Indian Journal of Science and Technology, Vol 8(S9), 501–505, May 2015
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
DOI: 10.17485/ijst/2015/v8iS9/68287
Solving a Fuzzy Linear Programming Problem with
Triangular Membership Function
Se-Ho Oh1*, Myungho Lee2 and Wooyoung Song2
Department of Industrial Engineering, Cheongju University, Korea; [email protected]
2
Department of Electronic Engineering, Cheongju University; Korea
1
Abstract
This paper deals with the linear problem under the equality constraints whose right hand side parameters are asymmetric
triangular fuzzy numbers. The right hand side values are defuzzified by using the triangular membership functions. Each
of these triangular functions is algebraically replaced with two piecewise linear functions. And the desired objective
value is also a fuzzy number. The membership function of the objective is assumed to be piecewise linear. Then, according
to Zimmermann’s approach, by using these membership functions and the max-min rule, the fuzzy linear problem is
transformed into the crisp linear programming problem. Consequently a fuzzy solution is obtained by solving this linear
programming problem.
Keywords: Fuzzy Linear Programming, Max-Min Rule, Piecewise Linear, Triangular Membership Function
1. Introduction
The fuzzy linear programming is a very useful and
practical decision making model for many real world
problems under uncertainty. These uncertain situations
can be formulated into the linear programming models
with imprecision parameters. Bellman and Zadeh first
proposed the concept of decision analysis in fuzzy environment2. They defined a fuzzy decision by using the
membership functions of constraints and the max-min
operator. Since then, many researchers have considered
various kinds of the fuzzy models5–8. The types are usually
differentiated depending on which parts of the problem are assumed to be fuzzy and on which membership
functions are introduced to describe imprecise parameters. Zimmermann suggested a tolerance approach for
symmetric model of fuzzy linear programming based
on max-min operator6. It is known as the first practical
approach to the linear programming problem with fuzzy
resource and objective. Many researchers proposed some
approaches to solve a wide range class of the fuzzy linear
*Author for correspondence
programming based on this decision rule4,9. This paper
deals with a linear programming problem under the linear equalities whose right-hand-side and objective are
fuzzy num
ber. The right-hand-side values are defuzzified by using
the triangular membership functions. Each triangular function can be replaced with two inequalities. And
the membership function of the objective is assumed to
be piecewise linear. Then, according to Zimmermann’s
approach, the fuzzy decision problem based on the maxmin rule is reduced to the crisp linear programming
problem. Consequently the fuzzy optimal solution with
highest membership degree can be obtained by solving the developed crisp linear programming problem.
Section 2 contains the process of defining the membership functions for the objective and the constraints.
It also shows how the max-min problem is converted
to the crisp linear programming problem. In section
3, a numerical example is given to illustrate the proposed method. Finally, the conclusion will be drawn in
section 4.
Solving a Fuzzy Linear Programming Problem with Triangular Membership Function
2. Transformation of a Fuzzy
Model to the Linear
Programming Problem
Then the membership function of the objective can be
algebraically described as follow:
In this paper, the following linear programming with
fuzzy resources and objective is considered:
min cT x
x≥0
(1)
where x ∈ R is the decision vector, Ai ∈ R is the i th
row vector of the constraint matrix, cT ∈ Rn is the objective vector and bi is the fuzzy resource whose value is in
[bi – pi, bi + qi] with the given upper and lower tolerance
pi, qi.
Let us assume that the decision-maker desired ­objective
value is a fuzzy number and that z0–zp is the tolerance of
the objective. It is defuzzified by a non-decreasing piecewise linear membership function shown in Figure 1. And
the ith right hand side is assumed to be fuzzy triangular
asymmetric number. Then its membership function is
defined to be triangular as shown in Figure 2.


0,
otherwise

 A x − bi
m ( Ai x ) = 1 + i
, bi − pi ≤ Ai x ≤ bi
pi

 Ai x − bi
, bi ≤ Ai x ≤ bi + qi
1 −
qi

n
Figure 1. Membership function of the objective
Figure 1. Membership function of the objective
(3)
By the way, this triangular membership function can be
split into two piecewise linear forms as follow:

0,
Ai x ≤ bi − pi

A
x
b
−

i
m1 ( Ai x ) = 1 + i
, bi − pi ≤ Ai x ≤ bi
p
i


1,
bi ≤ Ai x
(4)

1,
Ai x ≤ bi

 A x − bi
, bi ≤ Ai x ≤ bi + qi
m2 ( Ai x ) 1 + i
qi


0,
bi + qi ≤ Ai x
(5)
1.0
1.0
0
The i
Ai x ≤ bi
Figure 2. Membership function of the th
− Ai x ≤ −bi
constraint
(6)
(7)
If μ1(Aix) μ2(Aix) are taken as the membership
­functions respectively of two inequalities (6), (7)
μ(Aix)
1.0
ͳǤͲ
Theorem 1
{
}
{
}
max min m ( Ai x ) , i = 1, ... , m = max min m1 ( Ai x ) , m2 ( Ai x ) , i = 1, ... , m
0Ͳ
(2)
The algebraic representation of the ith constraint is as
follow:
n
‡”•Š‹’ˆ—…–‹‘‘ˆ–Š‡‘„Œ‡…–‹˜‡
( )
Ai x = bi , i = 1, 2, ... , m
0

