JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
437
Pareto Optimal Solution Analysis of Convex
Multi-Objective Programming Problem
Guoli Zhang
Department of Mathematics and Physics, North China Electric Power University, Baoding, China
Email: [email protected]
Hua Zuo
Department of Mathematics and Physics, North China Electric Power University, Baoding, China
Email: [email protected]
Abstract—The main method of solving multi-objective
programming is changing multi-objective programming
problem into single objective programming problem, and
then get Pareto optimal solution. Conversely, whether all
Pareto optimal solutions can be obtained through
appropriate method, generally the answer is negative. In
this paper, the methods of norm ideal point and
membership function are used to solve the multi-objective
programming problem. In norm ideal point method, norm
and ideal point are given to structure the corresponding
single objective programming problem. Then prove that for
any Pareto optimal solution there exist weights such that
Pareto optimal solution is the optimal solution of the
corresponding single objective programming problem. In
membership function method, firstly construct membership
function for every objective function, then establish the
single objective programming problem, after then solve the
single objective programming problem, finally prove that
for any Pareto optimal solution there exist weights such that
the Pareto optimal solution is the optimal solution of the
corresponding single objective programming problem. At
last, two examples are given to illustrate that the two
methods are effective in getting Pareto optimal solution.
Index Terms—convex multi-objective, Pareto
solution, M-Pareto optimal solution, weight
optimal
I. INTRODUCTION
In multi-objective programming problem, it is difficult
to find an optimal solution to achieve the extreme value
of every objective function, so that the decision maker is
searching for the compromise solution. Based on this idea,
the concepts of Pareto optimal solution and weakly
Pareto optimal solution are introduced into multiobjective programming problem [1]. The main method of
solving multi-objective programming problem is turning
multi-objective programming problem to single objective
programming problem, and we can get Pareto optimal
solution or weakly Pareto optimal solution. Then,
whether we can get all Pareto optimal solutions or weakly
Pareto optimal solutions through appropriate method,
Corresponding author: Guoli Zhang
Email: [email protected]
© 2013 ACADEMY PUBLISHER
doi:10.4304/jnw.8.2.437-444
generally the answer is negative. So it is a very important
topic to discuss in what condition we can get all Pareto
optimal solutions or weakly Pareto optimal solutions. In
this paper, we only discuss this problem for convex
multi-objective programming problem. Some scholars
have done a lot of work about the Pareto optimal solution
of multi-objective programming problem.
K. Suga, S. Kato, and K. Hiyama analyzed structure of
Pareto optimal solution sets, presented the analysis
process as well as the case study details, and showed the
method they proposed is effective at finding an
acceptable solution for multi-objective optimization
problems [2]. Y. Shang and B. Yu presented a constraint
shifting combined homotopy method for solving convex
multi-objective programming problem with both equality
and inequality constraints. This method does not need the
starting point to be an interior point or a feasible point.
The existence and convergence of a smooth path to an
effective solution were proved [3]. A. Eusebio and J. R.
Figueira presented a new algorithm for identifying all
supported non-dominated vectors in the objective space,
as well as the corresponding effective solutions in the
decision space for multi-objective integer network flow
problem. The proposed approach used the connectedness
property of the set of supported non-dominated effective
solution to find all integer solutions in maximal nondominated efficient facets [4]. B.A. Ghaznavi-ghosoni
and E. Khorram analyzed the relationships between ε efficient points of multi-objective optimization problem
and є -optimal solutions of the related scalarized problem
and obtained necessary and sufficient conditions for
approximating efficient points of a general multiobjective optimization problem via approximate solutions
of the scalarized problem [5]. M. D. Monfared, A.
Mohades and J. Rezaei introduced a new method for
ranking the solutions of an evolutionary algorithm’s
population, and the proposed algorithm was very suitable
for the convex multi-objective optimization problems [6].
Y. Gao introduced the notion of ε -quasi weakly
saddle point and obtained the necessary and sufficient
conditions for the existence of ε -quasi weakly efficient
solution in multi-objective optimization problem in terms
of Hadamard directional derivatives and limiting
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JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
subdifferentials [7]. Y. Liu, Z. Peng and Y. Tan analyzed
objective programming problem. Finally, the conclusions
the relations among absolutely optimal solutions,
are given in Section Ⅴ.
effective solutions and weakly effective solutions of
multi-objective programming problem [8]. F. He and Y.
II. PRELIMINARY KNOWLEDGE
Shang gave the dynamic constraint combination
A convex multi-objective optimization problem can be
homotopy method for minimal weakly efficient solutions
stated as follows:
for convex multi-objective programming problems on

