Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming Problem

TELKOMNIKA, Vol.16, No.6, December 2018, pp. 2782–2790
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v16i6.11584 2782
Using Alpha-cuts and Constraint Exploration
Approach on Quadratic Programming Problem
Yosza Dasril*1
, Zahriladha Zakaria2
, and Ismail Bin Mohd3
1,2
Center for Telecommunication Research & Innovation , Faculty of Electronics and Computer Engineering,
Universiti Teknikal Malaysia Melaka (UTeM), Hang Tuah Jaya, Malaysia
1,3
Center for Mathematical Research (INSPEM) , Universiti Putra Malaysia (UPM), Serdang, Malaysia
*
Corresponding author, e-mail: yosza@utem.edu.my1
, zahriladha@utem.edu.my2
,
ismail ayah irma@yahoo.com3
Abstract
In this paper, we propose a computational procedure to find the optimal solution of quadratic pro-
gramming problems by using fuzzy α-cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
α-cuts and constraint exploration to find the optimal solution of QPP.
Keywords: fuzzy set, triangular fuzzy number, feasible set, quadratic programming, alpha-cuts, positive
definite.
Copyright c 2018 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
The theory of quadratic programming problem (QPP) deals with problems of constrained
minimization, where the constraint functions are linear and the objective is a positive definite
quadratic function [1, 2]. Many engineering problems can be represented as QPP such as in
sensor network localization, principle component analysis and optimal power flow [3] and design
of digital filters. The design aspect of digital filters that can be handled by quadratic programming
problem efficiently is to choose the parameters of the filter to achieve a specified type of frequency
response [4]. Although there is a natural transition from the theory of linear programming to the
theory of nonlinear programming problem, there are some important differences between their
optimal solution. If the optimum solution of quadratic programming problem exists then it is either
an interior point or boundary point which is not necessarily an extreme point of the feasible region.
The QPP model involves a lot of parameters whose values are assigned by experts. However,
both experts and decision makers frequently do not precisely know the values of those parame-
ters. Therefore, it is useful to consider the knowledge of experts about the parameters as fuzzy
data [5]. Bellman and Zadeh [6] proposed the concept of decision making in fuzzy environment
while Tanaka et al, [7] adopted this concept for solving mathematical programming problems.
Zimmerman [8] proposed the first formulation of fuzzy linear programming. Ammar and Khali-
fah [9] introduced the formulation of fuzzy portfolio optimization problem as a convex quadratic
programming approach and gave an acceptable solution to such problem. The constraint explo-
ration has been proposed by [10] where the method is based on the violated constraints by the
unconstrained minimum of the objective function of QPP for exploring, locating and computing
the optimal solution of the QPP. In this paper, we extend the concept of constraint exploration
method to solve the QPP in fuzzy environment. By using this approaches the fuzzy optimal so-
lution of the QPP occurring in the real life situations can be obtained. This paper is organized
Received June 7, 2018; Revised October 8, 2018; Accepted November 12, 2018

2783 ISSN: 1693-6930
as follows. In Section 2, some basic notations, definitions and arithmetic operations between two
triangular fuzzy numbers are reviewed. In Section 3, formulation of the QPP and determination of
the unconstrained optimal solution in fuzzy environment are discussed. In Section 4, a method to
determine the optimal solution on the boundary of violated constraints is described. In Section 5,
a new approaches or algorithm for solving the QPP is proposed. To illustrate the capability of the
proposed method, numerical examples are solved in Section 6. The conclusion ends this paper.
2. Preliminaries
In this section, some necessary notations and arithmetic operations of fuzzy set theory
are reviewed.
2.1. Basic Definition
Definition 1 [11] The characteristic function µA of a crisp set A ⊆ X assigns a value either 0 or
1 to each member in X. This function can be generalized to a function µ ¯A such that the value
assigned to the element of the universal set X falls within a specified range i.e. µ ¯A : X → [0, 1].
The assigned value indicates the membership grade of the element in the set A. The function µ ¯A
is called the membership function and the set ¯A = {(x, µ ¯A(x)): x ∈ X} defined by µ ¯A for each
x ∈ X is called a fuzzy set.
Definition 2 [12] A fuzzy number ¯A = (a, b, c) is called a triangular fuzzy number if its membership
function is given by
µ ¯A(x) =