0,
cT x ≤ z p

 cT x − z 0
T
m c x = 1 −
, z p ≤ cT x ≤ z0
z
−
z
0
p


1
z 0 ≤ cT x
,

bi–pi
bi–pi
bi–qi
Aix–pi
‹‰—”‡ ʹǤ‡„‡”•Š‹’ˆ—…–‹‘‘ˆ–Š‡th –Š
Figure 2. Membership
function of the i .
x ≥0
x ≥0
Proof:
{
}
m ( Ai x ) = min m1 ( Ai x ) , m2 ( Ai x ) , i = 1, ... , m
ip function of the objective can be algebraically described as follow:
502
Vol 8 (S9) | May 2015 | www.indjst.org
Indian Journal of Science and Technology
Se-Ho Oh1, Myungho Lee and Wooyoung Song
{
}
{
min m ( Ai x ) | x ≥ 0 ∈Rn = min min m1 ( Ai x ) , m2 ( Ai x ) | x ≥ 0 ∈Rn
i
{
i
= min min m1 ( Ai x ) , min m2 ( Ai x ) | x ≥ 0 ∈Rn
i
i
{
}
= min m1 ( Ai x ) , m2 ( Ai x ) | x ≥ 0 ∈Rn
i
}
s.t 1 −
Bx ≤ d x ≥ 0 ∈ Rn }

0,
d j + rj ≤ B j x

 Bj x − dj
, d j ≤ B j x ≤ d j + rj
m B j x = 1 −
rj


1,
Bj x ≤ dj

(8)
( )
(9)
μ(Bjx) can be interpreted as the degree to which the
decision vector x satisfies the ith fuzzy inequality. Bellman
and Zadeh were interested in a solution which maximizes
the smallest unfulfillment i.e. the smallest value among
the unfulfillments of membership function2. The maxmin operator has embodied their idea. Thus the following
fuzzy decision problem is constructed.
}
max min m B j x , j = 1, ... , (2m +1) x ≥0
max l
Vol 8 (S9) | May 2015 | www.indjst.org
Consequently, a unique optimal solution with highest
membership degree can be achieved by solving the above
crisp linear programming problem.
The following
­procedure:
problem
illustrates
the
min 2 x1 + x2 + x3 + x 4 + 2 x5
~�
s.t − 2 x1 + x2 + x3 + x 4 + x5 = 12
− x1 + 2 x2 + x 4 + x5 = 5
~�
x1 − 3x2 + x3 + 4 x5 = 11
xi ≥ 0, ∀i proposed
(12)
z p = 50, p1 = 0.25, p2 = 0.01, p3 = 0.02, z0 = 48, q1 = 0.05, q2 = 0.1, q3 = 0.2
Consequently, the parameters redefined in (8) are as
­follows:
d1 = −48, d2 = 12, d3 = 5, d4 = 11, d5 = −12, d6 = −5, d7 = −11
r = 2, r2 = 0.05, r3 = 0.1, r4 = 0.2, r5 = 0.25, r6 = 0.01, r7 = 0.02
1
The membership functions are as follow:
(10)
By the way, (10) is the nonlinear programming
­ roblem. Fortunately the nonlinear property can be
p
avoided by constructing the following linear ­programming
problem.
(11)
If the right hand side vector is assumed to be crisp,
the optimal solution is (x1, x2, x3, x4, x5) = (9, 7, 23, 0, 0)
and z∗ = 48. Now each parameters of (1) and zp, z0 are
given as follows to define the membership functions of
system (8):
−z , j = 1
p

r
=
q
where j  j −1 , j = 2, ... , ( m +1)

 p j −(m+1) , j = (m + 2) , ... , (2m +1)
{( )
x≥0
Hence each of the (2m+1) rows of system (8) shall
be represented by a fuzzy set with membership function
μ(Bjx) of (9) with the tolerance rj redefined such as
≥ l, j = 1, ... , (2m +1)
3. Numerical Example and
Application
 − z0 
 − cT 




2m +1)×n 
(
where  A  ∈R
, d =  b  ∈R2m+1  − A 
 −b 
rj
Therefore the above fuzzy problem (1) may be
­transformed into the symmetric linear system (8).
Bj x − dj
−48 ≤ B1 x
0,

