A MOLP Method for Solving Fully Fuzzy Linear Programming with LR Fuzzy Parameters
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 782376, 10 pages
http://dx.doi.org/10.1155/2014/782376
Research Article
A MOLP Method for Solving Fully Fuzzy Linear
Programming with LR Fuzzy Parameters
Xiao-Peng Yang,1,2 Xue-Gang Zhou,1,3 Bing-Yuan Cao,1 and S. H. Nasseri4
1
School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong,
Higher Education Institutes, Guangzhou University, Guangzhou 510006, China
2
Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China
3
Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521, China
4
Department of Mathematics, Mazandaran University, Babolsar 47416-95447, Iran
Correspondence should be addressed to Bing-Yuan Cao;
Received 22 March 2014; Accepted 15 September 2014; Published 29 September 2014
Academic Editor: Yang Xu
Copyright © 2014 Xiao-Peng Yang et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kaur and Kumar, 2013, use Mehar’s method to solve a kind of fully fuzzy linear programming (FFLP) problems with LR fuzzy
parameters. In this paper, a new kind of FFLP problems is introduced with a solution method proposed. The FFLP is converted into
a multiobjective linear programming (MOLP) according to the order relation for comparing the LR flat fuzzy numbers. Besides, the
classical fuzzy programming method is modified and then used to solve the MOLP problem. Based on the compromised optimal
solution to the MOLP problem, the compromised optimal solution to the FFLP problem is obtained. At last, a numerical example
is given to illustrate the feasibility of the proposed method.
1. Introduction
The research on fuzzy linear programming (FLP) has risen
highly since Bellman and Zadeh [1] proposed the concept
of decision making in fuzzy environment. The FLP problem
is said to be a fully fuzzy linear programming (FFLP)
problem if all the parameters and variables are considered
as fuzzy numbers. In recent years, some researchers such
as Lofti and Kumar were interested in the FFLP problems,
and some solution methods have been obtained to the
fully fuzzy systems [2–4] and the FFLP problems [5–13].
FFLP problems can be divided in two categories: (1) FFLP
problems with inequality constraints; (2) FFLP problems with
equality constraints. If the FFLP problems are classified by
the types of the fuzzy numbers, they will include the next
three classes: (1) FFLP problems with all the parameters and
variables represented by triangular fuzzy numbers; (2) FFLP
problems with all the parameters and variables represented
by trapezoidal fuzzy numbers; (3) FFLP problems with all
the parameters and variables expressed by 𝐿𝑅 fuzzy numbers
(or 𝐿𝑅 flat fuzzy numbers).
Fuzzy programming method is a classical method to solve
multiobjective linear programming (MOLP) [14, 15]. In this
paper, the fuzzy programming method is modified and then
used to obtain a compromised optimal solution of the MOLP.
The modified fuzzy programming method is shown in Steps
4–10 of the proposed method in Section 3.
Dehghan et al. [2–4] employed several methods to find
solutions of the fully fuzzy linear systems. Hosseinzadeh Lotfi
et al. [6] used the lexicography method to obtain the fuzzy
approximate solutions of the FFLP problems. Allahviranloo
et al. [7] and Kumar et al. [5, 8] solved the FFLP problem by
use of a ranking function.
Fan et al. [12] adopted the 𝛼-cut level to deal with a
generalized fuzzy linear programming (GFLP) probelm. The
feasibility of fuzzy solutions to the GFLP was investigated and
a stepwise interactive algorithm based on the idea of design
of experiment was advanced to solve the GFLP problem.
2
Mathematical Problems in Engineering
Kaur and Kumar [9] introduced Mehar’s method to the
FFLP problems with 𝐿𝑅 fuzzy parameters. They consider the
following model:
𝑛
𝐿𝑅
𝑗=1
2.1. Basic Notations
Definition 1 (𝐿𝑅 fuzzy number, see [2]). A fuzzy number 𝑢̃ is
said to be an 𝐿𝑅 fuzzy number if
Maximize (or Minimize)
∑ ((𝑝𝑗 , 𝑞𝑗 , 𝛼𝑗 , 𝛽𝑗 )
2. Preliminaries
⊙ (𝑥𝑗 , 𝑦𝑗 , 𝛼𝑗 , 𝛽𝑗 ) )
𝐿𝑅
subject to
(1)
𝑛
∑ ((𝑎𝑖𝑗 , 𝑏𝑖𝑗 , 𝛼𝑖𝑗 , 𝛽𝑖𝑗 )𝐿𝑅
𝑗=1
⊙(𝑥𝑗 , 𝑦𝑗 , 𝛼𝑗 , 𝛽𝑗 ) ) ⪯, ≈, ⪰ (𝑏𝑖 , 𝑔𝑖 , 𝛾𝑖 , 𝛿𝑖 )𝐿𝑅 ,
𝑚−𝑥
{
) , 𝑥 ≤ 𝑚,
𝐿(
{
{
𝛼
{
{
𝑢̃ (𝑥) = {
{
{
{
{𝑅 ( 𝑥 − 𝑚 ) , 𝑥 ≥ 𝑚,
𝛽
{
𝛼 > 0,
(3)
𝛽 > 0,
where 𝑚 is the mean value of 𝑢̃ and 𝛼 and 𝛽 are left and right
spreads, respectively, and function 𝐿(⋅) means the left shape
function satisfying
𝐿𝑅
𝑖 = 1, 2, . . . , 𝑚,
where the parameters and variables are 𝐿𝑅 flat fuzzy numbers
and the order relation for comparing the numbers is defined
as follows.