(x−a)
(b−a) , a ≤ x ≤ b
(x−c)
(b−c) , b ≤ x ≤ c
0, otherwise,
and alpha-cuts corresponding to ¯A = (a, b, c) can be written as
¯A[α] = [a1(α), a2(α)], α ∈ [0, 1]
where a1(α) = [(b − a)α + a)] and a2(α) = [−(c − b)α + c].
Definition 3 [12] A triangular fuzzy number (a, b, c) is said to be non-negative fuzzy number if and
only if a > 0.
Definition 4 Let ¯A = (a1, b1, c1) and ¯B = (a2, b2, c2) be two triangular fuzzy numbers, then
(a) ( ¯A) ( ¯B) iff a1 ≤ a2, b1 − a1 ≤ b2 − a2, c1 − b1 ≤ c2 − b2.
(b) ( ¯A) ( ¯B) iff a1 ≥ a2, b1 − a1 ≥ b2 − a2, c1 − b1 ≥ c2 − b2.
(c) ( ¯A) = ( ¯B) iff a1 = a2, b1 − a1 = b2 − a2, c1 − b1 = c2 − b2.
2.2. Fuzzy Arithmetic
The following concepts and results are introduced in [11, 13]. Let ¯A[α] = [a−
α , a+
α ] and
¯B[α] = [b−
α , b+
α ] be two closed, bounded, intervals of real numbers. If ∗ denotes addition, sub-
straction, multiplication, or division, then [a−
α , a+
α ] ∗ [b−
α , b+
α ] = [β, δ] where
[β, δ] = {a ∗ b|a−
α ≤ a ≤ a+
α , b−
α ≤ b ≤ b+
α }.
If ∗ is division, we must assume that zero does not belong to [b−
α , b+
α ]. We may simplify
the above equation as follows:
1. Addition
[a−
α , a+
α ] ⊕ [b−
α , b+
α ] = [a−
α + b−
α , a+
α + b+
α ].
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TELKOMNIKA ISSN: 1693-6930 2784
2. Substraction
[a−
α , a+
α ] [b−
α , b+
α ] = [a−
α − b+
α , a+
α − b−
α ].
3. Division
[a−
α , a+
α ] [b−
α , b+
α ] = [a−
α , a+
α ] ⊗ [ 1
b+
α
, 1
b−
α
].
4. Multiplication
[a−
α , a+
α ] ⊗ [b−
α , b+
α ] = [β, δ]
where
β = min{a−
α · b−
α , a−
α · b+
α , a+
α · b−
α , a+
α · b+
α } and
δ = max{a−
α · b−
α , a−
α · b+
α , a+
α · b−
α , a+
α · b+
α }.
Remark 1 Multiplication may be further simplified as follows. For ¯A = (a, b, c) and ¯B = (x, y, z)
be a non-negative triangular fuzzy numbers, then
¯A ⊗ ¯B =



(ax, by, cz), a ≥ 0
(az, by, cz), a < 0, c ≥ 0
(az, by, cx), c < 0
Lemma 1 [14]. Suppose that f(x), x ∈ Rn
is an ordinary real valued function, and ¯A be the set
of all closed and bounded fuzzy numbers. If ¯r = (r1, r2, r3) ∈ ¯A then ¯r satisfied:
1. {x|x ∈ , ¯r(x) = 1} = ∅
2. if we define f(¯r) α∈[0,1] f(¯rα) then
f(¯r1)
α
= f(¯rα) = ∧x∈¯rα
f(x), ∨x∈¯rα
f(x)
3. f(¯rα) ∈ ¯A.
3. Problem Formulation
A quadratic function on Rn
to be considered in this paper, which is defined by
f(x1, · · · , xn) =
1
2
n
j=1
n
i=1
xidijxj +
n
j=1
cjxj (1)
where q, ci and dij, (i, j = 1, ..., n) are constant scalar quantities. Equation (1) can be written in
vector-matrix notation as
f(x) =
1
2
xT
Dx + cT
x (2)
in which D = (dij)n×n, c = (c1, ..., cn)T
, and x = (x1, ..., xn)T
.
Without loss of generality, we consider D to be a positive definite symmetric matrix and if
D is a positive definite, then f(x), which is given by (2), can be called a positive definite quadratic
function. The set all feasible solutions, so-called the feasible region, which will be considered in
this paper, is a closed set defined by
F = {x|x ∈ (R)n
, Ax ≤ b, x ≥ 0} (3)
where A an (mxn) matrix and b is a vector in Rm
.
Since f(x) given by (1) is positive definite quadratic function, then f(x) is strictly convex
in x, therefore f(x) attains a unique minimum at
x(0)
= −D−1
c (4)
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming... (Y. Dasril)

