Nonparametric control chart for the range

• US 6,718,221 B1
• Filed: 05/21/2003
• Issued: 04/06/2004
• Est. Priority Date: 05/21/2002
• Status: Expired due to Fees

• Current Assignee: University Of Kentucky Research Foundation (University of Kentucky)
• Original Assignee: University Of Kentucky Research Foundation (University of Kentucky)
• Inventors: Stromberg, Arnold J.
• Primary Examiner(s): Picard, Leo
• Assistant Examiner(s): Masinick, Michael D.
• Application Number:
  US10/442,330
• Time in Patent Office: 321 Days
• Field of Search: 700/95, 700/108, 700/109, 700/110
• US Class Current: 700/109
• CPC Class Codes: G06F 18/214 Generating training patterns; Bootstrap methods, e.g. bagging or boosting

Abstract

The method comprises establishing the number of subsets of a dataset that have a range of the difference between any two datapoints within the dataset, and computing a control chart for the range based thereon. In another aspect, a software program for accomplishing the method of the present invention is provided. The method of the invention allows monitoring variability of a product being produced by a particular piece of machinery, of a process conducted by the machinery, or of a product stream generated thereby, accurately detecting changes in variability in real time. The true distribution of the data is reflected, and the desired result is achieved without requiring an inordinate number of computations.

Claims

1. A method for establishing a control limit for variability of a manufacturing process parameter, comprising:

measuring the parameter over a predetermined period of time;

assembling a dataset having a plurality of datapoints, wherein each datapoint represents an individual measured value of the parameter;

rank-ordering the datapoints within the dataset;

selecting at least one subset of the dataset, wherein the subset is a predetermined number of datapoints including a high measured value and a low measured value defining a range of measured parameter values;

calculating all possible numbers of a subsample of the dataset having the range defined by the subset;

repeating the previous steps of selecting a subset and calculating all possible numbers of subsamples of the dataset having the range defined by the subset, until all possible numbers of ranges have been calculated for all possible subsets of the predetermined size within the dataset to define a set of ranges;

rank-ordering the defined set of ranges; and

establishing the control limit for the manufacturing process, defined by an upper limit and a lower limit wherein the upper and lower limits are a predetermined percentile of the rank-ordered set of ranges.

2. The method of claim 1, wherein the control limit is established using a jackknife method.

3. The method of claim 1, wherein the control limit is established using a bootstrap method.

4. The method of claim 2, wherein the possible numbers of a subsample of the dataset having the range defined by the subset are calculated in accordance with the formula:

$\left(\begin{array}{c}h-g-1\\ n-2\end{array}\right)<br><br>$

where h is the highest measured value within a range, g is the lowest measured value within a range, and n is the number of observations within the subset.

5. The method of claim 3, wherein the relationship between the highest measured value and the lowest measured value in a range is defined by the equation g=h where h is the highest measured value within a range and g is the lowest measured value within a range.

6. The method of claim 3, wherein the relationship between the highest measured value and the lowest measured value in a range is defined by the equation g=h−

1 where h is the highest measured value within a range and g is the lowest measured value within a range, and the control limits are defined by the range of 2ⁿ−

2 subsets, where n is the number of observations within the subset.

7. The method of claim 3, wherein the possible numbers of a subsample of the dataset having the range defined by the subset are calculated in accordance with the formula:

$\underset{c=1}{\overset{n-1}{\sum <br><br>}}<br><br>\underset{d=1}{\overset{n-c}{\sum <br><br>}}<br><br>\left(\begin{array}{c}n\\ c\end{array}\right)<br><br>\left(\begin{array}{c}n-c\\ c\end{array}\right)<br><br>{\left(h-g-c-d+1\right)}^{n-c-d}$

where n is the number of observations within the subset, c is the number of times the lowest measured value appears in the subset, d is the number of times the highest measured value appears in the subset, h is the highest measured value within a range and g is the lowest measured value within a range.

8. A computer software program for establishing a control limit for variability of a manufacturing process parameter, wherein the software performs the steps of:

rank-ordering a plurality of datapoints within a dataset, wherein each datapoint represents an individual measured value of the parameter;

selecting at least one subset of the dataset, wherein the subset is a predetermined number of datapoints including a high measured value and a low measured value defining a rank-ordered range of measured parameter values;

calculating all possible numbers of a subsample of the dataset having the range defined by the subset;

repeating the previous steps of selecting a subset and calculating all possible numbers of subsamples of the dataset having the range defined by the subset, until all possible numbers of ranges have been calculated for all possible subsets of the predetermined size within the dataset to define a set of ranges;

rank-ordering the defined set of ranges;

establishing the control limit for the manufacturing process, defined by an upper limit and a lower limit wherein the upper and lower limits are defined as a predetermined percentile of the rank-ordered set of ranges; and

comparing a subsequent datapoint against said control limit to ensure that the parameter is within a predetermined variability range.

9. The software program of claim 8, wherein the control limit is established by a jackknife method.

10. The software program of claim 8, wherein the control limit is established by a bootstrap method.

11. The software program of claim 9, wherein the possible numbers of a subsample of the dataset having the range defined by the subset are calculated in accordance with the formula:

$\left(\begin{array}{c}h-g-1\\ n-2\end{array}\right)<br><br>$

where h is the highest measured value within a range, g is the lowest measured value within a range, and n is the number of observations within the subset.

12. The software program of claim 10, wherein the relationship between the highest measured value and the lowest measured value in a range is defined by the equation g=h where h is the highest measured value within a range and g is the lowest measured value within a range.

13. The software program of claim 10, wherein the relationship between the highest measured value and the lowest measured value in a range is defined by the equation g=h−

1 where h is the highest measured value within a range and g is the lowest measured value within a range, and the control limits are defined by the range of 2ⁿ−

2 subsets, where n is the number of observations within the subset.

14. The software program of claim 10, wherein the possible numbers of a subsample of the dataset having the range defined by the subset are calculated in accordance with the formula:

$\underset{c=1}{\overset{n-1}{\sum <br><br>}}<br><br>\underset{d=1}{\overset{n-c}{\sum <br><br>}}<br><br>\left(\begin{array}{c}n\\ c\end{array}\right)<br><br>\left(\begin{array}{c}n-c\\ c\end{array}\right)<br><br>{\left(h-g-c-d+1\right)}^{n-c-d}$

where n is the number of observations within the subset, c is the number of times the lowest measured value appears in the subset, d is the number of times the highest measured value appears in the subset, h is the highest measured value within a range and g is the lowest measured value within a range.

15. A method for establishing a control limit for variability of a manufacturing process parameter, comprising:

measuring the parameter over a predetermined period of time;

assembling a dataset having a plurality of datapoints, wherein each datapoint represents an individual measured value of the parameter;

rank-ordering the datapoints within the dataset;

defining a predetermined subset size;

for each possible value for the range of the predetermined subset size, calculating the number of subsamples within the dataset which provide the range defined by the subset to define a set of ranges;

rank-ordering the set of all ranges; and

establishing control limits defined by an upper control limit and a lower control limit for the manufacturing process, wherein the upper and lower control limits are a predetermined percentile of the rank-ordered set of all ranges.