find x = ( x1 , x2 , , xn )T
unbounded sets and proved the existence and

convergence of the homotopy path [9]. X. Niu, T. Zhan
which minimizes f1 ( x), f 2 ( x), , f m ( x) 
(1)
and L. Xu studied the large scale multi-objective

subject to

programming with trapezoidal structure, which was

g j ( x) ≤ 0; j =
1, 2, , p
decomposed into several subproblems. The relations of

effective solutions between the large-scale multiwhere x is an n -dimensional vector of decision
objective programming and its subproblems have been
variables, f1 ( x), f 2 ( x), ⋅⋅⋅, f m ( x) are convex functions
studied and the existence of effective solution has been
investigated [10]. X. Zhou proved the connectedness of
defined on X , and=
0; j 1, 2, , p} is
X { x | g j ( x) ≤ =
cone-efficient solution set for cone-quasiconvex multiconvex set. Problem (1) is also called convex multiobjective programming in topological vector spaces [11].
objective programming problem.
Y. Li and Q. Zhang obtained some Kuhn-Tucker type
Definition 1. x* ∈ X is said to be a Pareto optimal
optimality conditions for a class of non-smooth multisolution of problem (1), if and only if there does not exist
objective
semi-infinite
programming
involving
another x ∈ X such that f i ( x) ≤ f i ( x* ),=
i 1, 2, ⋅⋅⋅, m ,
generalized convexity and some non-smooth non-convex
functions. [12]. Y. An and H. Liu used Taylor’ formal of
with strict inequality holding for at least one i .
the vector-valued function and Gordan Slection principle
Definition 2. x* ∈ X is said to be a weakly Pareto
to prove several necessary and sufficient conditions of
optimal solution of problem (1), if and only if there does
weakly Pareto optimal solution for convex multinot
exist
another
such
that
x∈ X
objective programming, and turned the problem how to
f i ( x) < f i ( x* ),=
i 1, 2, ⋅⋅⋅, m .
determine the weakly efficient solution of convex multiobjective programming into the problem of judging
We denote that R+m =
{w ∈ R m w ≥ 0} , where
whether an equation had non-negative non-zero solution
i 1, 2, ⋅⋅⋅, m .
=
w ( w1 , w2 , ⋅⋅⋅, wm )T , w ≥ 0 ⇔ wi ≥ 0,=
[13].
Lemma 1 [15]. Suppose that S is a nonempty convex set
There are a lot of methods of turning multi-objective
in normed linear space, and f i ( x),=
i 1, 2 ⋅⋅⋅, m are convex
programming problem to single objective programming
problem. In norm ideal point method, for the given
functions defined on S . Then convex function
weights, the optimal solution of the corresponding single
inequalities F ( x) < 0, x ∈ S are incompatible if and only
objective programming problem is Pareto optimal
if there exists w ∈ R+m \ {0} such that wT F ( x) ≥ 0, ∀x ∈ S ,
solution of multi-objective programming problem. In
membership function method, for the given weights, the
where
=
F ( x) ( f1 ( x), f 2 ( x), ⋅⋅⋅, f m ( x))T ,
optimal solution of the corresponding single objective
F ( x) < 0 ⇔ f i ( x) < 0,=
i 1, 2 ⋅⋅⋅, m .
programming problem is M-Pareto optimal solution of
multi-objective programming problem, which is similar
III. THE DISCUSSION OF PARETO OPTIMAL SOLUTION
to Pareto optimal solution [14]. Conversely, when taking
all weights, whether all Pareto optimal solutions or MIn order to get Pareto optimal solution or weakly
Pareto optimal solutions can be got, this problem for the
Pareto optimal solution of problem (1), the main method
convex multi-objective programming problem will be
is changing multi-objective programming problem into
discussed.
single objective programming problem. Next, two
The rest of this paper is organized as follows. For clear, methods are used to discuss whether all Pareto optimal
Section II gives the preliminary knowledge of convex
solutions can be obtained.
multi-objective programming problem and a lemma used