2785 ISSN: 1693-6930
which is called unconstrained minimum of f(x). As mention in Section 1, x(0)
can be an interior
point or boundary point of feasible region. However, there is one more possibility that is x(0)
can be an exterior point. Therefore, if x(0)
∈ F, then x(0)
becomes the optimal solution of the
QPP [1]. Another advantage of strictly convex properties of f(x) is that, if x(0)
is an exterior
point, then definitely, x∗
, the optimal solution of the considered problem is on the boundary of
the feasible region. Therefore, x∗
must be located on one of the active or equality constraints or
on the intersection of several active (equality) constraints [2, 10]. In the conventional approach,
the values of the parameters of QPP models must be well defined and precise. However, in
real life world environment this is not a realistic assumption. In the real problems there may
exist uncertainty about the parameters. In such a situation, the parameters of QPP with m fuzzy
constraints and n fuzzy variables may be formulated as follows:
Minimize ¯Z(¯x) =
1
2
n
j=1
n
i=1
¯xi
¯dij ¯xj +
n
i=1
¯ci ¯xi (5)
subject to
m
i=1
n
j=1
¯aij ¯xj
m
i=1
¯bi (6)
where ¯c = (¯ci)n×1, ¯A = (¯aij)m×n, ¯(b) = (¯bi)m×1, ¯D = ( ¯dij)n×n is a positive definite and symmetric
matrix of fuzzy numbers and all variables, ¯x = (¯x1, · · · , ¯xn) are non-negative fuzzy numbers.
Definition 5 [15], Any set of xi which satisfies the set of the constraints in (6) is called feasible
solution for (5)-(6). Let F be the set of all feasible solutions of (6). We shall say that x∗
∈ F is an
optimal feasible solution for (5)-(6) if ¯Z(x∗
) ¯Z(x), ∀x ∈ F.
Remark 2 The fuzzy optimal solution of QPP problem (5)-(6) will be a triangle fuzzy number
¯x∗
[α] = [x∗
1(α), x∗
2(α)] if its satisfies the following conditions.
1. ¯x∗
is a non-negative fuzzy number
2. ¯A ⊗ ¯x∗ ¯b
3. ¯x∗
1(α) monotonically increasing, 0 ≤ α ≤ 1
4. ¯x∗
2(α) monotonically decreasing, 0 ≤ α ≤ 1
5. ¯x∗
1(1) ¯x∗
2(1)
By using α-cuts notation [11], the fuzzy QPP of Equation (5)-(6) can be written as follows:
Minimize ¯Zα(¯x) =
1
2
n
j=1
n
i=1
[(d−
ij)α, (d+
ij)α]¯xi ¯xj +
n
i=1
[(c−
i )α, (c+
i )α]¯xi + [(q−
)α, (q+
)α] (7)
subject to
m
i=1
n
j=1
[(a−
ij)α, (a+
ij)α]¯xj
m
i=1
[(b−
i )α, (b+
i )α] (8)
all variables are non-negative, and α ∈ [0, 1].
By separation terms ¯Z(α)−
and ¯Z(α)+
of Equation (7), we have two types of the fuzzy
QPP to be solved, as follows:

Min ( ¯Zα(x))−
=
n
i=1
n
j=1
x−
j di,jx−
j +
n
j=1
cT −
j x−
j
s.t
m
i=1
n
j=1
ai,jx−
j
m
i=1
b−
i