(1) 𝐿(𝑥) = 𝐿(−𝑥);
(2) 𝐿(0) = 1 and 𝐿(1) = 0;
(i) 𝑢̃ ⪯ ̃V if and only if R(̃
𝑢) ≤ R(̃V),
(ii) 𝑢̃ ⪰ ̃V if and only if R(̃
𝑢) ≥ R(̃V),
(3) 𝐿(𝑥) is nonincreasing on [0, ∞).
(iii) 𝑢̃ ≈ ̃V if and only if R(̃
𝑢) = R(̃V).
Here 𝑢̃ and ̃V are two arbitrary 𝐿𝑅 flat fuzzy numbers.
In our study, we consider a new kind of FFLP problems
with 𝐿𝑅 flat fuzzy parameters as follows:
min (or max)
Naturally, a right shape function 𝑅(⋅) is similarly defined
as 𝐿(⋅).
Definition 2 (𝐿𝑅 flat fuzzy number, see [9, 16]). A fuzzy
number 𝑢̃, denoted as (𝑚, 𝑛, 𝛼, 𝛽)𝐿𝑅 , is said to be an 𝐿𝑅
flat fuzzy number if its membership function 𝑢̃(𝑥) is given
by
̃ = 𝑐̃1 ⊗ 𝑥̃1 ⊕ 𝑐̃2 ⊗ 𝑥̃2 ⊕ ⋅ ⋅ ⋅ ⊕ 𝑐̃𝑛 ⊗ 𝑥̃𝑛
𝑧 (𝑥)
subject to
𝑎̃𝑖1 ⊗ 𝑥̃1 ⊕ 𝑎̃𝑖2 ⊗ 𝑥̃2 ⊕ ⋅ ⋅ ⋅ ⊕ 𝑎̃𝑖𝑛 ⊗ 𝑥̃𝑛 ≤ ̃𝑏𝑖 ,
(2)
𝑖 = 1, 2, . . . , 𝑚,
̃
𝑥̃𝑗 ≥ 0,
𝑗 = 1, 2, . . . , 𝑛,
where the parameters and variables are 𝐿𝑅 flat fuzzy numbers
and the order relation shown in Definition 4 is different from
the one above.
In this paper, we modify the classical fuzzy programming
method. The FFLP is changed into a MOLP problem solved
by the modified fuzzy programming method. We get the
compromised optimal solution to the MOLP and generate the
corresponding compromised optimal solution to the FFLP.
The rest of the paper is organized as follows. In Section 2,
the basic definitions and the FFLP model are introduced. In
Section 3, we propose a MOLP method to solve the FFLP
problems. Some results are discussed from the solutions
obtained by the proposed method. In Section 4, a numerical
example is given to illustrate the feasibility of the proposed
method. In Section 5, we show some short concluding
remarks.
𝑚−𝑥
{
),
𝐿(
{
{
𝛼
{
{
{
{
{
{
{
𝑢̃ (𝑥) = {𝑅 ( 𝑥 − 𝑛 ) ,
{
{
𝛽
{
{
{
{
{
{
{
{1,
𝑥 ≤ 𝑚, 𝛼 > 0,
𝑥 ≥ 𝑛, 𝛽 > 0,
(4)
𝑚 ≤ 𝑥 ≤ 𝑛.
Definition 3 (see [5, 9]). An 𝐿𝑅 flat fuzzy number 𝑢̃ =
(𝑚, 𝑛, 𝛼, 𝛽)𝐿𝑅 is said to be nonnegative 𝐿𝑅 flat fuzzy number
if 𝑚 − 𝛼 ≥ 0 and is said to be nonpositive 𝐿𝑅 flat number if
𝑛 + 𝛽 ≤ 0.
We define 𝑢̃ = (𝑚, 𝑛, 0, 0)𝐿𝑅 as an 𝐿𝑅 fuzzy number with
membership function
𝑢̃ (𝑥) = {
1, 𝑚 ≤ 𝑥 ≤ 𝑛,
0, otherwise,
and denote (0, 0, 0, 0)𝐿𝑅 as 0̃.
(5)
Mathematical Problems in Engineering
3
2.2. Ar (...truncated)