A. Norm Ideal Point Method
in the following. In Section III, three theorems are given
to proof that all Pareto optimal solutions and M- Pareto
For the problem (1), firstly give ideal value f i for
optimal solutions can be got through norm ideal point
every objective function f i ( x) , which satisfies
method and membership function method for convex
multi-objective programming problem, and Pareto
, f ( f1 , f 2 , ⋅⋅⋅, f m ) is called
f i ≤ min f i ( x),=
i 1, 2, ⋅⋅⋅, m=
x∈ X
optimal solution is equal to M- Pareto optimal solution.
ideal point, after then introduce the norm ⋅ , finally get
Section IV present examples to illustrate that there exist
the feasible solution which is having the nearest distance
weights such that Pareto optimal solution or M-Pareto
optimal solution of multi-objective programming problem
with the given ideal point f in the norm.
is the optimal solution of the corresponding single
© 2013 ACADEMY PUBLISHER
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
439
Use absolute value norm to structure the corresponding
single objective programming ( Pf ) :
m
min ∑ wi f i ( x) − f i
x∈ X
(2)
i =1
where
=
w ( w1 , w2 , ⋅⋅⋅, wm )T ∈ R+m \ {0} .
Because f i ≤ min f i ( x),=
i 1, 2, ⋅⋅⋅, m , ( Pf ) can be
x∈ X
m
simplified to min ∑ wi ( f i ( x) − f i ) .
x∈ X
i =1
For the given ideal point f and weights w ∈ R+m \ {0} ,
the optimal solution of ( Pf ) is weakly Pareto optimal
*
solution of problem (1) [16]. Conversely, if x is a
weakly Pareto optimal solution of problem (1), whether
there exists w such that x* is the optimal solution of the
corresponding
single
objective
programming
problem ( Pf ) . The following theorem is given.
Theorem 1. If x* is weakly Pareto optimal solution of
problem (1), then there exists w ∈ R+m \ {0} such that
x*
is the optimal solution of the corresponding single
objective programming problem ( Pf ) .
Proof. Suppose that x* is weakly Pareto optimal solution
of problem (1), then there does not exist x ∈ X such that
f i ( x) < f i ( x* ),=
i 1, 2, ⋅⋅⋅, m .
So there does not exist x ∈ X such that
f i ( x) − f i < f i ( x* ) − f i=
, i 1, 2, ⋅⋅⋅, m .
This implies that the following inequalities are
incompatible:
f i ( x) − f i − ( f i ( x* ) − f i ) < 0,=
i 1, 2, ⋅⋅⋅, m .
Because f i ( x),=
i 1, 2, ⋅⋅⋅, m are convex functions defined
on X , f i ( x) − f i − ( f i ( x* ) − f i ) are also convex functions
defined on X .
By
lemma
1,
there
exists
m
=
w ( w1 , w2 , ⋅⋅⋅, wm ) ∈ R+ \ {0} such that
m
∑ w [ f ( x) − f
i =1
i
i
i
− ( f i ( x* ) − f i )] ≥ 0 .
This means that
m
m
∑ w ( f ( x) − f ) ≥ ∑ w ( f ( x ) − f ) .
*
i
i
i
=i 1 =i 1
i
i
i
*
So x is optimal solution of problem ( Pf ) , and the
theorem is proved.
Attention should be paid that Pareto optimal solution
must be weakly Pareto optimal solution, which implies
that if all weakly Pareto optimal solutions can be obtained,
then all Pareto optimal solutions can be obtained. This
also shows that theoretically all Pareto optimal solutions
can be obtained through changing weights.
B. Membership Function Method
Firstly structure membership function µi ( f i ( x)) for
every objective function f i ( x) , then use µi ( f i ( x)) as the
new objective functions to structure the new multi© 2013 ACADEMY PUBLISHER
objective programming problem, and then turn the new
multi-objective programming problem to single objective
programming problem through some appropriate methods,
finally solve the single objective programming problem
to get the optimal solution, which is also the M-Pareto
optimal solution of the original multi-objective
programming problem.
Now, structure the membership function for every
objective function as follows:
f i ( x ) − f i1
µi ( f i ( x=
))
, i 1, 2, ⋅⋅⋅, m ,
=
f i * − f i1
