Kazerooni, A. et al "Application of Fuzzy Set Theory in Flexible Manufacturing System Design"
Computational Intelligence in Manufacturing Handbook
Edited by Jun Wang et al
Boca Raton: CRC Press LLC,2001

©2001 CRC Press LLC

5

Application of Fuzzy Set
Theory in Flexible
Manufacturing

System Design

5.1 Introduction

5.2 A Multi-Criterion Decision-Making Approach
for Evaluation of Scheduling Rules

5.3 Justification of Representing Objectives with
Fuzzy Sets

5.4 Decision Points and Associated Rules

5.5 A Hierarchical Structure for Evaluation
of Scheduling Rules

5.6 A Fuzzy Approach to Operation Selection

5.7 Fuzzy-Based Part Dispatching Rules in FMSs


and fuzzy set theory.

Scheduling criteria or performance measures are used to evaluate the system performance under
applied scheduling rules. Examples of scheduling criteria include

production throughput

,

makespan

,

system
utilization

,

net profit

,

tardiness

,

lateness

,

University of South Australia

L. H. S. Luong

University of South Australia

F. T. S. Chan

University of Hong Kong

©2001 CRC Press LLC

• Minimizing lateness/tardiness
• Minimizing makespan
• Maximizing system/machine utilization
• Minimizing WIP (work in process)
• Maximizing throughput
• Minimizing average flow time
• Minimizing maximum lateness/tardiness
Hutchison and Khumavala [1990] stated that production rate (i.e., the number of parts completed per
period) dominates all other criteria. Chryssolouris et al. [1994] and Yang and Sum [1994] selected total
cost as a better overall measure of satisfying a set of different performance measures.
One of the most important considerations in scheduling FMSs is the right choice of appropriate
criteria. Although the ultimate objective of any enterprise is to maximize the net present value of the
shareholder wealth, this criterion does not easily lend itself to operational decision making in scheduling
[Rachamadugu and Stecke 1994]. An example of conflict in these objectives is minimizing WIP and
average flow time necessitates lower system utilization. Similarly, minimizing average flow time necessi-
tates a high maximum lateness, or minimizing makespan can result in higher mean flow time. Thus,
most of the above listed objectives are mutually incompatible, as it may be impossible to optimize the
system with respect to all of these criteria. These considerations indicate that a scheduling procedure that


mean tardiness

and

number of tardy parts

are important too.
But from a scheduling point of view, all criteria do not possess the same importance. Depending on
the situation of the shop floor, importance of criteria or performance measures varies over the time.
Virtually no published paper has considered performance measures bearing different important weights.
They have evaluated the results by considering the same importance for all performance measures.

5.2 A Multi-Criterion Decision-Making Approach

for Evaluation of Scheduling Rules

Scheduling rules are usually involved with combination of different decision rules applied at different
decision points. Determination of the best scheduling rule based on a single criterion is a simple task,
but decision on an FMS is made with respect to different and usually conflicting criteria or performance
measures. The simple way to consider all criteria at the same time is assigning a weight to each criterion.
It can be defined mathematically as follows [Hang and Yon 1981]: Assume that the decision-maker assigns
a set of important weights to the attributes,

W

= {

w











=…
==
∑∑
11
1

©2001 CRC Press LLC

where

x

ij

is the outcome of the

i

th

alternative (

performance measure
or criterion and

w

j
is the important weight of the

j

th

performance measure. Usually the weights of
performance measures are normalized so the

Σ

w

j
= 1. This method is called simple additive weighting
(SAW) and uses the simulation results of an alternative and regular arithmetical operations of multipli-
cation and addition.
The simulation results can be converted to new values using fuzzy sets and through building mem-

performance measure. Therefore,

x

ij

in Equation 5.1 is replaced with its membership value

mvx

ij

.
Equation (5.2)
Considering the objectives,

A

1

,

A

2

, . . .

, A



D

(

x

) is the degree to which

x

satisfies the objectives, and the solution, of course, is the highest {

D

(

x

)|

x

∈

X

}. For unequal important weights

α

j

th

perfor-
mance measure

w

i

. For this model, the following process is used:
1. Select the smallest membership value of each alternative

X

i

related to all performance measures
and form

D

(

x

).
2. Select the alternative with the highest member in


∑∑
max
11
1
Dx A x A x A x x X
m
()
=
()
∩
()
∩…∩
()
∈
12
,,
Dx A xA x A x x X
m
()
=
{
() ()
…
()
}
∈min
12
,,
Dx A x A x A x x X
aa







=… =…






max max and 11

©2001 CRC Press LLC

5.3 Justification of Representing Objectives with Fuzzy Sets

Unlike ordinary sets, fuzzy sets have gradual transitions from membership to nonmembership, and can
represent both very vague or fuzzy objectives as well as very precise objectives [Yager 1978]. For example,
when considering net profit as a performance measure, earning $200,000 in a month is not simply earning
twice as much as $100,000 for the same period of time. With $100,000 the overhead cost can just be
covered, while with $200,000 the research and development department can benefit as well. Membership
functions can show this kind of vagueness. The membership functions play a very important role in
multi-criterion decision-making problems because they not only transform the value of outcomes to a
nondimensional number, but also contain the relevant information for evaluating the significance of
outcomes. Some examples of showing outcomes with membership values are depicted in Figure 5.1.

