Chang, Shing I "A Hybrid Neural Fuzzy System for Statistical Process Control"
Computational Intelligence in Manufacturing Handbook
Edited by Jun Wang et al
Boca Raton: CRC Press LLC,2001

©2001 CRC Press LLC

18

A Hybrid Neural Fuzzy
System for Statistical

Process Control

18.1 Statistical Process Control

18.2 Neural Network Control Charts

18.3 A Hybrid Neural Fuzzy Control Chart

18.4 Design, Operations, and Guidelines
for Using the Proposed Hybrid Neural
Fuzzy Control Chart

18.5 Properties of the Proposed Hybrid Neural
Fuzzy Control Chart

18.6 Final RemarksAbstract

A quality characteristic (QC) is a measure of quality on a product or service. Examples of QC are
weight of a juice can, length of a cylinder part, the number of errors made during payroll operations,
etc. A QC can be mathematically defined as a random variable, which is a function that takes values from
a population or distribution. Denote a QC as random variable

x

. If a population

Ω

only contains discrete
members, that is,

Ω

= {x

1

, x

2

, …, x

n

}, then QC


x
∈
Ω

={

x

| L

≤

x

≤

U}. For example,

x

is the weight
of a juice can with a target weight of 8 oz.
The central limit theorem (CLT) implies that the sample mean of a continuous random variable



,
–

x

, is approximately normal distributed, i.e.,

N

(

µ

,

σ

2

/

n

)
where

µ

and

remains in control, 99.73% of the sample population will fall
within the two control limits. On the other hand, if either

µ

or

σ

shifts from its target, this will increase
the probability that sample points plot outside the control limits, which indicates an out-of-control
condition.
A pair of control charts are often used simultaneously to monitor QC

x

— one for the mean

µ

and
the other for the standard deviation

σ

. The goal is to make sure the process characterized by QC

x

is

starts to shift to

µ

1

. One of the
most often used control charts, chart, can be used to detect this situation. On the other hand, at
time

t

2

, the mean is on target but the standard deviation has increased from

σ

0

to

σ

1

where

σ


σ
– 3
n
µ
σ
+3
n
X

©2001 CRC Press LLC

Most control chart improvements over the years have been focused on detecting process mean shifts,
with a few exceptions that are discussed in the following section.
Shewhart R, S, and S

2

charts are the first statistical control charts for monitoring process variance
changes. Johnson and Leone (1962a, 1962b) and Page (1963) later proposed CUSUM charts based on
sample variance and sample range. As an alternative, Crowder and Hamilton (1992) developed an expo-
nential weighted moving average (EWMA) scheme based on the log transformation of the sample variance

ln

(S

2

). Their experimental results show that the EWMA chart outperforms the Shewhart S


Hwarng and Hubele (1991, 1993) trained a backpropagation pattern recognition classifier to detect
six unnatural control chart patterns — trend, cycle, stratification, systematic, mixture, and sudden shift.
Their results were promising in recognizing various special causes in out-of-control situations.
Smith (1994) and Smith and Yazici (1993) described a combined X-bar and R chart backpropagation
model to investigate both mean and variance shifts. They found their networks performed 50% better
in average error rate when compared to Shewhart control charts. However, the majority of the wrong

FIGURE 18.1

In-control and out-of-control scenarios in SPC. (From Montgomery, D.C., 1996,

Introduction to
Statistical Quality Control

, 2nd ed. p. 131. Reproduced with the permission of John Wiley & Sons, Inc.)
Assignable cause three
is present; process is
out-of-control
Assignable cause two
is present; process is
out-of-control
Assignable cause one
is present; process is
out-of-control
Only chance causes of
variation present;
process is in 
control
LSL
Process quality characteristic, x

0
t
1
t
2
t
3

©2001 CRC Press LLC

classification is of type I error. That is, the network signals too many out-of-control false alarms when
the process is actually in control.
Chang and Aw (1994) proposed a four-layer backpropagation network and a fuzzy inferencing system
for detecting process mean shifts. Their network outperforms conventional Shewhart control charts in
terms of both type I and type II errors, while Pugh’s and Smith’s charts have larger type I errors than
that of the 3

σ

chart. Further, Chang and Aw’s scheme has the advantage of identifying the magnitude
of shifts. None of the Shewhart-type charts, or the other neural network charts, offer this feature. Chang
and Ho (1999) further introduced a two-stage neural network approach for detecting and classifying
process variance shifts. The performance of the proposed method is comparable to that of the other
control charts for detecting variance changes as well as being capable of estimating the magnitude of the
variance change, which is not supported by the other control charts. Furthermore, Ho and Chang (1999)
integrated both neural network control chart schemes and compared this with many other approaches
for monitoring process mean and variance shifts. In this chapter, we will summarize the proposed hybrid
neural fuzzy system for monitoring both process mean and variance shifts, provide guidelines and
examples for using this system, and list the properties.