where f i * = min f i ( x) , f i1 = max f i ( x) , =
i 1, 2, ⋅⋅⋅, m .
x∈ X
x∈ X
Without loss of generality, suppose f i * < f i1 ,=
i 1, 2, ⋅⋅⋅, m .
Definition 3. x* ∈ X is said to be a M-Pareto optimal
solution of problem (1), if and only if there does not exist
another x ∈ X such that µi ( f i ( x)) ≥ µi ( f i ( x* )) ,
=
i 1, 2, ⋅⋅⋅, m , with strict inequality holding for at least
one i .
Definition 4. x* ∈ X is said to be a weakly M-Pareto
optimal solution of problem (1), if and only if there does
not exist another x ∈ X such that µi ( f i ( x)) > µi ( f i ( x* )) ,
=
i 1, 2, ⋅⋅⋅, m .
It is obvious that M-Pareto optimal solution of problem
(1) must be weakly M-Pareto optimal solution of problem
(1).
Furthermore, the following conclusion is true.
Theorem 2. x* is M-Pareto optimal solution of problem
(1), if and only if x* is Pareto optimal solution of
problem (1).
Proof: “ ⇒ ”
If x* is M-Pareto optimal solution of problem (1), then
by definition 3, there does not exist another x ∈ X , with
strict inequality holding for at least one i , such that
i 1, 2, ⋅⋅⋅, m .
µi ( f i ( x)) ≥ µi ( f i ( x* )),=
So there does not exist another x ∈ X , with strict
inequality holding for at least one i , such that
f i ( x ) − f i1 f i ( x * ) − f i1
, i 1, 2, ⋅⋅⋅, m .
≥
=
f i * − f i1
f i * − f i1
Therefore, there does not exist another x ∈ X such
that f i ( x) ≤ f i ( x* ),=
i 1, 2, ⋅⋅⋅, m , with strict inequality
holding for at least one i , so x* is Pareto optimal
solution of problem (1).
“⇐ ”
If x* is Pareto optimal solution of problem (1), then
there does not exist another x ∈ X such that
f i ( x) ≤ f i ( x* ),=
i 1, 2, ⋅⋅⋅, m , with strict inequality holding
for at least one i by definition 1.
So there does not exist another x ∈ X , with strict
inequality holding for at least one i , such that
f i ( x) − f i1 ≤ f i ( x* ) − f i1=
, i 1, 2, ⋅⋅⋅, m .
This implies that there does not exist another x ∈ X ,
with strict inequality holding for at least one i , such that
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
f i ( x ) − f i1 f i ( x * ) − f i1
, i 1, 2, ⋅⋅⋅, m .
≥
=
f i * − f i1
f i * − f i1
Therefore, there does not exist another x ∈ X such
that
µi ( fi ( x)) ≥ µi ( f i ( x* )),=
i 1, 2, ⋅⋅⋅, m ,
with strict inequality holding for at least one i , so x* is
M-Pareto optimal solution of problem (1).
In order to turn multi-objective programming problem
into single objective programming problem, consider the
linear weighted model ( Pµ ) :
m
max ∑ wi µi ( f i ( x))
x∈ X
(3)
i =1
where
w ( w1 , w2 , ⋅⋅⋅, wm ) ∈ R+m \ {0} .
=
For the given w ∈ R+m \ {0} , the optimal solution of
problem ( Pµ ) is weakly M-Pareto optimal solution of
problem (1) [16].
Conversely, if x* is weakly M-Pareto optimal solution
of problem (1), whether there exists w ∈ R+m \ {0} such
*
that x is the optimal solution of the corresponding
single objective programming problem ( Pµ ) . We will
discuss this problem below.
Because µi ( f i ( x)) is nonincreasing linear function
about f i ( x) and f i ( x) is convex, µi ( f i ( x)) is concave
function [17]. We have the following theorem.
Theorem 3. If x* is weakly M-Pareto optimal solution of
problem (1), then there exists w ∈ R+m \ {0} such that x*
is the optimal solution of the corresponding problem
( Pµ ) .
Proof. Suppose that x* is weakly M-Pareto optimal
solution of problem (1), then there does not exist x ∈ X
such that
µi ( fi ( x)) > µi ( fi ( x* )),=
i 1, 2, ⋅⋅⋅, m .
This implies that the following inequalities are
incompatible:
i 1, 2, ⋅⋅⋅, m .
µi ( f i ( x* )) − µi ( f i ( x)) < 0,=
concave about f i ( x) , then the above conclusion still
holds.
IV. AN ILLUSTRATIVE EXAMPLE
A. Norm Ideal Point Method
Consider the following linear multi-objective
programming problem:
min{− x + 2 y, 2 x + y}

