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An Example
- In a converting operation, a plastic film is
combined with paper coming off a spooled reel.
- As the two come together, they form a moving
sheet that passes as a web over a series of rollers.
- The operation runs in a continuous feed, and the
thickness of the plastic coating is an important
product characteristic.
- Coating thickness is monitored by a highly
automated piece of equipment that uses 10
heads to take 10 measurements across the web at
half-hour intervals.
- Head # 8:30 9: - 1 2.08 2. - 2 2.26 2. - 3 2.13 2. - 4 1.94 1. - 5 2.30 2. - 6 2.15 2. - 7 2.07 1. - 8 2.02 2. - 9 2.22 2. - 10 2.18 2.
- Ave. 2.14 2. - s 0.111 0. - 17:00 17:30 18: - 1.98 2.08 2. - 2.30 2.12 2. - 2.31 2.11 1. - 2.12 2.22 2. - 2.08 2.00 2. - 2.10 1.95 2. - 2.15 2.15 1. - 2.35 2.14 2. - 2.12 2.28 2. - 2.26 2.31 2. - 2.18 2.14 2. - 0.121 0.113 0.
- It is now appropriate to use some of the methods that will be described in Chapter 10 (such as check sheets, Pareto analysis, or brainstorming) to attempt to reduce the common causes of variation in the never- ending quest to decrease the difference between process performance and customer needs.
Individuals and Moving Range Charts
It is not uncommon to encounter a situation where
only a single variable value can be periodically
observed for control charting.
Perhaps measurements must be taken at relatively
long intervals, or the measurements are
destructive and/or expensive; or perhaps they
represent a single batch where only one
measurement is appropriate, such as the total
yield of a homogenous chemical batch process.
Whatever the case, there are circumstances when
data must be taken as individual units that cannot
conveniently be divided into subgroups.
Single measurements of variables are considered a subgroup of size one. Hence, there is no variability within the subgroups themselves, and an estimate of the process variability must be made in some other way. An estimate of variability is based on the point-to-point variation in the sequence of single values, measured by the moving range (the absolute value of the difference between each data point and the one that immediately preceded it):
i i- 1
R = x − x
An average of the moving ranges is used as
the centerline for the moving range portion
of the chart and as a basis of an estimate of
the overall process variation:
k - 1
R
Centerline (Moving Range) R
- Where k is the number of single measurements.
- As it is impossible to calculate to moving range for the first subgroup because none precede it, there are only k- range measurements; so the sum of the R values is divided by k-1.
8.6 Individuals and Moving Range
Charts
For the individuals portion of the control chart, the centerline is the average of the single measurements. We find control limits by adding and subtracting three times the standard deviation of the single measurements, estimated by Rbar/d 2 : Centerline(x) = xbar = Σx/k (8.54) UCL(x) = xbar + 3(Rbar/d 2 ) (8.55) Using the factor E 2 to represent 3/d 2 , the expression for the upper control limit becomes UCL(x) = xbar + E 2 Rbar (8.56) Where E 2 depends on subgroup size.
For the individuals portion of the control chart, the centerline is the average of the single measurements. We find control limits by adding and subtracting three times the standard deviation of the single measurements, estimated by: 2 d R Where E depends onsubgoup size. UCL(x) x E R d 3 Using the factor E for d R UCL(x) x 3 k x Centerline (x) x 2 2 2 2 2 = + = + = = ∑
- A chemical company produces 2,000-gallon batches of a chemical product, A-744, once every two days.
- The product is a combination of six raw materials, of which three are liquids and three are powdered solids.
- Production takes place in a single tank, agitated as the ingredients are added, and for several hours thereafter.
- Shipments of A-744 to the customer are made in bins as single lots when the batches are finished.
- The chemical company is concerned with the density of the finished product, which it measures in grams per cubic centimeter.
- As batches are constantly stirred during production, the density is assumed to be relatively uniform throughout each batch.
- Therefore, management decides that density will be measured by only one reading per batch. During a 60-day period, 30 batches of A-744 are produced.
The process appears to be in a state of statistical control, since there are no points beyond the control limits and no other signs of any trends or patterns in the data.
Correlation in the Moving Range. The moving ranges tend to be correlated. Large moving range values tend to be followed by other large moving range values tend to be followed by other large moving range values, and small moving range values tend to be followed by other small moving range values. Because of this, users must be cautious in applying rules for a lack of control dealing with patterns in the data.
It is usually best to be conservative when applying the rules concerning patterns in the data other than points beyond the control limits that indicate a lack of control in moving range charts. For example, instead of 8 consecutive values above or below the centerline indicating a lack of control, we might require 10 or 12. Knowledge and experience are the best guides in establishing policy in this case.
When this happens, control limits should be
based on the median of the moving range
values, rather than those based on the
average of the moving Range values. is
calculated using the following procedure:
1. Arrange the subgroup ranges from low to high.
2. If there are an odd number of subgroup ranges, select the
middle subgroup range as the median range.
3. If there are an even number of subgroup ranges, select the
two middle most subgroup ranges and compute their
average. This average is the median range.
e R
M
e R M
If inflated control limits are suspected, control
limits based on the median of the moving ranges
should be calculated and compared to those
based on the average moving range; the narrower
of the two sets should be used. Control limits for
the moving range portion, based on the median,
can be computed using:
