Application functions for fuzzy multiple
objective programs ∗
Christer Carlsson
ccarlsso@ra.abo.fi
Robert Fullér
rfuller@ra.abo.fi
Abstract
In multiple objective programs [MOP], application functions are established
to measure the degree of fulfillment of the decision maker’s requirements (achievement of goals, nearness to an ideal point, satisfaction, etc.) about the objective
functions ([4, 13]) and extensively used in the process of finding ”good compromise” solutions. In this paper, generalizing the principle of application function
to fuzzy multiple objective programs [FMOP], we define a large family of application functions for FMOP and illustrate our ideas by a simple biobjective
program.
1
Application functions for MOP
Consider a multiple objective program
max f1 (x), . . . , fk (x)
x∈X
(1)
where fi : IRn → IR are objective functions, x ∈ IRn is the decision variable, and X is
a subset of IRn without any additional conditions for the moment.
An application function hi for MOP (1) is defined as [Zim78, Del90]
hi : IR → [0, 1]
such that hi (t) measures the degree of fulfillment of the decision maker’s requirements
about the i-th objective by the value t. In other words, with the notation of
Hi (x) = hi (f (x)),
Hi (x) may be considered as the degree of membership of x in the fuzzy set ”good
solutions” for the i-th objective. Then a ”good compromise solution” to (1) may
be defined as an x ∈ X being ”as good as possible” for the whole set of objectives.
∗
in: P.Eklund, ed., Proceedings of MEPP’93 Workshop, June 14-18, 1993, Mariehamn, Finland,
Reports on Computer Science & Mathematics, Ser. B. No 17, Åbo Akademi, Åbo, 1994 10-16.
1
mi
Mi
fi (x)
Figure 1: The case of linear membership function.
Taking into consideration the nature of Hi (.), i = 1, . . . k, it is quite reasonable to look
for such a kind of solution by means of the following auxiliary problem
max H1 (x), . . . , Hk (x)
(2)
x∈X
For max H1 (x), . . . , Hk (x) , which may be interpreted as a synthetical notation of
a conjuction statement (maximize jointly all objectives) and Hi (x) ∈ [0, 1], it is reasonable to use a t-norm T [Sch63] to represent the connective AND. In this way (2)
turns into the single-objective problem
max T (H1 (x), . . . , Hk (x)).
x∈X
(3)
There exist several ways to introduce application functions [Kap90]. Usually, the
authors consider increasing membership functions (the bigger is better) of the form

if t ≥ Mi
 1
vi (t) if mi ≤ t ≤ Mi
hi (t) =

0
if t ≤ mi
where mi [≥ minx∈X fi (x)] denotes the reservation level representing minimal requirement and Mi [≤ maxx∈X fi (x)] is the desirable level on the i-th objective.
Let’s recall some definitions. Fuzzy sets of the real line are called fuzzy quantities.
A fuzzy number ã is a fuzzy quantity with a continuous, finite-supported, fuzzy-convex
and normalized membership function ã : IR → [0, 1]. The family of all fuzzy numbers
will be denoted by F(IR). An α-level set of a fuzzy quantity ã is a non-fuzzy set
denoted by [ã]α and is defined by
[ã]α = {t ∈ IR|ã(t) ≥ α}
for α ∈ (0, 1] and [ã]α = cl(supp ã) for α = 0. A triangular fuzzy number ã denoted
by (a, α, β) is defined as ã(t) = 1 − |a − t|/α if |a − t| ≤ α, ã(t) = 1 − |a − t|/β if
|a − t| ≤ β and ã(t) = 0 otherwise, where a ∈ IR is the modal value and α, β are the
left and right spreads of ã, respectively.
Let ã, b̃ ∈ F(IR) with [ã]α = [a1 (α), a2 (α)], [b̃]α = [b1 (α), b2 (α)] and let ω : F(IR) →
IR be a defuzzyfier in F(IR).
We suppose that crisp inequality relations between fuzzy numbers are defined by
defuzzifiers, i.e.
ã ≤ b̃ iff ω(ã) ≤ ω(b̃).
2
We metricize F(IR) by the metrics [Kal87],
1
1/p
α
α p
Dp (ã, b̃) =
d([ã] , [b̃] )
0
for 1 ≤ p ≤ ∞, especially, for p = ∞ we get
D∞ (ã, b̃) = sup d([ã]α , [b̃]α ),
α∈[0,1]
where d denotes the classical Hausdorff metric in the family of compact subsets of
IR2 . We shall use the following inequality relation between fuzzy numbers [Goe86]
ã ≤ b̃ iff
ω(ã) =
1
r(a1 (r) + a2 (r))dr ≤ ω(b̃) =
0
2
1
r(b1 (r) + b2 (r))dr
(4)
0
Application functions for FMOP
Consider a fuzzy multiple objective program [FMOP]
max f˜1 (x), . . . , f˜k (x)
x∈X
(5)
where f̃i : IRn → F(IR) (i.e. a fuzzy-number-valued function) and f̃i is to be maximized in the sense of a given crisp inequality relation ≤ between fuzzy numbers
(defined by a defuzzifier ω).
An application function for FMOP (5) is defined as
hi : F(IR) → [0, 1]
such that hi (µ) measures the degree of fulfillment of the decision maker’s requirements
about the i-th objective by the (fuzzy) value µ. In other words, with the notation of
Hi (x) = hi (f (x)),
Hi (x) may be considered as the degree of membership of x in the fuzzy set ”good
solutions” for the i-th fuzzy objective. As in the crisp case, we consider only increasing
membership functions (the bigger is better) of the form