subject to − x − y + 2 ≤ 0 
(4)

3x − 2 y − 1 ≤ 0 
− 2 x + 3 y − 6 ≤ 0 
For convenience, the feasible region of the problem is
denoted as X .
5
4.5
4
C(3,4)
3.5
Variable y
440
3
-x-y+2=0
2
A(0,2)
3x-2y-1=0
1.5
B(1,1)
1
0.5
0
-1
-0.5
0
0.5
1
1.5
Variable x
2
2.5
3
3.5
4
Figure 1. The graph of feasible region.
The vertices of feasible region formed by constraints
are A (0,
=
=
2), B (1,1),
=
C (3, 4) , and the function
values of the two objective functions in vertices are given
in the following table I:
TABLE I.
THE FUNCTION VALUES OF VERTICES
A
B
C
Because µi ( f i ( x)),=
i 1, 2, ⋅⋅⋅, m are concave functions,
µi ( f i ( x* )) − µi ( f i ( x)) are convex functions defined on X .
-2x+3y-6=0
2.5
2x + y
2
3
10
−x + 2 y
4
1
5
Because the particularity of linear programming, it is
easy to know min{−=
x + 2 y} 1, min{2
=
x + y} 2 , and all
x∈ X
x∈ X
By lemma 1, there exists
=
w ( w1 , w2 , ⋅⋅⋅, wm ) ∈ R+m \ {0}
Pareto optimal solutions of problem (4) are
such that
=
A (0,
=
2), B (1,1) .
m
wi [µi ( f i ( x* )) − µi ( f i ( x))] ≥ 0 .
Next, ideal point method is used to solve the multi∑
i =1
objective programming problem (4), and illustrate that
Thus
there exists w such that each Pareto optimal solution of
m
m
*
problem (4) is the optimal solution of the corresponding
∑ wi µi ( fi ( x )) ≥ ∑ wi µi ( fi ( x)) .
single objective programming problem. In the ideal point
=i 1 =i 1
*
method, we take ideal point f = ( f1 , f 2 ) , where
Therefore x is the optimal solution of problem ( P ) ,
µ
and the theorem is proved.
In the above, the membership function of objective
function is set by using simple linear function. According
to the properties of the composition of convex function, if
the membership function of f i ( x) is nonincreasing and
© 2013 ACADEMY PUBLISHER
=
f1 min{− x + 2 y=
} 1 , =
f 2 min{2 x +=
y} 2 . So the
x∈ X
x∈ X
linear multi-objective programming problem (4) can be
turned into the following single objective programming
problem:
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
441
2
3
, and the multi-objective
=
, w2
5
5
programming problem (4) can be turned into the
following single objective programming problem:
Variable y
min w1[(− x + 2 y ) − 1] + w2 [(2 x + y ) − 2]
2
3

min [(− x + 2 y ) − 1] + [(2 x + y ) − 2]