5.4 Decision Points and Associated Rules


rule

2

/ . . . /

rule

p

in which rule

k

is a decision rule applied at DP

k

1

≤
k
≤

mvx

ij

, where

i

varies from 1 to

n

and

j

varies from 1 to

m

.

Π

wc

i


Equation (5.8)Dwcwsx
i
ijjij
j
m
=××
















=
∑
max Π
1

©2001 CRC Press LLC


%

©2001 CRC Press LLC

where if the PM
j
is to be maximized
if the PM
j
is to be minimized
MAW method:
i = 1, . . . , n Equation (5.9)
Max–Min method:
i = 1,...,n, j = 1,...,m Equation (5.10)
FIGURE 5.2 Hierarchical structure for evaluation of scheduling rules.
s
s
j
j
=
=





1
1–
D wc w mvx
i







max min / Π
©2001 CRC Press LLC
Max–Max method:
Equation (5.11)
where it is assumed that Σw
j
= 1. The value inside the outermost parenthesis of each of the above equations
shows the overall scores of all scheduling rules with respect to the related method.
5.5.1 Important Weight of Performance Measures and Interval Judgment
The first task of the decision-maker is to find the important weight of each performance measure. Saaty
[1975, 1977, 1980, 1990] developed a procedure for obtaining a ratio scale of importance for a group of
m elements based upon paired comparisons. Assume there are m objectives and it is desired to construct
a scale and rate these objectives according to their importance with respect to the decision, as seen by
the analyst. The decision-maker is asked to compare the objectives in pairs. If objective i is more important
than objective j then the former is compared with the latter and the value a
ij
from Table 5.1 shows how
objective i dominates objective j (if objective j is more important than objective i, then a
ji
is assigned).
The values a
ij
and a
ji

In the evaluation of scheduling rules process, it is necessary for the decision-maker to find the consistency
of his/her decision on assigning intensity of importance to the performance measures. This is done by
first constructing the matrix of the lower limit values [A]
l
and the matrix of the upper limit values [A]
u
,
Equation 5.14 below, then calculating the consistency index, CI, for each of the matrices:
D wc w mvx i n j m
ij
ij ij
=××
()






=… =…max max Π 11,,, ,,
A
lu l u
u
l
lu
u
l
u
l
mm















1
1
1
1
1
1
1
1
1
12 12 1 1
12
12
22
1
1
2

} by dividing its elements by the sum of all elements (ΣY
i
) to construct
the column-wise vector w
i
:
i = 1,..., m Equation (5.17)
TABLE 5.1 Intensity of Importance in the Pair-Wise Comparison Process
Intensity of Importance Definition
Value of a
ij
1 Equal importance of i and j
2 Between equal and weak importance of i over j
3 Weak importance of i over j
4 Between weak and strong importance of i over j
5 Strong importance of i over j
6 Between strong and demonstrated importance of i and j
7 Demonstrated importance of i over j
8 Between demonstrated and absolute importance of i over j
9 Absolute importance of i over j
A
ll
l
l
ll
l
m
m
mm
[]










1
1
1
1
11
1
12 1
12
2
12
L
L
MM M
L
A
uu
u
u
uu
u
m













1
1
1
1
11
1
12 1
12
2
12
L
L
MM M
L
XA
iij
j
m

=














=
∑
1
©2001 CRC Press LLC
2. Find vector {F
i
} by multiplying each matrix of Equation 5.14 by {w
i
}:
i = 1,..., m Equation (5.18)
3. Divide F
i
by w
i
to construct vector {Z

Sometimes max–min method will lead to bad decisions. For example, in a situation where a
combination of scheduling rules leads to a poor value for one performance measure but extremely
satisfactory values for the other ones, the method rejects the combination. The advantage of this
method is that the selected combination of rules does not lead to a poor value for any performance
measure.
TABLE 5.2 The Random Index (RI) for the Order of Comparison Matrix
m 123456789101112
RI 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.58
FAw
iijj
j
m
{}
=×
()
=
∑
1
Z
F
w
i
i
i
{}
=



