpk = 2.00 = only 3.4 units per million outside design specification
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Figure 10-17
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Figure 10-17 Page # 370
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Design of Experiments
Taguchi suggested a more concise set of experiments by changing levels of factors to measure their influence on output and identifying best levels for each factor.
Identify controllable factors that could influence variation
Set each factor to 2 or more levels
Measure the variation in the process
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| Experiment | ||||
| Factor | (I) | (II) | (III) | (IV) |
| a | 1 | 1 | 2 | 2 |
| b | 1 | 2 | 1 | 2 |
| c | 1 | 2 | 2 | 1 |
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Page #372
design of experiments Performing experiments by changing levels of factors to measure their influence on output and identifying best levels for each factor.
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A methodology that is used to show how well parts being produced fit into a range specified by design limits is ….?
Capability analysis
Six Sigma
Range Chart
Mean Chart
None of the above
Concept Check
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Answer: a. Capability analysis
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You want to prepare a p-chart and you observe 200 samples with 10 in each, and find 5 defective units. What is the resulting “proportion defective”?
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2.5
0.0025
0.00025
Can not be computed on data above
Concept Check
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Answer: c. 0.0025 (5/(200x10)=0.0025)
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You want to prepare an x-bar chart. If the number of observations in a “subgroup” is 10, what is the appropriate “factor” used in the computation of the UCL and LCL?
1.88
0.31
0.22
1.78
None of the above
Concept Check
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Answer: b. 0.31 (from Table 10-3)
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You want to prepare an R chart. If the number of observations in a sample is 5, what is the appropriate “factor” used in the computation of the LCL?
0
0.88
1.88
2.11
None of the above
Concept Check
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Answer: a. 0 (from Table 10-3)
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Statistical process control and control charts focus on detecting departures from stability in a process.
Variation types are random and assignable.
Sample mean control charts are used to monitor the process mean.
Sample range control charts are used to monitor process dispersion or spread.
Individual unit X control charts are used for single observations (n = 1).
Moving range control charts monitor the dispersion or spread of the differences between consecutive observations.
p-charts are used to monitor the proportion of defective items.
c-charts are used to monitor the number of defects per unit product.
If a sample statistic falls outside control limits or its series has a pattern, then the process is out of control.
Run test is used to determine if a nonrandom pattern exists.
Summary
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Briefly explain the statistical process control (SPC) process.
Explain how control charts are designed and the concepts that underlie their use.
Select and create an appropriate SPC charts.
Explain the use of capability analysis.
Analyze the capability of a process.
Describe Six Sigma quality.
Explain how design of experiments can be used to improve processes.
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Learning Checklist
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oleObject1.bin
Type of
business
Inspection
points
Characteristics
Fast Food
Cashier
Counter area
Eating area
Building
Kitchen
Accuracy
Appearance, productivity
Cleanliness
Appearance
Health regulations
Hotel/motel
Parking lot
Accounting
Building
Main desk
Safe, well lighted
Accuracy, timeliness
Appearance, safety
Waiting times
Supermarket
Cashiers
Deliveries
Accuracy, courtesy
Quality, quantity
