Du, R. et al "Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Set Theory"
Computational Intelligence in Manufacturing Handbook
Edited by Jun Wang et al
Boca Raton: CRC Press LLC,2001

©2001 CRC Press LLC

14

Monitoring and
Diagnosing
Manufacturing
Processes Using Fuzzy

Set Theory

14.1 Introduction

14.2 A Brief Description of Fuzzy Set Theory

14.3 Monitoring and Diagnosing Manufacturing
Processes Using Fuzzy Sets

14.4 Application Examples

14.5 ConclusionsAbstract

Monitoring and diagnosis play an important role in modern manufacturing engineering. They help to

*This work was completed when Dr. Du visited The Chinese University of Hong Kong.

©2001 CRC Press LLC

phrases that have similar or slightly different meanings, such as fault detection, fault prediction, in-
process verification, on-line inspection, identification, and estimation.
Monitoring and diagnosing play a very important role in modern manufacturing. This is because
manufacturing processes are becoming increasingly complicated and machines are much more auto-
mated. Also, the processes and the machines are often correlated; and hence, even small malfunctions or
defects may cause catastrophic consequences. Therefore, a great deal of research has been carried out in
the past 20 years. Many papers and monographs have been published. Instead of giving a partial review
here, the reader is referred to two books. One by Davies [1998] describes various monitoring and diagnosis
technologies and instruments. The reader should also be aware that there are many commercial moni-
toring and diagnosis systems available. In general, monitoring and diagnosis methods can be divided
into two categories: a model-based method and a feature-based method. The former is applicable where
a dynamic model (linear or nonlinear, time-invariant or time-variant) can be established, and is com-
monly used in electrical and aerospace engineering. The book by Gertler [1988] describes the basics of
model-based monitoring. The latter uses the features extracted from sensor signals (such as cutting forces
in machining processes and pressures in pressured vessels) and can be used in various engineering areas.
This chapter will focus on this type of method.
More specifically the objective of this chapter is to introduce the reader to the use of fuzzy set theory
for engineering monitoring and diagnosis. The presented method is applicable to almost all engineering
processes and systems, simple or complicated. There are of course many other methods available, such
as pattern recognition, decision tree, artificial neural network, and expert systems. However, from the
discussions that follow, the readers can see that fuzzy set theory is simple and effective method that is
worth exploring.
This chapter contains five sections. Section 14.2 is a brief review of fuzzy set theory. Section 14.3
describes how to use fuzzy set theory for monitoring and diagnosing manufacturing processes. Section
14.4 presents several application examples. Finally, Section 14.5 contains the conclusions.


), then given any element in

X

, say

x

, there will be either

x
∈
A

or

x
∉

A

′

, its membership
function,

µ

(

A

′

), varies between 0 and 1, that is

µ

(

A

) = [0, 1]. In other words, there are cases in which
the instance of the event

x


′

). Furthermore, the fuzzy set is denoted
as

x

/

µ

A

’

(

x

),

∀
x


µ

(

x

) = 0 means

x

is impossible while

µ

(

x

) = 1 implies

x

is certainly true. In addition, the fuzzy membership function may take various forms
such as a discrete tablet,

x

:


)

µ

(

x

2

)…

µ

(

x

n

) Equation (14.1)
or a continuous step-wise function,

©2001 CRC Press LLC

Equation (14.2)
where

a

⊆
X

, we have
(a) union:

µ

(

A

∪

B

) = max{

µ

(

A

),

µ

(

A

∩

B

) = min{

µ

(

A

),

µ

(

B

)},

∀


a

,

b

,

c

,

d

} and fuzzy events,

f

=

a

/ 1 +

b

/ 0.7 +

cg

,

f

∩
g

and

f

.
Solution: Using Equations 14.2 through 14.4, it is easy to see

f

∪
g


/ 1 +

b / 0.6 + c / 0.3 + d / 0.1
f = a / 0 + b / 0.3 + c / 0.5 + d / 0.9
FIGURE 14.1 Illustration of crisp and fuzzy concept.
µ
x
xa
xa
ba
axb
bxc
dx
dc
cxd
dx
()
=
≤
<≤
<≤
<≤
<