,

x

2

, …, and

x

n

in the first input method are independent of each other. In the proposed system,

n

is chosen
as five, that is, each plotting point consists of a sample of five observations. Traditional Shewhart-type
control charts normally use this input method.
A moving window of five observations is used for the second method to select incoming observations.
For example, the first sample point consists of observations

x

1

,

x


decision making. Unlike the “true” individual observation input method, the moving range method must
wait until the fifth observation to complete the first sample point to start using the proposed chart. After
this point, it is on pace with the “true” individual observation input method in that it uses the most
recent and four immediately passed observations. The reason for maintaining a few observations in a
sample point is due to the need to evaluate process variation. An individual observation does not provide
such information.
Transformation is also a key component in the data input module. As we will discuss later, both neural
networks were trained “off-line” from simulated observations. In order to make the proposed schemes work
for various applications, data transformation is necessary to standardize the raw data into the value range
that both neural network components can work with. Formulas for data transformation are as follows:
X

©2001 CRC Press LLC

18.3.1.1 Transformation for M-NN Input

Equation (18.1)
where

i

is the index of observations in a sample or window;

t

is the index for the sample period, and
and

s


18.3.2 Data Processing Module

The heart and soul of the proposed system is a module composed of two independently developed neural
networks: M-NN and V-NN. M-NN, developed by Chang and Aw (1996), is a 5–8–5–1 four-layer neural
network for detecting process mean shift. On the other hand, Chang and Ho’s (1999) V-NN is a 5–12–12–1
neural network for detecting process variance shift. Data from transformation formulas (Equations 18.1
and 18.2) are fed into M-NN and V-NN, respectively. Both neural networks have single output nodes.
M-NN’s output values range from –1 to +1. A value that falls into a negative range indicates a decrease in
process mean value, while a positive M-NN output value indicates a potential increase in process mean
shift. On the other hand, V-NN’s output ranges from 0 to 1 with larger values meaning larger shifts. Note
that both neural networks were trained off-line using simulations. By incorporating the trained weight
matrices, one can start using the proposed method. The only setup required is to estimate both process
mean and variance for transformation. The central limit theorem guarantees that transformed data is

FIGURE 18.2

A schematic diagram of C-NN (combined neural network) control chart. (Adapted from Ho and
Chang, 1999, Figure 3, p. 1891.)
Sample Observations
Individual Observations
Trans-
formation
M-NN
V-NN
Cutoff
value(s)
Cutoff
value(s)
Mean/
Variance


similar to the simulated data used for training. Thus the proposed method can be applied to many
applications with various data types as long as they can be defined as QC

x

. Before M-NN and V-NN
are introduced in detail, we first summarize calculation and training of any feedforward, multiple-layer
neural networks as follows.

18.3.2.1 Computing in a Neural Network

The most commonly implemented neural network is the multilayer backpropagation network, which
adapts weights according to the steepest gradient descent rule along a nonlinear transformation function.
The reason for this popularity is due to the versatility of its paradigm in solving diverse problems, and
its strong mathematical foundation. An example of a multilayer neural network is shown in Figure 18.3.
In neural networks, information propagates from input nodes (or neurons) through the system’s weight
connections in the middle layers (or hidden layers) of nodes, finally passing out the last layer of nodes
— the output nodes.
Each node, for example node

j

in the hidden and output layers, contains the input links with weights

w

ij

, an activation function (or transfer function)


pi
is the output of node

i

from the previous layer.
Many activation functions, e.g., sigmoidal and hyperbolic-tangent functions, are available. We choose
to use the sigmoidal function
Equation (18.4)
where

c

is a coefficient that adjusts the abruptness of the function.

FIGURE 18.3

An example of a multilayer neural network.
Input Layer
Hidden Layers
Output Layer
VfI
IwV
jj
jijpi
i


, o = 1, 2, . . . ,

n

o

, for the output layer nodes as follows:
Equation (18.5)
where

f' (I)

is the first-order derivative of the activation function f(I); t
o
is the desired target value; and
V
o
is the actual output for output node o. We then update the weights connected between hidden layer
nodes and output layer nodes:
w
ho
(new) = w
ho
(old) +
ηδ
o
V
h
+ α[∆w

between node j in the lower hidden layer
and node h in the current hidden layer can be updated as follows:
w
jh
(new) = w
jh
(old) +
ηδ
h
V
j
+ α[∆w
jh
(old)], Equation (18.8)
FIGURE 18.4 Node j and its input–output values in a multilayer neural network.
V
p1
V
V
Node j at Current
Layer
Nodes in the Next

Layer
W
1j
Transfer
Function
f
W

fI w V V w=
′
()
′
=+
()()
′
=
′
=
′
∑∑
11
051 1.± ,