Statistical Process Control

1.20

Statistical Process Control D. H. F. LIU (1994)

Use of statistical techniques to detect variations in product quality and consistency dates back to Walter Shewhart's work at Bell Laboratories in the early 1900s. His work resulted in the development of statistical quality charts (Shewhart charts). These charts are still used for analyzing patterns in product variability. In the 1940s and 1950s, W. Edwards Deming's work in statistical quality control (SQC) methodology evolved into a 14-point management program for quality improvement. His approach emphasized the ap­ plication of statistical principles to control the production process. This field is known as statistical process control (SPC). Problem-solving techniques associated with SPC in­ clude: • Analysis of processes for stability and the effects of process modifications (control charts and capability in­ dices) • Defining problems and setting priorities (Pareto charts) • Identifying causes for good and bad performance (cause and effect diagrams—fishbones) • Quantifying relationships between process or product variables and other variables (scatter plots or other corre­ lation tools)

SAMPLE SIZE (n) • UCL

CENTERING CHART

DISPERSION CHART

FIG. 1.20a

Shewhart chart for a subgroup sampling method. The centering (X) chart helps determine if the centering of the process is stable. The Dispersion (R) chart helps determine if the spread of the process is stable.

TABLE 1.20b

Control Chart Types for a Single Point Sampling Method Chart Family

Chart Type

Description

Centering Chart

Individual Geometric Mean

Uses individual point values to provide a trend. Uses the geometric mean to provide exponential smoothing. Uses the moving average to remove noise and indicate an overall trend.

Moving Average Dispersion Chart

Range (R chart)

Courtesy of Rosemount, Inc.

138

Commonly called an R chart. Plots the range which is the difference between the highest and lowest sample in the range. The number of samples for calculating the range is configured by the user.

1.20 Statistical Process Control CONTROL CHARTS AND CAPABILITY INDICES The most widely used SPC tools are control charts. On these charts the measurements of product samples are plotted to show their centering (XBar chart) and dispersion values (R chart). The centering value is the average of the samples. The dispersion value is the frequency distribution from that average. These charts are used to distinguish random, un­ controllable variations in measured parameters from varia­ tions that are controllable and should be corrected. To con­ trol product quality, the aim is to keep the product variation within the random pattern. CONTROL CHARTS

139

Tables 1.20e, 1.20f, and 1.20g give the equations for single point centering chart limits and single point disper­ sion chart values. Tables 1.20h, 1.20i, 1.20j, and 1.20k give the equations for subgroup centering chart values and chart limits. INTERPRETATION OF CHARTS The XBar chart (Table 1.20d) or Shewhart chart, used in combination with rule-based checks, can detect when a process output has undergone statistical abnormality—that

TABLE 1.20c

Figure 1.20a shows a centering and a dispersion chart. Both charts have similar components: centerline, upper limit, lower limit, and sample size. The centerline, also called the average, is shown with a solid line. The average of all the samples should be at this value. The upper and lower limits are shown by dotted lines. If the average is achieved, the averages of n samples should be less than this upper limit and greater than this lower limit. Variations within the limits are considered to be normal. Variations outside the limits are considered to be abnormal. Tables 1.20b, 1.20c, and 1.20d show various control strategies for both the single-point sampling method and the subgroup sampling method. Also described are points to be stored for various types of centering charts.

Single Point Centering Chart Point Values Centering Chart Type

Point to be Stored

Individual

New value

Geometric Mean

(last point value) (1-W) + (new value) (W), where W is the specifed weight of a new point

Moving Average

Average of the new individual and the last n individuals, where n is the number of samples for range

Courtesy of Rosemount, Inc.

TABLE 1.2M

Control Chart Types for a Subgroup Sampling

Method

Chart Family

Chart Type

Description

Centering Chart

Average (XBar)

Commonly called an XBar chart. Plots the average of data in the specified range. All subgroup samples are saved until the specified size is met. The average is then calculated, and the value written to disk.

Median

Plots the median value in the same manner as the average chart plots the average value.