if M̃i ≤ µ
 1
hi (µ) =
v (µ) if m̃i ≤ µ ≤ M̃i
 i
0
if µ ≤ m̃i
where m̃i [≥ minx∈X f˜i (x)] denotes the reservation level representing minimal requirement and M̃i [≤ maxx∈X f˜i (x)] is the desirable level on the i-th objective. We suggest
the use of the family of application functions


1
if ω(M̃i ) ≤ ω(f̃i (x))


Hi (x) =
1 − D(f̃i (x), M̃i )/D(m̃i , M̃i ) if ω(m̃i ) ≤ ω(f̃i (x)) ≤ ω(M̃i )



0
if ω(f̃i (x)) ≤ ω(m̃i )
3
1
1
Figure 2: The modal values of the objective functions.
where D is a metric (e.g Dp ) in F(IR). Thus, similarly to the crisp case, FMOP (5)
turns into the single-objective problem
max T (H1 (x), . . . , Hk (x)).
x∈X
(6)
It is clear that the closer the value of the objective function of problem (6) to one the
closer the fuzzy functions to their independent optima.
3
Example
We illustrate the use of applications functions by a simple biobjective program.
Let us consider the following FMOP
max f˜1 (x), f˜2 (x)
(7)
0≤x≤1
where f˜1 (x) = (1 − x2 , x2 , x), f˜2 (x) = (x, x, x) (i.e. fuzzy numbers of triangular type)
and we maximize the objective functions in the sense of relation (4).
It is easy to see (Fig.2.) that the modal values of the objective functions are in conflict.
Suppose that the decision maker wants to have as much gain on objectives as
possible, i.e. m̃i = minx∈X f˜i (x) and M̃i = maxx∈X f˜i (x). It is easy to compute that
M̃1
M̃2
m̃1
m̃2
=
=
=
=
f˜1 (1/14) = (1 − 1/142 , 1/142 , 1/14) (see Fig.3.),
f˜2 (1)) = (1, 1, 1),
f˜1 (1) = (0, 1, 1),
f˜2 (0) = (0, 0, 0),
We note that f˜1 attends its maximum (in the sense of inequality relation (4)) at
1/142 in [0, 1], and not at zero, where f1 (the function of its modal values) attends its
maximal value. Using the definition of the metric D∞ we have
max 2|x2 − 1/142 |, |x2 − x + 13/142 |
H1 (x) = 1 −
,
2|1 − 1/142 |
4
and
H2 (x) = 1 − |x − 1|
if x ∈ [0, 1] and H1 (x) = H2 (x) = 0 otherwise.
1/14
1
Figure 3. The defuzzified values of the objective function f˜1 .
By using the minimum operator to represent the connective AND in (6) the original
FMOP turns into the following single objective MP
max 2|x2 − 1/142 |, |x2 − x + 13/142 |
min 1 −
, 1 − |x − 1| → max
2|1 − 1/142 |
subject to x ∈ [0, 1].
Remark. The crisp biobjective problem
max 1 − x2 , x
0≤x≤1
where the objective functions are the modal values of the fuzzy objectives of (7) and
the minimum operator is used for the connective AND in (3) has unique solution (see
Fig.2.)
√
5−1
x∗ =
.
2
Our optimal solution to FMOP (7) is different from x∗ because the fuzziness has
essentially changed the application function for the first objective.
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