Reliable Computing 10: 389–400, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Solutions for the Portfolio Selection Problem
with Interval and Fuzzy Coefﬁcients
National Institution for Academic Degrees and University Evaluation, 1–29–1, Gakuen nishimachi,
Kodaira, Tokyo, 187–8587, Japan, e-mail: firstname.lastname@example.org
(Received: 14 June 2002; accepted: 20 May 2003)
Abstract. In this paper we consider portfolio selection problem with interval and fuzzy objective
function coefﬁcients as a kind of multiple objective problems including uncertainties. For this problem
two kinds of efﬁcient solutions are introduced: possibly efﬁcient solution as an optimistic solution,
necessarily efﬁcient solution as a pessimistic solution. Investigating the properties of two efﬁciency
conditions by means of preference cones and feasible region, we discuss that the two kinds of solutions
can be identiﬁed with the sets of combinations of lower or upper bounds of intervals.
Uncertainty and multiple criteria are the essentially important factors in decision
making. From a practical viewpoint it is usually difﬁcult to determine exactly
the coefﬁcients in mathematical programming problems due to various kinds of
uncertainties. However, it is sometimes possible to estimate the perturbations of
coefﬁcients by intervals, fuzzy numbers or possibilistic distributions.
For such decision making situations, in the past decades a number of papers
of mathematical programming problems with uncertain coefﬁcients have described
decision making situations including uncertainties –. Inuiguchi  surveyed
the advantages and disadvantages of such mathematical programming approaches
compared with stochastic programming. Some newly developed ideas and tech-
niques in fuzzy mathematical programming were reviewed.
In the setting of multiple objective programming with fuzzy or possibilistic
coefﬁcients, there are two kinds of efﬁcient solution sets, i.e., a set of possibly
efﬁcient solutions and a set of necessarily efﬁcient solutions are deﬁned as fuzzy
sets whose membership grades represent the possibility and necessity degrees to
which the solution is efﬁcient or Pareto optimal . Therefore, tests to check the
possible efﬁciency and necessary efﬁciency should be discussed elaborately when
a feasible solution is given.
Regarding interval case, where all possibilistic coefﬁcients degenerate into inter-
val coefﬁcients, some important results were obtained , : Possibly efﬁcient
solutions can be identiﬁed by a combination of the lower or upper bounds of the
coefﬁcients in the interval matrix, necessarily efﬁcient solutions can be identiﬁed