XAMPLE
2: Given a discrete space X = {a, b, c, d} and a fuzzy event f ⊆ X,
f = a / 1 + b / 0.7 + c / 0.5 + d / 0.1,
find the probability mass function of Y = f.
Solution: First, the possibility function of f is:
µ
(a) = 1,
µ
(b) = 0.7,
µ
(c) = 0.5,
µ
(d) = 0.1
This is equivalent to:
µ
({a, b, c, d}) = 1,
µ
({b, c, d}) = 0.7,
µ
({c, d}) = 0.5,
µ
({d}) = 0.1
Assuming P(ƒ) ≤
µ
(ƒ), and
P(a) = p
a
, P(b) = p
b
, P(c) = p

≥ 0, i = a, b, c, d
Solving this set of equations, we have:
0.3 ≤ p
a
≤ 1
0 ≤ p
b
≤ 0.7
0 ≤ p
c
≤ 0.5
0 ≤ p
d
≤ 0.1
Therefore, the probability mass function of f is
m(a): [0.3, 1], m(b): [0, 0.7], m(c): [0, 0.5], m(d): [0, 0.1]
or
m = {a}: 0.3, {a, b}: 0.2, {a, b, c}: 0.4, {a, b, c, d}: 0.1
In general, suppose that A ⊆ X is a discrete fuzzy event, namely
©2001 CRC Press LLC
A = x
1
/
µ
(x
1
) + x
2
/
µ

Π({x
i
, x
i+1
, …, x
n
}) =
µ
(x
i
) Equation (14.7)
If P(A) ≤ Π(A), ∀ A ∈ 2
X
, then we have
for i = 2, …, n Equation (14.8a)
Equation (14.8b)
Solving Equation 14.8 results in
1 –
µ
(x
2
) ≤ P(x
1
) ≤ 1 Equation (14.9a)
0 ≤ P(x
i
) ≤
µ
(x
i

0.8 + b / 0.6 + d / 0.2, X = {a, b, c, d}, then the mass assignment would be
m
f
= a: 0.2, {a, b}: 0.4, {a, b, c}: 0.2; ∅: 0.2
In this case, we need to normalize the mass assignment by using the formula:
µ
(x
i
) =
µ
(x
i
) /
µ
(x
1
), i = 2, 3, .., n Equation (14.11)
and then do the mass assignment. For the above example, the normalization results in f* = a / (0.8/0.8)
+ b / (0.6/0.8) + d / (0.2/0.8) = a / 1 + b / 0.75 + d / 0.25, and the corresponding mass assignment is
m
f*
= a: 0.25, {a, b}: 0.5, {a, b, c}: 0.25
Px x
k
k
n
i
()
≤
()

g
= {L
i
: l
i
} and m
g′
= {M
i
: m
i
} and form a matrix
where Equation (14.12)
Then, the truth mass function m
(g / g′)
is given below:
Equation (14.13)
where, l
i
.m
j
denotes the element multiplication. The following example illustrates how a conditional mass
function is obtained.
E
XAMPLE
3: Let
g = a/1 + b/0.7 + c/0.2
g
′
= a/0.2 + b/1 + c/0.7 + d/0.1

()
∑
1
X
MTLMlm
ijij
=
()
{}
/:., TL M
tML
fML
u
ij
ji
ji
/
()
=
⊆
∩=










()
=
()
=
=






















∑
∑
∑

= {b}: 0.3, {b, c}: 0.5, {a, b, c}: 0.1, {a, b, c, d}: 0.1
The following matrix can be formed:
*
The cardinality of a set is its size. For example, given a set A = [a, b, c], card(A) = 3.
{b}
0.3
{b,c}
0.5
{a,b,c}
0.1
{a,b,c,d}
0.1
{a}
0 0 0.01 0.000750.3
{a,b}
0.15 0.125 0.0333 0.0250.5
{a,b,c}
0.06 0.1 0.02 0.0150.2
M =
{}
=
∩
()
()