Range (R chart)

Commonly called an R chart. Plots the range, which is the difference between the highest and lowest sample in the subgroup. Samples are collected until the subgroup is complete, then sorted in order of size. The average or median is determined for the centering chart, and then the range is calculated.

Range with Sigma

Plots the R values in the same manner as in the range chart, but the limits are calculated using a derivation value (sigma) that was entered during configuration.

Root Mean Squared

Used for subgroups of 12 to 25 points. Uses the root mean square (RMS) to calculate the limits and centerline.

Root Mean Squared with Sigma

Plots points in the same manner as the RMS chart, but the limits and the centerline are calculated using a deviation value (sigma) that was entered during configuration.

Standard Deviation

Plots the standard deviation of the subgroup points.

Dispersion Chart

Courtesy of Rosemount, Inc.

140

Control Theory

TABLE 1.20e

Equations for Single Point Centering Chart Limits Centering Chart Type

Control Limit

Individual— "Limit Calculation Type" Range

Individual— "Limit Calculation Type" Standard Deviation

UCL = AVG X + LCL = AVG X -

UCL = AVG X +

LCL = AVG X -

Parameters

3 (AVG R)

ΓΪ28 ΓΪ28

3 yjt (X. -" AVG X)

3 V! ( x '- - AVG X)

2

n = 2 Xj = sample value X = subgroup average

n - 1

UCL = AVG X +

V. _w w

3

(Λλ ° K)

(AVG

1.128

Geometric Mean

Moving Average

2

n —1

3

LCL = AVG X -

R = range value X = subgroup average

3 (AVG R)

ΡΪ

V, - w " W

(AVG R) v '

R = range value W = weight of the point X = subgroup average

1.128

UCL = AVGX + (1.88) (R)

R = range value X = subgroup average

LCL = AVGX - (1.88) (R)

Courtesy of Rosemount, Inc.

TABLE 1.20f

Single Point Dispersion Chart Values Dispersion Chart Type

Range Value Equation

Parameters

Range

R (f°r

° n l y tW° s a m P l e s ) = lR™* - R n l R (for more than two samples) = Rmax " R n

Rn = minimum range value Rmax = maximum range value

Courtesy of Rosemount, Inc.

TABLE 1.20g

Equations for Single Point Dispersion Chart Limits Dispersion Chart Type

Control Limit Equation

Range

UCL = (3.27) (AVG R) LCL = (0) (AVG R) CL = AVG R

Courtesy of Rosemount, Inc.

Parameters

R = average range value

1.20 Statistical Process Control

141

TABLE 1.20h

Equations for Subgroup Centering Chart Values Centering Chart Type

Average, X

Equation

Parameters

x, + x2 + x3 · · · + xn

X n

individual sample values subgroup average number of values in the subgroup

Courtesy of Rose mount, Ine.

TABLE 1.20Ϊ

Equations for Subgroup Centering Chart Limits Centering Chart Type

Average

Control Limit Equation

Parameters

UCL = AVG X + (A2) (AVG R) LCL - AVG X - (A2) (AVG R)

n = number of values in the subgroup R = range of the subgroup A2 = value from Table l .20j

Courtesy of Rosemount, Inc.

TABLE 1.20J

TABLE 1.20k

Constants for Subgroup

Centering Chart Limits

Equation for Subgroup Dispersion Chart Value

n

A2

n

—

—

ll

0.29

2

1.88

12

0.27

3

1.02

13

0.25

4

0.73

14

0.24

5

0.58

15

0.22

6

0.48

16

0.21

7

0.42

17

0.20

8

0.37

18

0.19

9

0.34

19

0.19

10

0.31

20

0.18

&2

Courtesy of Rosemount, Inc.

is, a shift from its normal statistical behavior. On noisy processes a simple time trend does not make this kind of deviation visually obvious, but simple rule checks can. Centering charts generated from single point measure­ ments (Table 1.20b) indicate that a process is not in statisti­ cal control if any one of the following is true:

Dispersion Chart Type

Equation

Range

Parameters

range of the subgroup (values sorted highest to lowest)

Courtesy of Rosemount, Inc.

