265
Notes
*
CHAPTER 1 INTRODUCTION
Basic concepts oF multiple oBjective programming (mop) proBlem
Most multi-objective programming (MOP) problems can mathematically be repre-
sented as:
xx x
Ax b
x
ff f
st
0
max(), (),...,
()
..
k12
[]
(N1.1)
Example: Two objectives and two variables
=
x
xx
(,
)
12
x
x
fx
x
fx
x
st
xx
max()
max()
..
0, 3
11
2
22
1
12
=+
=−
≤≤
or
x
x
fx
x
fx
x
st
x
x
max()
max()
..
03
03
12
21
1
2
=+
=−
≤≤
≤≤
or
x
x
fx
x
fx
x
st
x
x
xx
max()
max()
..
10
01
3
3
0, 0.
12
21
1
2
12
=+
=−
≥≥
Note that the Pareto optimal solution is also called the non-inferior or non-dominated
solution.
Developing the criteria and designing the fuzzy linguistic scale — The rst
step is dening the decision goals and developing criteria for the specic research
question. Linguistic variables take on values dened in the term set (set of linguis-
tic terms). Figure N1.1 displays a triangular fuzzy number (TFN). Linguistic terms
are subjective categories for linguistic variables. The values of a linguistic vari-
able are words or sentences in a natural or articial language. A triangular fuzzy
number
xA
and
=Almu(,
,)
on R may be a TFN if its membership function
[]
µℜ
x()
:0
,1
A
is equal to the following equation:
µ=
−− ≤≤
−− ≤≤
x
xl ml lxm
ux um mxu
()
()/( ),
()/( ),
0o
therwise
A
(N1.2)
*
The contents are a result of Professor Tzeng taking part of his related teaching courses “Research
Methods for Problem-Solving” outline in each part.
266 Notes
(Note: “Pareto Optimal Solutions” is also named “Non-inferior Solutions” or Non-
dominated solutions” or called “Efciency Solutions.”)
From Equation (N1.2), the diagonal l and u denote the lower and upper bounds of
the fuzzy number
A
, and m is the modal value for
A
. The TFN can be denoted by
A
= (l,m,u). The operational laws of TFNs
=Almu(,
,)
11
11
and
=Al
mu
(,
,)
22
22
are
displayed as Equations (N1.3) through (N1.7).
Addition of fuzzy number :

⊕= ⊕=++ +AA lmulmu llmm
uu
(, ,) (, ,)(,
,)
12111222 1212
12
(N1.3)
Multiplication of fuzzy number :

⊗=
=>>>
AA lmulmu
ll mm uu ll mm uu
(, ,) (, ,)
(, ,)for, 0; ,0;, 0
12111222
12 12 12 12 12 12
(N1.4)
Subtraction of fuzzy number Θ:

Θ= Θ=
−−−
AA lmulmu lumm
ul
(, ,)(, ,)(,
,)
12 111222 1212
12
(N1.5)
Division of fuzzy number :

∅=
=>>>
AAlmulmu
lu mm ul ll mm uu
(, ,) (, ,)
(/ ,/,/)for ,0;, 0;
,0
12111222
12 1212 12 12 12
(N1.6)
Reciprocal of fuzzy number:
== >>>
−−
Almu umlllmmuu(, ,) (1/,1/ ,1/)for, 0; ,0;, 0
1
111
1
111121212
(N1.7)
We use this kind of expression to evaluate two shopping websites by nine basic linguistic
terms (natural language) for measuring perceptions and feelings. Examples are beautiful,
good, perfect, very high inuence, high inuence, low inuence, very low inuence, and
no inuence on a fuzzy level scale as shown in Table N1.1 and Figure N1.2.
2
x
2
(x)f
(3,3) 3
Decision
Space Objective
Space
(0,0) (3,0)
1
x
Negative ideal point
-3
D (3,–3)
Positive ideal point
A (6,0)
B (3,3)
1
(x)f
Pareto Optimal
Solutions
C (0,0)
(0,3)
FIGURE N1.1 Basic concept of decision space and decision space in MODM.
267Notes
extension principle For Fuzzy arithmetic operations
Let
m
and
n be two fuzzy numbers and z denote a specic event. The membership
functions of the four basic arithmetic operations for
m
and
n can be dened by


zmxnyx
yz
() sup{min((), ())| };
mn
xy,
µ= +=
+
(N1.8)
zmxnyx
yz
() sup{min((), ())| };
mn
xy,


µ= −=
(N1.9)
zmxnyx
yz
() sup{min((), ())| };
mn
xy,


µ= ×=
×
(N1.10)


zmxnyx
yz
() sup{min((), ())| };
mn
xy,
µ= ÷=
+
(N1.11)
Next, we provide another method to derive the fuzzy arithmetic operations based
on the concept of α-cut arithmetic. Let
m= mm,m[,
lmu
] and
=nn
nn
[,
,]
lmu
be two
fuzzy numbers in which the superscripts l, m, and u denote the inmum, mode, and
supremum, respectively. The standard fuzzy arithmetic operations can be dened
using the concepts of α-cut as follows:

m nmnm n() () [()(), () ( )];
ll
uu
α+ α= α+ αα
(N1.12)

m nmnm n() () [()(), () ( )];
lu
ul
α− α= α− αα
−α
(N1.13)

m nmmnn() () [(), ()][1(),1()];
lu
ul
α÷ α≈ αα×α α (N1.14)

α× α≈m nM
N() () [,] (N1.15)
TABLE N1.1
Linguistic Scales for Importance (Example)
Linguistic Influence Linguistic Value
Perfect (1, 1, 1)
Very high (VH) (0.5, 0.75, 1)
High (H) (0.25, 0.5, 0.75)
Low (L) (0, 0.25, 0.5)
Very low (VL) (0, 0, 0.25)
None (No) (0, 0, 0)
A
(x)
A
1.0 -----------------------
-
0
l
m
ux
~
µ
~
A
FIGURE N1.2 Membership function of triangular fuzzy number.

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