

Ratio Mathematica Volume 46, 2023

Solving fuzzy linear programming
problems by using the fuzzy exponential

barrier method

A.Nagoor gani*
R.Yogarani†

Abstract

In order to resolve the fuzzy linear programming problem, the fuzzy
exponential barrier approach is the major strategy employed in this ar-
ticle. To overcome the problems with fuzzy linear programming, this
method uses an algorithm. In this concept, a fuzzy inequality con-
straint is produced since the objective functions are convex.numerical
examples are provided.
Keywords:fuzzy exponential barrier function, fuzzy exponential bar-
rier convergence,fuzzy optimality solution.
2020 AMS subject classifications: 54E20, 54H25. 1

*Research Department of Mathematics, Jamal Mohamed College(Autonomous),Affiliated to
Bharathidasan University,Tiruchirappalli-620020,Tamil Nadu,India. ganijimc@yahoo.co.in.

†Research Department of Mathematics, Jamal Mohamed College(Autonomous), Affiliated to
Bharathidasan University,Tiruchirappalli-620020, Tamil Nadu, India.yogaranimaths@gmail.com.
1Received on September 15, 2022. Accepted on December 15, 2022.Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1088. ISSN: 1592-7415. eISSN: 2282-8214. ©A.Nagoor gani et
al. This paper is published under the CC-BY licence agreement.

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A.Nagoor gani and R.Yogarani

1 Introduction
Fuzzy set theory has aided mathematical modelling and control theory. Un-

solved decision-making problems can be applied in real-world situations using a
variety of techniques. The fuzzy set theory developed by A. et al. [2012] has pre-
sented in 1965. Fuzzy linear programming was first established in 1997 by Tanaka
and Asai [1984], and fuzzy numbers were published in Dubois [1983]. A function
was proposed by Hsieh [1999] to cope with fuzzy arithmetical operations. Using a
technique that divides, adds, and multiplies fuzzy integers in addition to subtract-
ing and adding them. The fuzzy linear programming problem, as formulated by
Mahadevi et al. [2009], A. and Yogarani [2021] and Fiacco and V. [1990] Zimmer-
mann [1978] likewise has a duality. The process of defuzzifying involves convert-
ing fuzzy values into clear crispvalues. Since a few years ago, these techniques
have been thoroughly researched and applied in fuzzy systems Mahadevi et al.
[2006]. A representative value from a given set, according to some characters,
was the main objective of these processes. The defuzzification method establishes
a link between all fuzzy sets and all real numbers. Given the high cost ofan in-
feasible solution, Moengin and Parwadi [2011] et al. Fiacco and Anthony [1976]
developed the exponential barrier function technique. The fuzzy exponential bar-
rier approach is a different way to solve fuzzy linear programming problems. The
fuzzy exponential barrier technique includes a fuzzy objective function that spec-
ifies a severe penalty for violating the fuzzy requirements in order to approximate
fuzzy constraints. a fuzzy barrier function in fuzziness optimization problems that
has a value that grows exponentially with the separation from the viable zone. The
fuzzy exponential barrier parameter, a positive decreasing parameter used in this
process, establishes how close to the original fuzzy unconstrained problem is. In
section 2 of this study, some fundamental ideas of fuzzy set theory and the alge-
braic operation of triangular fuzzy numbers are provided. A fuzzy exponential
barrier technique is used to build an algorithm in section 3 to tackle the problem
of fuzzy linear programming problem. Section 4 includes an example.

2 Preliminaries
2.1 : Fuzzy Set
If M̃ = {(x,µ̃M(x)) : x ∈ M,µM̃(x) ∈ [0,1]} defines a fuzzy set M̃ The pair
(x,µ

M̃
(x))that makes up the membership function has two elements: x, which is

a member of the classical set M̃,and the µ
M̃
(x) belongs to the interval [0,1].

2.2 : Triangular fuzzy number
The triangular fuzzy number is the fuzzy set M̃ = (M1,M2,M3),M1 ≤ M2 ≤

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Solving fuzzy linear programming problems by using the fuzzy exponential
barrier method

M3 ∈ R, if the membership function of M̃ is defined by

µ
M̃
(y) =




y−M1
M2−M1

, M1 ≤ y ≤ M2
m3−y
m3−M2

, M2 ≤ y ≤ M3
0, otherwise

3 : Fuzzy exponential barrier method
Primal-dual of the fuzzy linear programming problems
maxZ̃ = f̃t(ỹ)

subject to Mjỹ ≥ Ñk,j = 1,2.
Where M1 ∈ Rm×n, f̃, ỹ ∈ Rn, Ñk ∈ Rm

f̃t = (f1,f2,f3), Ñ = (n1,n2,n3) (1)

minZ̃ = Ñ(ỹ)

MTj (x̃) ≤ f̃,M2 = MT1 ∈ Rm×n, ÑTk , f̃
tt = f̃,

We may consider the rank of the matrix M1 without losing generality. The tri-
angular fuzzy number is used to specify the fuzzy components in the preceding
problem. Consider that every problem has at least one feasible solution (1)
The fuzzy exponential barrier function is considered by
Ẽ(ỹ,γ) : Rn → Rfor every scalar δ > 0 as follows.