1. One or more points fall outside the control limits. 2. Seven or more consecutive points fall on the same side of the centerline. 3. Ten of eleven consecutive points fall on the same side of the centerline. 4. Three or more consecutive points fall on the same side of the centerline and all are located closer to the control limit than to the centerline. Centering charts generated from the subgroup sampling method indicate abnormal fluctuations when: 1. Any single subgroup value is more than three standard deviations away from the centerline (or setpoint). 2. Two consecutive subgroup values are more than two standard deviations away from the centerline, on the same side of the centerline.

142

Control Theory

3. Three consecutive subgroup values are more than one standard deviation away from the centerline, on the same side of the centerline. 4. Two out of three consecutive subgroup values are more than two standard deviations away from the centerline, with all three values on the same side of the centerline. 5. Five consecutive subgroup values are on the same side of the centerline.

Cpu ■»

USL - X 3σ

Cpu - .67

Another process violation detection chart, called the moving range chart, differs from an Xbar chart in that each datum is an actual scanned data point rather than a subgroup average of several samples.

Cpu = 1.00

PROCESS CAPABILITY

Cpu - 1.33

Upon achieving a state of statistical control, a process should be evaluated as to whether it is capable of meeting the specification(s) or requirement(s). Usually a minimum of 20 to 25 subgroups is desirable to get a representative picture of the process. In order to characterize the capability of the process, its natural deviation is compared to the width of the specifications. This includes the evaluation of both the upper specification limit (USL) and the lower specification limit (LSL) independently to account for the location of the average process reading relative to the specifications. If there is only one limiting specification (LSL or USL), then the capability only needs to be calculated on that side. The standard deviation (σ) used to evaluate the common level of process variation is calculated by A

R

1.20(1)

σ=—

d2 where Λ indicates that the value is an estimate, d2 is a scaling factor which is based on n samples in the subgroup, and R is the average range. The values of d2 for 5,10,15,20, and 25 samples (n) are respectively 2.326, 3.078, 3.472, 3.735, and 3.931. A common technique for reporting process capability is the use of a ratio called the Cp index. The C p index is the ratio which is obtained by dividing the distance to specifica­ tion by the distance to the common cause variation. The Cp index is calculated as follows: Cp for the upper side: Cpu = USL-X 3&

1.20(2)

where X is the subgroup average. Cp for the lower side: Cpl =

X-LSL :— 3σ

1.20(3)

The minimum of these two values will be the worst of the two cases and is reported as the Cp or Cpk index. A Cp value over 1.33 would indicate a capable (accept­ able, saying that 99.73% of individual readings are within specification) process, while a Cp index of less than 1.0 would indicate a "noncapable" process. A Cp index be-

Cpu = 2.00

SPECIFICATION (USL)

FIG. 1.201 Cpu values for four different process states: Cpu less than 1.0, a noncapable process; Cpu between 1.0 and 1.33, a marginal process; Cpu equal to 1.33 or greater, a capable process. tween 1.0 and 1.33 would indicate a marginal process. Cpu values for four different process states are shown in Figure 1.201. For the latter cases, the next step is to improve the process. There are three general ways to increase the values oftheC p index: 1. Move the specifications or setpoints (if they have not been properly established). 2. Move the process average, if the specification has only one limit (unilateral), or center the process average if the process has two limits (bilateral). 3. Reduce the common cause variation. IMPLEMENTING SPC CONCEPTS

The implementation of real-time SPC places increased de­ mands on instrumentation, communication networks, and computer technology. Developments in microchip technol­ ogy offer improved accuracy and stability for primary sen­ sors. The use of digital transmitters eliminates the error contribution of analog-to-digital conversion. Microproces­ sor-based "smart'' transmitters contribute to better accuracy and higher reliability by their improved rangeability. Auto­ matic pressure and temperature compensation, remote cali­ bration, and self-diagnostics also contribute to better data quality. The range of process variables that can be measured has