5
5
subject to − x − y + 2 ≤ 0


(5)

4
7
8

3x − 2 y − 1 ≤ 0
= x+ y−


5
5
5

− 2x + 3y − 6 ≤ 0


(11)
subject to − x − y + 2 ≤ 0

The objective function of problem (5) can be written as

3x − 2 y − 1 ≤ 0
(− w1 + 2 w2 ) x + (2 w1 + w2 ) y − w1 − 2 w2

− 2x + 3y − 6 ≤ 0

In order to take minimum in point A = (0, 2) ,

according to the characteristic of linear programming, the


slope of objective function only need to satisfy
The
optimal
solution
of
single
objective
programming
w − 2 w2
(6)
< −1
− w1 + 2 w2 > 0 and 1
problem (11) is B = (1,1) , which is also the Pareto
2 w1 + w2
optimal solution problem (4). So there exists weights
or
2
3
w − 2 w2 2
such that B = (1,1) is the optimal
w1 =
, w2
(7) =
>
− w1 + 2 w2 > 0 and 1
5
5
2 w1 + w2 3
solution of the corresponding single objective
1
4
programming problem (11).
Specially, take=
, and the multi-objective
w1 =
, w2
5
5
To sum up, for all Pareto optimal solutions of multiprogramming problem (4) can be turned into the
objective programming problem (4), there exist weights
following single objective programming problem:
such that the Pareto optimal solution is the optimal
solution of the corresponding single objective
1
4

min [(− x + 2 y ) − 1] + [(2 x + y ) − 2]
programming problem. And the weight w is not unique.
5
5

From the above example, it is easy to see that if weight to
7
6
9

= x+ y−
satisfy inequality (6) or (7) the Pareto optimal solution

5
5
5
A = (0, 2) can be obtained. To get Pareto optimal solution

(8)
subject to − x − y + 2 ≤ 0

B = (1,1) the weight only need to satisfy inequality (9) or

3x − 2 y − 1 ≤ 0

(10). So it is easy to choose the weight w and get all
− 2x + 3y − 6 ≤ 0

Pareto optimal solutions of multi-objective programming

problem.


B. Membership Function Method
The optimal solution of single objective programming
In the following, membership function method is used
problem (8) is A = (0, 2) , which is also the Pareto
to solve the multi-objective programming problem.
optimal solution of problem (4). So there exists weights
Consider
the
following
linear
multi-objective
1
4
programming
problem:
such that A = (0, 2) is the optimal
=
w1 =
, w2
5
5
min{−3 x − y, x − 2 y, 4 x + 3 y}
solution of the corresponding single objective
subject to − 2 x − y + 6 ≤ 0 
programming problem (8).
(12)

2 x + 3 y − 18 ≤ 0 
In order to take minimum in point B = (1,1) , the slope

2x − 3y − 6 ≤ 0
of objective function only need to satisfy
7
w − 2 w2
(9)
<0
− w1 + 2 w2 > 0 and −1 < 1
A(0,6)
6
2 w1 + w2
5
or
4
2x+3y-18=0
w − 2 w2 3
(10)
<
− w1 + 2 w2 < 0 and 1
2 w1 + w2 2
3
C(6,2)
2
Specially, take=
w1
-2x-y+6=0
1
2x-3y-6=0
B(3,0)
0
-1
-1
0
1
2
3
Variable x
4
5
6
7
Figure 2. The graph of feasible region.
f1 =
−3 x − y ,
f2 =
x − 2 y, f3 =
4 x + 3 y . The feasible region of the
problem is denoted as X .
For
© 2013 ACADEMY PUBLISHER
convenience,
denote
that
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JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
− f + 30 
− f1 − 6
− f +3
)
+ w2 2
+ w3 3
14
15
18

subject to − 2 x − y + 6 ≤ 0
(14)


2 x + 3 y − 18 ≤ 0

2x − 3y − 6 ≤ 0
TABLE II.