Ẽ(ỹ,γ) = f̃t(ỹ)−γ
m∑
i=1

e(Bi(ỹ))
β

(2)

Consider Ẽ : Rn → (−∞,∞) is a function given by

Bi(ỹ) =Mjỹ − Ñk ≤ 0,Mjỹ − Ñk = 0forallj,
Mjỹ − Ñk > 0,Mjỹ − Ñk 6= 0forallj.

3.1 : Fuzzy exponential barrier lemma
This lemma is based on the local and global behaviour of the unconstraintsmaxi-
mizer of the fuzzy exponential barrier function. Statement: An exponential barrier
parameter γk is a fuzzy sequence that increased in size. The fuzzy linear program-
ming problems and fuzzy exponential barrier function then for each k condition
are below.

(i).Ẽ(ỹk+1,γk+1) ≥ Ẽ(ỹk,γk)
(ii).Ẽ(ỹk ≤ Ẽ(ỹk+1)
(iii).f̃t(ỹk) ≥ f̃t(ỹk+1)
(iv).f̃t(ỹ∗) ≤ Ẽ(ỹk,γk) ≤ f̃t(ỹk)

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A.Nagoor gani and R.Yogarani

Proof :

(i).Ẽ(ỹk+1,γk+1) =f̃t(ỹk+1)−γk+1
m∑
i=1

e(β(Mjỹ
k+1−Ñk)) ≥ f̃t(ỹk)

−γk
m∑
i=1

e(β(Mjỹ
k−Ñk))

= Ẽ(x̃k,γk)

(ii).f̃t(ỹk)−γk
m∑
i=1

e(β(Mjỹ
k+1−Ñk)) ≤ f̃t(ỹk+1)γk+1

m∑
i=1

e(β(Mjỹ
k+1−Ñk)) (3)

f̃t(ỹk+1)−γk+1
m∑
i=1

e(β(Mjỹ
k+1−Ñk)) ≤ f̃t(ỹk)−γk

m∑
i=1

e(β(Mjỹ
k−Ñk)) (4)

The inequalities adding (3) and (4), we get

f̃t(ỹk)−γk
m∑
i=1

e(β(Mjỹ
k−Ñk))+

f̃t(ỹk+1)−
m∑
i=1

e(β
k+1(Mjỹ

k−Ñk)) ≤ f̃t(ỹk+1)−
m∑
i=1

e(β
k+1(Mjỹ

k+1−Ñk))

+ f̃t(ỹk)−γk
m∑
i=1

e(β(Mjỹ
k−Ñk))...(f̃t(ỹk)

+ f̃t(ỹk+1))− (f̃t(ỹk+1)

+ (f̃t(ỹk)) + γk
m∑
i=1

e(β
k(Mjỹ

k−Ñk))

+ γk+1
m∑
i=1

e(β
k+1(Mjỹ

k−Ñk)))

≤ γk
m∑
i=1

e(β
k(Mjỹ

k−Ñk))

+ γk+1
m∑
i=1

e(β
k+1(Mjỹ

k−Ñk))).

However (f̃t(ỹk) + f̃t(ỹk+1))− (f̃t(x̃k+1) + f̃t(x̃k)) = 0.

Then (γk +γk+1)γk
∑m

i=1 e
(βk(Mjỹ

k−Ñk)) ≥ (γk +γk+1)
∑m

i=1 e
(βk+1(Mjỹ

k−Ñk))

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Solving fuzzy linear programming problems by using the fuzzy exponential
barrier method

Since δk ≤ δk+1, Ẽ(ỹk+1) ≥ Ẽ(ỹk).

(iii). From the proof of (i), it can be obtained that

f̃t(ỹk)−γk
∑m

i=1 e
(β(Mjỹ

k−Ñk)) ≥ f̃t(ỹk+1)−γk+1
∑m

i=1 e
(β(Mjỹ

k+1−Ñk)).

Ẽ(ỹk) ≤ Ẽ(ỹk+1). Then f̃t(ỹk+1) ≥ f̃t(ỹk).

(iv). From the proof of (ii)
∑m

i=1 e
(β(Mjỹ

k−Ñk)) ≤
∑m

i=1 e
(β(Mjỹ

k−Ñk)), f̃t(ỹk) ≥
f̃t(ỹk+1).

f̃t(ỹ∗) ≥ f̃t(ỹk)−γk
m∑
i=1

e(β(Mjỹ
k−Ñk)) = Ẽ(ỹk,γk) ≥ f̃t(ỹk).

3.2 : A Fuzzy exponential barrier convergence theorem
Suppose that the fuzzy linear programming problem and the fuzzy exponential
barrier functions are both continuous functions. In a fuzzy linear programming
problem, an increasing series of positive fuzzy exponential barrier parameters k is
necessary such that {γk} → ∞,k → ∞ to ỹ of z̃there is an optimal solution. If
the limit pointxis within the boundaries of the range of ỹof z̃, then it exists.

