文档介绍：Fuzzy Sets and Systems 109 (2000) 35–53
ate/fss
Evolutionary algorithm solution to fuzzy problems:
Fuzzy linear programming
James J. Buckleya ;∗, Thomas Feuringb;1
aDepartment of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
bInstitut fur Informatik, Westfalische Wilhelms-Universitat Munster, Einsteinstrae 62, 48149 Munster, Germany
Received July 1997; received in revised form January 1998
Abstract
In this paper we wish to ÿnd solutions to the fully fuzziÿed linear program where all the parameters and variables are
fuzzy numbers. We ÿrst change the problem of maximizing a fuzzy number, the value of the objective function, into a
multi-objective fuzzy linear programming problem. We prove that fuzzy
exible programming can be used to explore the
whole undominated set to the multi-objective fuzzy linear program. An evolutionary algorithm is designed to solve the fuzzy
exible program and we apply this program to two applications to generate good solutions.
c 2000 Elsevier Science .
All rights reserved.
Keywords: Mathematical programming; Fuzzy linear programming; Evolutionary algorithms
1. Introduction where the Ci, Aij and Bi are all triangular fuzzy num-
bers and the X i are also triangular fuzzy numbers. We
In this paper we wish to construct solutions to the will code the FFLP as
fully fuzziÿed linear program (written FFLP)
max Z = C X (4)
max Z = C X + ···+ C X (1)
1 1 n n s:t: AX 6B; X ¿0; (5)
s:t: Ai1 X 1 + ···+ Ain X n6Bi; 16i6m; (2) t t
where C =(C1;:::;Cn), X =(X 1;:::;X n), B =
(B ;:::;B ) and A =[A ]am × n matrix of fuzzy
X i¿0 for all i; (3) 1 m ij
numbers. Before we discuss the contents of the paper,
let us introduce the basic notation to be employed.
∗ Corresponding author. Tel.: +1-205-934-2154; fax: +1-205- We place a bar over a capital letter to denote a fuzzy
934-9025. subset of the real numbers. So A, B, C, X , etc., are
E-mail address: ******@math.