THE FUNCTION VALUES OF VERTICES
The above model can also be described by the
following formula (15):
f1
f2
f3
3
1
4
1
2
3

A
-6
-12
18
max( w1 − w2 − w3 ) x + ( w1 + w2 − w3 ) y 
B
-9
3
12
14
15
18
14
15
18

C
-20
2
30
6
3
30

+ (− w1 + w2 − w3 )

14
15
18
Because the particularity of linear programming, it is
 (15)
subject to − 2 x − y + 6 ≤ 0

obvious that

2
x
3
y
18
0
+
−
≤
min f1 =
−20, max f1 =
−6 ,

x∈ X
x∈ X
2x − 3y − 6 ≤ 0

min f 2 =
−12, max f 2 =
3,

x∈ X
x∈ X

=
min f 3 12,
=
max f 3 30 .

x∈ X
x∈ X
In
order
to
take
maximum
in
point
,
the
slope
A
=
(0,
6)
Next, for every objective function, membership
of
objective
function
only
need
to
satisfy
function can be structured as follows:
3
1
4
− f −6
w1 − w2 − w3
,
µ1 ( f1 ) = 1
14
15
18
14
−
>0,
1
2
3
− f2 + 3
+
−
w
w
w
1
2
3
,
µ2 ( f 2 ) =
14
15
18
15
or satisfy simultaneously
− f + 30
.
3
1
4
µ3 ( f 3 ) = 3
w1 − w2 − w3 < 0
18
14
15
18
The membership function values in vertices are given
and
in the following table III:
3
1
4
w1 − w2 − w3
15
18 < −2 .
TABLE III.
− 14
THE MEMBERSHIP FUNCTION VALUES OF VERTICES
1
2
3
w1 + w2 − w3
14
15
18
µ1 ( f1 )
µ2 ( f 2 )
µ3 ( f 3 )
Here
take
A
0
1
23
3
1
4
w1 − w2 − w3
B
0
1
3 14
14
15
18
−
=
−4 .
C
1
0
1
2
3
1 15
w1 + w2 − w3
14
15
18
From the definition of M-Pareto optimal solution, the
1
5
279
Specially, take=
, and the
w1 =
, w2 =
, w3
M-Pareto optimal solutions of problem (12) are
5
7
280
=
A (0,
=
6), B (3,
=
0), C (6, 2) .
multi-objective programming problem (12) can be turned
Next, membership function method is used to solve the
into the following single objective programming problem:
multi-objective programming problem (12), and illustrate
228
57
491 
max( −
x−
y+
)
that there exists w such that the M-Pareto optimal
1080
1080
280 

solution of problem (12) is the optimal solution of the
subject to − 2 x − y + 6 ≤ 0
(16)

corresponding single objective programming problem.

2 x + 3 y − 18 ≤ 0
According to model ( Pµ ) , the multi-objective

2x − 3y − 6 ≤ 0

programming problem (12) can be turned into the
The optimal solution of single objective programming
following single objective programming problem:
problem (16) is A = (0, 6) , which is also the M-Pareto
max( w1 µ1 ( f1 ) + w2 µ 2 ( f 2 ) + w3 µ3 ( f 3 )) 

optimal solution of problem (12). So there exists weights
subject to − 2 x − y + 6 ≤ 0

(13)

1
10
279
2 x + 3 y − 18 ≤ 0
=
w1 =
, w2
=
, w3
, such that A = (0, 6) is the

5
14
280

2x − 3y − 6 ≤ 0
optimal solution of the corresponding single objective
Because of the way of structuring membership function
programming problem (16).
of objective function f1 , f 2 and f 3 , the above model
In order to take maximum in point B = (3, 0) , the slope
can be written as
of objective function only need to satisfy
The vertices of feasible region formed by constraints
are A (0,
=
=
6), B (3,
=
0), C (6, 2) , and the function
values of the three objective functions in vertices are
given in the following table II:
© 2013 ACADEMY PUBLISHER
max( w1
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
3
1
4
w1 − w2 − w3 < 0
14
15
18
443
2
1
18
=
, w2 =
, w3
3
7
35
and
and the multi-objective programming problem (12) can
be turned into the following single objective
3
1
4
w1 − w2 − w3
programming problem:
15
18
−2 < − 14
<0,
2
2
151 
1
2
3
x−
y+
max(
)
w1 + w2 − w3
105
105
210 
14
15
18

or satisfy simultaneously
subject to − 2 x − y + 6 ≤ 0 
(18)