1.20 Statistical Process Control also increased. Advances in on-line analytical instru­ mentation have made it possible to measure a variety of physical properties and chemical compositions which could not be directly detected in the past. (For more information, refer to the Process Measurement volume of this handbook.) Quality data input (measurement) is useful for on-line SPC only if the control system network can transfer the process information rapidly enough for real-time data ma­ nipulation. The use of MAP (Manufacturing Automation Protocol) for integration of different vendors' products into a common communications network facilitates streamlined data transmission (see Chapter 7). The MAP OSI (Open System Interconnection) model defines communication net­ work functions in seven layers, allowing software programs to be utilized over different networks. The ability to interface a personal computer with the control system also increases the feasibility of adding real­ time SPC to existing plants. Intelligence at the I/O interface level enables routine operating functions to be handled lo­ cally, while conserving higher-level processing capability for SPC-type applications. DATA STORAGE, COMPUTATION, AND DISPLAY

The ability to store statistical records of individual process data points is essential for generating and analyzing SPC control charts. Historical information is particularly useful for statistical analysis of cause-and-effect relationships us­ ing fishbone diagrams. Figure 1.20m shows the Pareto chart that enables diagnosis of the most likely causes of off-spec production runs at component level by searching the histori­ cal database. The desirability of performing identical mathematical studies as well as providing real-time data points warrants a dedicated SPC computer for many chemical process indus­ try applications. Figure 1.20n is a scatter plot showing the use of a linear regression algorithm to explore the relation­ ship between a selected product property (particle size) and an input variable (mixing time). Knowing this relationship, the operator can adjust the problem variable (mixing time) to restore the product (particle size) to meet specifications. Statistical calculations translate into practical control re­ sults through the interpretation of statistical control charts. Configurable CRT displays are prerequisites to maximizing the use of on-line SPC charting capabilities. Automatic SPC alarm notifications and summary displays can isolate the problem input variables. A computerized technique called on-line diagnostics is used to determine and anticipate the causes of plant operat­ ing problems before they actually occur. This diagnostic technique is basically a mix of SPC and fault-diagnostic principles which are programmed into an expert system that

CAUSE*

PERCENT

QTY

CAUSO

23.1

25

CARBO

21.3

Σ3

FILAID

14.8

16

PREIMO

13.9

15

COACO

9.3

10

ALUMO

8.3

9

POCHO

5.6

6

PRCHLO

3.7

4

143

ί ■ ■ V

WM

■nhi n^.fo.J

•THE CAUSES ARE SPECIFIC TO THE PROCESS AND THE EQUIPMENT USED

FIG. 1.20m

A historical data search, presented in the form of a Pareto chart, can help identify the most likely causes of off-spec production runs. UNCONTROLLED PROCESS VARIABLE (PARTICLE SIZE)

5.00 A

4.50 A

4.00 A 3.50 A CONTROLLED VARIABLE (MIXING TIME)

INTERCEPT 5.086

~i—I—i—i—i—i—i—i—i—i—i—i—i—»·

20 22 24 26 28 30 32 34 36 38 40 42 44 46

FIG. 1.20η

Scatter plots can be used to investigate relationships between process variables and can indicate how much a controlled variable (mixing time) should be adjusted to bring an uncontrolled variable (particle size) into line.

operated on-line. Thereby, process operators can anticipate and react to potential problems before they occur.

Bibliography Deming, W.E., Out of Crisis, Cambridge, Massachusetts: MIT Center of Advanced Engineering Study, 1986. Doherty, J., "Detecting Problems with SPC," Control, November 1990, pp. 70-73. Grant, E.L., and Leavenworth, R.S., Statistical Quality Control, 6th edition, New York: McGraw-Hill Book Co., 1988. Holmes, D.S., "Time Series Analysis Overcomes SPC Problems," Con­ trol, February 1991, pp. 36-38. Scherkenback, W.W., The Deming Route to Quality and Productivity, Roadmaps and Roadblocks, Rockville, Maryland: Mercury Press, 1986. Shewhart, W.A., Economic Control of Quality Manufactured Product, New York: Van Nostrand Company, 1980. Wheeler, DJ., and Chambers, D.S., Understanding Statistical Process Control, Knoxville, Tennessee: Keith Press, 1986. Wolske, B.K., "Implementing SPC Concepts on a Real-Time Basis," Chemical Processing, March 1988, pp. 52-56.