11
= 0. For the element in the second row
and second column, since {a, b} ∩ {b, c} = {b}, card(L
2
∩ M
2
) = card({b}) = 1, card(M
2
) = card({b, c})
= 2, m
22
= (1/2)(0.5)(0.5) = 0.125. The other components can be determined in the same way. Based on
the matrix, it is easy to find P(g/g′) = 0 + 0 + 0.01 + … + 0.015 = 0.53980.
We can also determine the fuzzy degree of g given g′. It is a pair: the possibility of g/g′ is defined as
Π(g/g
′
) = max(g ∩ g
′
) Equation (14.16)
and the necessity of g/g′ is defined as
π(g/g
′
) = 1 – Π(g/g
′
) Equation (14.17)
This is analogous to the probability support pair and provides the upper and lower bounds of the
conditional fuzzy set.
E
XAMPLE
5: Following Example 3, find its possibility support pair.

years of study, it is commonly accepted that tool wear can be determined by Taylor’s equation:
VT
n
= C Equation (14.18)
where V is the cutting speed (m/min), T is the tool life (min), n is a constant determined by the tool
material (e.g., n = 0.2 for carbide tools), and C is a constant representing the cutting speed at which the
©2001 CRC Press LLC
tool life is 1 minute (it is dependent on the work material). Figure 14.2 shows a typical example of tool
wear development, and the end of tool life is determined at VB = 0.3 mm for carbide tools (VB is the
average flank wear), or VB
max
= 0.5 mm (VB
max
is the maximum average flank wear). However, it is also
found that the tool may wear out much earlier or later depending on various factors such as the feed,
the tool geometry, the coolant, just to name a few. In other words, there is an uncertainty of occurrence.
Such an uncertainty can be described by the probability mass function shown in Figure 14.3. As shown
in the figure, the states of tool wear can be divided into three categories: initial wear (denoted as A),
normal tool (denoted as B), and accelerated wear (denoted as C). Their occurrences are a function of time.
On the other hand, it is noted that the state of tool wear may be manifested in various shapes depending
on various factors, such as the depth of cut, the coating of the cutter, the coolant, etc. Consequently,
even though the state of tool wear is the same, the monitoring signals may appear differently. In order
words, there is an uncertainty of appearance. Therefore, in tool condition monitoring, the question to
be answered is not only how likely the tool is worn, but also how worn is the tool. To answer this type
of problem, it is best to use the fuzzy set theory.
FIGURE 14.2 Illustration of tool wear.
FIGURE 14.3 Illustration of the tool wear states and corresponding fuzzy sets.
t
VB = 0.3
m

can be represented by a vector x = [x
1
, x
2
, …, x
n
]. Note that although the numeric values are most
common, the attributes may also be integers, sets, or logic values. Owing to the complexity of the process
and the cost, it is not unusual that the attributes do not directly reveal the process conditions. Conse-
quently, decision-making must be carried out. There have been many decision-making methods; the
fuzzy set theory is one of them and has been proved to be effective.
Mathematically, the unified model shown in Figure 14.4, as represented by the bold lines, can be
described by the following relationship:
y • R = x Equation (14.19)
where R is the relationship function, which represents the combined effect of the process, sensing, and
signal processing. Note that R may take different forms such as a dynamic system (described by a set of
differential equations), patterns (described by a cluster center), neural network, and fuzzy logic. Finally,
it should be noted that the operator “•” should not be viewed as simple multiplication. Instead, it
corresponds to the form of the relationship.
The process of monitoring and diagnosing manufacturing processes consists of two phases. The first
phase is learning. Its objective is to find the relationship R based on available information (learning from
samples) and knowledge (learning from instruction). Since the users must provide information and
instruction, the learning is a supervised learning. To facilitate the discussions, the available learning
samples are organized as shown in Table 14.1.
FIGURE 14.4 A unified model for monitoring and diagnosing manufacturing processes.
y
Manufacturing
process
Sensing
Signal