Reliable Computing 10: 357–367, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Fuzzy Linear Programming with Interactive
Division of Mathematical Science for Social Systems, Department of Systems Innovation,
Graduate School of Engineering Science, Osaka University, 1–3, Machikaneyama, Toyonaka,
Osaka 560–8531, Japan, e-mail: firstname.lastname@example.org
Department of Electronics and Information Systems, Graduate School of Engineering,
Osaka University, 2–1 Yamadaoka, Suita, Osaka 565–0871, Japan,
(Received: 15 October 2002; accepted: 22 February 2003)
Abstract. In this paper, we treat fuzzy linear programming problems with uncertain parameters whose
ranges are speciﬁed as fuzzy polytopes. The problem is formulated based on fractile optimization
model using a necessity measure. It is shown that the problem can be reduced to a semi-inﬁnite linear
programming problem and that a solution algorithm based on a relaxation procedure can be applied.
A simple numerical example is given to illustrate the solution procedure.
In order to treat optimization problems under uncertainty, stochastic, interval and
fuzzy programming techniques have been proposed . In stochastic program-
ming, the probability distribution of uncertain coefﬁcient vector is assumed to be
estimated from historical data or from some statistical inference. On the other hand,
in interval and fuzzy programming, a possible range of each uncertain coefﬁcient
has been subjectively speciﬁed as an interval or a fuzzy number by the ﬁeld expert.
We focus on fuzzy linear programming which includes interval linear programming
as a special case. For example, the interval [a
b] is the fuzzy set whose membership
function is 1 for all x in [a
b] and zero otherwise.
Since the possible range of uncertain coefﬁcient vector is given component by
component, the interaction among uncertain parameters cannot be considered at
all in the conventional interval and fuzzy linear programming problems. However,
in real world problems, we often know approximate values of sum of uncertain
parameters and difference and ratio between two uncertain parameters. In general,
This research is partly supported by the Ministry of Education, Science, Sports and Culture,
Grant-in-Aid Scientiﬁc Research for Encouragement of Young Scientists, No. 12780333.
The corresponding author.