Proof :
Let ỹ be a limit point of {ỹk}

Ẽ(ỹk, γ̃k) = f̃t(ỹk)−γk
m∑
i=1

e(β(Mjỹ
k−Ñk)) ≤ f̃t(ỹ∗)

lim
k→∞

f̃t(ỹk) = f̃t(ỹ), lim
k→∞

e(β(Mjỹ
k−Ñk)) ≤ 0

From the lemma 3.2 (iv) we get,

lim
k→∞

Ẽ(ỹk, γ̃k) = f̃t(y∗)

ỹ is feasible.
3.2 fuzzy exponential barrier function algorithm
1. Acknowledge the fuzzy objective function and constraints of the problem and
repharse the problem standard forms to reflect them. write max z̃ = f̃t(ỹ). Sub-
ject to Mjỹk − Ñk ≤ 0.

2.The fuzzy exponential barrier function defined by

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A.Nagoor gani and R.Yogarani

Ẽ(ỹ, γ̃) = f̃t(ỹ)−γ
m∑
i=1

e(β(Mjỹ
k−Ñk)).

3.3. Maxinequality constraints with the fuzzy exponential barrier functions

maxẼ(ỹ, γ̃) = f̃t(ỹ)−γ
m∑
i=1

e(β(Mjỹ
k−Ñk)).

4.First-order needed conditions for optimality γ →∞, which are used to find
the best solution to the specified fuzzy linear programming problems.

5.Find Ẽ(ỹk, γ̃k) = maxx≥0 Ẽ(ỹ, γ̃), then minimize ỹk and γ = 1,k =
1,2, ......,k = I then stop.
Alternatively, cancontinue to step 5.
The fuzzy exponential barrier approach employs the same procedure as the earlier
algorithm for the dual fuzzy linear programming problems.

4. Numerical example The primal fuzzy linear programming problem

min z̃ = (3.75,4,4.25)ỹ1 + (2.75,3,3.25)ỹ2

2ỹ1 + 3ỹ2 ≥ (5.75,6,6.25),
4ỹ1 + ỹ2 ≥ (3.75,4,4.25).

Solution : Figure 1: Finding an infeasible solution for primal fuzzy linear pro-
gramming problem Using the fuzzy exponential barrier function algorithm we get,
The fuzzy exponential barrier method is given by

Ẽ(ỹ,γ) = ((3.75,4,4.25)ỹ1 + (2.75,3,3.25)ỹ2 −γ
m∑
i=1

e(Bi(y))
β

According to this strategy, you can convert fuzzy linear programming problems
into a standard form of unconstrained problems.

ỹ1 = (0.45,0.8,1.15)−
1.2104

γ
, ỹ2 = (0.0727,0.6,1.1273)−

1.3951

γ

x̃1 =
0.0060

γ
+ (0.35,0.60,0.85), x̃2 = (0.9518,1.60,2.25) +

0.0180

γ
,

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Solving fuzzy linear programming problems by using the fuzzy exponential
barrier method

For different values of γj,j → ∞ the optimal values of ỹ1(γ), ỹ2(γ) are calcu-
lated as listed in Tables (1) and (2) we get,
Table1: Primal fuzzy linear programming problem solution(i)
No δk ỹ1 ỹ2
1 10 (0.3290,0.6790,1.0290) (-0.0668,0.4605,0.9878)
2 102 (0.4379,0.7879,1.1379) (0.0587,0.5860,1.1133)
3 103 (0.4488,0.7988,1.1488) (0.0713,0.5986,1.1259)
4 104 (0.4499,0.7999,1.1499) (0.0726,0.5999,1.1272)
5 105 (0.45,0.8,1.15) (0.0727,0.6,1.1273)

Table 2: Dual fuzzy linear programming problem solution(i)
No γj ỹ1 ỹ2
1 10 (0.3506,0.6006,0.8506) (0.9518,1.6018,2.2518)
2 102 (0.3501,0.6001,0.8501) (0.9502,1.6002,2.2502)
3 103 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500)
4 104 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500)
5 105 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500)

The optimal value of the given problem (1) can be obtained by

ỹ1 = (0.45,0.8,1.15), ỹ2 = (0.0727,0.6,1.1273),min z̃ = (2.9908,7.2,11.4092).

The corresponding given problem of the optimal values for x̃1, x̃2 are to be ob-
tained by

x̃1 = (0.35,0.6,0.85), x̃2 = (0.95,1.6,2.25),max z̃ = (4.25,7.2,10.15).

3 Conclusions

This article outlines a method for solving the primal-dual fuzzy linear pro-
gramming problem more effectively by combining the fuzzy exponential barrier
function and the fuzzy exponential barrier parameter.The table for the primal-dual
problem demonstrates how quickly the best solution can be reached using the
fuzzy exponential barrier primal-dual algorithm we developed when the Fuzzy
Exponential barrier parameter is used.

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A.Nagoor gani and R.Yogarani

Figure 1: Finding an infeasible solution for primal fuzzy linear programming
problem

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