 B x + 48

m ( B1 x ) = 1 − 1
, −50 ≤ B1 x ≤ −48
2

B1 x ≤ −50
1,

where B1 x = −2 x1 − x2 − x3 − x 4 − 2 x5
Indian Journal of Science and Technology
503
Solving a Fuzzy Linear Programming Problem with Triangular Membership Function
0,
12.05 ≤ B2 x

 B x − 12

m ( B2 x ) = 1 − 2
, 12 ≤ B2 x ≤ 12.05
0.05

B2 x ≤ 12
1,

where B2 x = −2 x1 + x2 + x3 + x 4 + x5
0,
5.1 ≤ B3 x

 B x −5
, 5.0 ≤ B3 x ≤ 5.1
m ( B3 x ) =  1 − 3
0. 1


1,
B3 x ≤ 5.0
where B3 x = − x1 + 2 x2 + x 4 − x5
0,
11.2 ≤ B4 x

 B x − 11

m ( B4 x ) = 1 − 4
, 11 ≤ B4 x ≤ 11.2
0. 2

B4 x ≤ 11
1,

where B4 x = x1 − 3x2 + x3 − 4 x5
1,
12 ≤ B5 x

 B x + 12

, 11.75 ≤ B5 x ≤ 12
m ( B5 x ) = 1 − 5
2

0,
B5 x ≤ 11.75

where B5 x = 2 x1 − x2 − x3 − x 4 − x5
1,
5 ≤ B6 x

 B x+5

, 4.95 ≤ B6 x ≤ 5
m ( B6 x ) = 1 − 6
2

0,
B6 x ≤ 4.95

where B6 x = x1 − 2 x2 − x 4 + x5
1,
11 ≤ B7 x

 B x + 11

, 10.98 ≤ B7 x ≤ 11
m ( B7 x ) = 1 − 7
0.02

0,
B7 x ≤ 10.98

where B7 x = x1 + 3x2 − x3 − 4 x5
The above fuzzy programming problem (12) is reduced
to the following linear programming problem(13).
504
Vol 8 (S9) | May 2015 | www.indjst.org
max
l
s.t 2x1 + x2 + x3 + x 4 + 2 x5 − 2l ≥ 48
2 x1 − x2 − x3 − x 4 − x5 − 0.05l ≥ −12.05
x1 − 2 x2 − x 4 + x5 − 0.1l ≥ −5.1
− x1 + 3x2 − x3 − 4 x5 − 0.2l ≥ −11.2
−2 x1 + x2 + x3 + x 4 + x5 − 0.25l ≥ 11.75
− x1 + 2 x2 + x 4 − x5 − 0.01l ≥ 4.99
x1 − 3x2 + x3 + 4 x5 − 0.02l ≥ 11.98
l ≥ 0, xi ≥ 0, ∀i
(13)
The optimal solution of (13) is
(
)
l∗ = 0.54, x1∗ , x2∗ , x3∗ , x 4∗ , x5∗ = (9.29, 7.17, 23.31, 0.00, 0.00)
.
4. Conclusions
This paper has developed a method to solve the fuzzy
­linear programming problem with equality constraints.
The solution based on the max-min rule is obtained by
solving the crisp linear programming problem derived
from the Zimmermann’s symmetric model. In the overestimated system, a family of approximated solution with
acceptable degree are often more meaningful than the
unique least square solution1. Therefore our approach may
be useful in the over-constrained decision environment
if the approximated solution is well defined to reflect the
weight of each constraint. For further research, the generalization of the membership functions to the piecewise
concave linear ones will be desirable.
5. References
1. Anton H. Elementary linear algebra. 9th ed. John Wiley and
Sons Inc; 2005.
2. Bellman RE, Zadeh IA. Decision-making in a fuzzy
­environment. Manag Sci. 1970; 17:141–64.
3. Gasimov RN, Yenilmez K. Solving fuzzy linear ­programming
problems with linear membership functions. Turh Journal
Math. 2002; 267:375–96.
4. Fang S, Puthenpura S. Linear optimization and extensions.
Prentice-Hall International Inc; 1993.
5. Guu S, Wu YK. Two-phase approach for solving the fuzzy
linear programming problems. Fuzzy Set Syst. 1999;
107:191–5.
6. Kamyad A, Hassanzadeh N, Chaji J. A new vision
on ­solving of fuzzy linear programming. IEEE
Conference on Computational Intelligence and Software
Engineering; 2009.
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Se-Ho Oh1, Myungho Lee and Wooyoung Song
7. Tang W, Luo Y. A new method of fuzzy linear ­programming
problems. IEEE International Conference on Business
Intelligence and Financial Engineering. 2009; 178–81.
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­relational equations. Fuzzy Set Syst. 2008; 159:23–39.
Vol 8 (S9) | May 2015 | www.indjst.org
10. Zadeh I A. Fuzzy sets. Iformation and control. 1965;
8:338–53.
11. Zimmermann H J. Fuzzy programming and linear
­programming with several objective functions. Fuzzy Set
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Kluwer Academic Publisher.
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