3
1
4
2 x + 3 y − 18 ≤ 0
w1 − w2 − w3 > 0

14
15
18
2 x − 3 y − 6 ≤ 0 
and
The optimal solution of single objective programming
3
1
4
problem (18) is C = (6, 2) , which is also the M-Pareto
w1 − w2 − w3
2
15
18
< .
0 < − 14
optimal solution of problem (12). So there exists weights
1
2
3
w1 + w2 − w3 3
2
1
18
14
15
18
=
w1 =
, w2 =
, w3
, such that C = (6, 2) is the
3
7
35
Here take
optimal solution of the corresponding single objective
3
1
4
w1 − w2 − w3
programming problem (18).
15
18
− 14
=
−1 .
Therefore, for all M-Pareto optimal solutions of multi1
2
3
w1 + w2 − w3
objective programming problem (12), there exist weights
14
15
18
such that M-Pareto optimal solution is the optimal
Specially, take
solution of the corresponding single objective
2
1
18
programming problem.
, w2 =
, w3
=
w1 =
From the two examples above, it is easy to see that
5
7
35
weight
w is not unique. In fact, choose weight w in a
and the multi-objective programming problem (12) can
large range, the corresponding Pareto optimal solution or
be turned into the following single objective
M-Pareto optimal solution can be obtained. This also
programming problem:
shows that the two methods discussed in this paper are
4
4
11 
x−
y + )
max( −
very effective to obtain the Pareto optimal solution for
105
105
14

convex multi-objective optimization problem.
subject to − 2 x − y + 6 ≤ 0 
(17)
2 x + 3 y − 18 ≤ 0 
V. CONCLUSION

2 x − 3 y − 6 ≤ 0 
This paper discusses the Pareto optimal solution and
The optimal solution of single objective programming
M-Pareto optimal solution for convex multi-objective
problem (17) is B = (3, 0) , which is also the M-Pareto
programming problem. Generally, all Pareto optimal
solutions and M-Pareto optimal solutions cannot be
optimal solution of problem (12). So there exists weights
obtained through changing weights. But for convex
2
1
18
=
w1 =
, w2 =
, w3
, such that B = (3, 0) is the
multi-objective programming problem, all Pareto optimal
5
7
35
solutions and M-Pareto optimal solutions can be obtained
optimal solution of the corresponding single objective
through taking out all the weights. At last, linear
programming problem (17).
programming examples are given to illustrate that for any
In order to take maximum in point C = (6, 2) , the
Pareto optimal solution there exist weights such that
slope of objective function only need to satisfy
Pareto optimal solution is the optimal solution of the
3
1
4
corresponding single objective programming problem.
w1 − w2 − w3 > 0
And from the examples we can see that the weight is not
14
15
18
unique. It is not difficult to choose weight such that the
and
Pareto optimal solution is the optimal solution of the
3
1
4
w1 − w2 − w3
corresponding single objective programming problem. It
15
18 > 2 .
− 14
shows that the two methods discussed in this paper not
1
2
3
only have very important theoretic significant but also
w1 + w2 − w3 3
14
15
18
have widely practical value.
Here take
3
1
4
ACKNOWLEDGMENT
w1 − w2 − w3
14
15
18
1.
−
=
This work is supported by Hebei Province Science
1
2
3
w1 + w2 − w3
Foundation (A2012502061).
14
15
18
Specially, take
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© 2013 ACADEMY PUBLISHER
=
w1
444
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