Fuzzy Multiple Objective Decision Making

145

12

Fuzzy Interdependent

Multi-Objective

Programming

Since Bellman and Zadeh (1970) originally proposed the concepts of decision mak-

ing in a fuzzy environment, much research has been proposed to guide study in

the eld of fuzzy multi-objective programming (FMOP). The rst step of FMOP

is to view objectives and constraints as fuzzy sets and characterize them by their

individual membership functions. Then, a crisp (non-fuzzy) solution is generated

by transforming FMOP into multi-objective programming (MOP) and determining

the optimal solution to achieve the highest degree of satisfaction in the decision set.

For further discussions, readers can refer to Zimmermann (1978); Verners (1987);

Martinson (1993); and Lee and Li (1993). As with MOP, the problem of FMOP can

be dened by calculating the following model:



…





…



=∈ ≤=

∈xX

xx x

XxXx

ff f

st gkm

max/ min {(), (),,()}

.. {|() 0, 1,

,}

.

n

k

12

(12.1)

Much effort has been directed to this problem, both in theory and in practice (Sakawa,

1993; Sakawa et al., 1995; Shibano et al., 1996; Shih et al., 1996; Ida and Gen, 1997;

Shih and Lee, 1999). What seems lacking, however, is considering the problem of

interdependence between objectives. As we know, the supportive and the conict-

ing objectives usually occur in realistic decision-making problems. From the view

of optimization, because the optimal solution may be different while objectives are

interdependent, the problem of interdependence between objectives in FMOP prob-

lems should not be overlooked.

Carlsson and Fullér (1994, 1996) proposed two methods to reshape the membership

function to deal with the problem of FMOP with interdependence. Östermark (1997)

extended their method to consider temporal interdependence between objectives.

However, several shortcomings of their methods should be overcome before employ-

ing fuzzy interdependent multi-objective programming (FIMOP) in practice.

First, one method proposed by Carlsson and Fullér (1995) does not precisely mea-

sure the supportive or the conicting grade between the objectives and can deal

only with one-dimensional decision space. In contrast, a later method (Carlsson

and Fullér (2002) can be employed only in linear fuzzy independent multi-objective

programming (LFIMOP). Since real-life problems are usually complex, a general

method should be devised for dealing with all kinds of FIMOP problems.

146 Fuzzy Multiple Objective Decision Making

In this chapter, we propose another FIMOP to overcome these problems. First, a

new index is developed to measure the interdependent grade between fuzzy objec-

tives precisely. This method is suitable not only for the many-dimensional decision

spaces but also for nonlinear FIMOP problems. A numerical example is used to

demonstrate the method and compared with conventional FMOP. On the basis of

the numerical results, we concluded that the proposed method can extend FMOP for

considering the issue of interdependences between fuzzy objectives.

The problem of interdependence with objectives in multi-objective programming

is discussed in Section 12.2. Fuzzy interdependent multi-objective programming

is proposed in Section 12.3. In Section 12.4, we present a numerical example to

demonstrate the proposed method and compare the results with the conventional

FMOP model. Discussions are presented in Section 12.5 and conclusions are in the

nal section.

12.1 INTERDEPENDENCE WITH OBJECTIVES

The main problem of the conventional MOP model is the impractical assumption

of independence of objectives. To demonstrate the impact, we can transform the

MOP model into a single-objective programming (SOP) model by using the follow-

ing compromise programming (Yu, 1985):

…

=−

=∈ ≤=

∗

yyy

XxXx

rp

st gkm

min (; )||||

.. {|() 0, 1,

,}

,

p

k

(12.2)

where r(y; p) is a measurement of regret from y to

∗

y according to the

l

p

-norm, y

denotes the objective vector, and

∗

y denotes the ideal point vector.

Assume a two-objective problem and let p = ∞. Consider a case depicted in

Figure12.1. The optimal solution should be y

∞

if and only if f

1

(x) is independent

of f

2

(x). However, if f

1

(x) supports

xf

ˆ

()

2

, the optimal solution should transfer from

y

∞

to

∞

y

ˆ

.

Therefore, if we want to extend MOP to interdependent multi-objective

programming (IMOP), we should consider the interdependence between objectives,

Decision

Space

1

x

2

x

2

()f x

2

ˆ

()f x

Outcome/Objective

Space

1

()

f

x

y

ˆ

y

y

ˆ

y

FIGURE 12.1 Optimal solution between independent and interdependent objectives.