(9)
with all variables are non negative, and α ∈ [0, 1], and
Min ( ¯Zα(x))+
=
n
i=1
n
j=1
x+
j di,jx+
j +
n
j=1
cT +
j x+
j
s.t
m
i=1
n
j=1
ai,jx+
j
m
i=1
b+
i



(10)
with all variables are non negative, and α ∈ [0, 1].
Any set of ¯x(α) = [x(α)−
, x(α)+
] which satisﬁes the set of the constraints in (6) is called
feasible solutions. Let F in Equation (3) be the set of all feasible solutions, we shall say that
¯x∗
= [x∗
(α)−
, x∗
(α)+
] resides inside of F is an optimal feasible solution provided ¯Zα(x∗
) ¯Zα(x)
for all ¯x ∈ F.
Since ¯zα(x) given by (7) is a positive deﬁnite quadratic function, then ¯zα(x) is strictly
convex in ¯xα(x) = [x(α)−
, x(α)+
], therefore ¯zα(x) attains a unique minimum at
n
j=1
di,jx−
j =
n
j=1
c−
j , and
n
j=1
di,jx+
j =
n
j=1
c+
j (11)
for (i = 1, · · · , m).
If we denoted x(0)
= [x(0)
(α)−
, x(0)
(α)+
] as the fuzzy unconstrained minimum of (5) then
we have
x(0)−
= −
n
j=1
(di,j)−1
c−
j , and x(0)+
= −
n
j=1
(di,j)−1
c+
j . (12)
4. Searching The Equality Constraint Point
This section will describe how to search a point on the equality constraint which becomes
a candidate of optimal solution to the problem (7) - (8). This method will be applied to the equality
constraint which are violated by x(0)
, which is AT
j xj > bi, (i = 1, · · · , n, j = 1, · · · , m). Let us
consider the constraint of the QPP given by
ai,1x1 + ai,2x2 + · · · + ai,kxk ≤ bi, (k ≤ n) (13)
aj,1x1 + aj,2x2 + · · · + aj,kxk ≤ bj. (14)
and their equality constraints is given by
ai,1x1 + ai,2x2 + · · · + ai,kxk = bi, (15)
aj,1x1 + aj,2x2 + · · · + aj,kxk = bj. (16)
The fuzzy equality constraints can be written as

2787 ISSN: 1693-6930
ai,1(x−
1 , x+
1 ) + ai,2(x−
2 , x+
2 ) + · · · + ai,k(x−
k , x+
k ) = [b−
i , b+
1 ] (17)
aj,1(x−
1 , x+
1 ) + aj,2(x−
2 , x+
2 ) + · · · + aj,k(x−
k , x+
k ) = [b−
j , b+
j ]. (18)
or
ai,1x−
1 + ai,2x−
2 + · · · + ai,kx−
k = b−
i ,
ai,1x+
1 + ai,2x+
2 + · · · + ai,kx+
k = b+
i ,
(19)
aj,1x−
1 + aj,2x−
2 + · · · + aj,kx−
k = b−
j ,
aj,1x+
1 + aj,2x+
2 + · · · + aj,kx+
3 = b+
j .
(20)
Clearly, the point
¯b−
i
a1,1
(1 − ω1 − · · · − ωk−1),
¯b−
i
a1,2
ω1, · · · ,
¯b−
i
a1,k
ωk−1 (21)
and
¯b+
i
a1,1
(1 − ω1 − · · · − ωk−1),
¯b+
i
a1,2
ω1, · · · ,
¯b+
i
a1,k
ωk−1 (22)
which lies on the Equation (15) is uniquely determined since there is one to one correspondence
between the point and its respected ωi, (i = 1, ..., k − 1). Therefore, by substituting the point in
Equation (21) into quadratic function (8), we can obtain the function with ωi as the independent
variable from which the unconstrained minimum of f(ωi) can be achieved through minimizing
f(ωi) with respect to ωi. If ω∗
i , (i = 1, ..., k − 1) denotes the unconstrained minimum of f(ωi), then
we obtain the point
¯b−
i
a1,1
(1 − ω∗
1 − · · · − ω∗
k−1),
¯b−
i
a1,2
ω∗
1, · · · ,
¯b−
i
a1,k
ω∗
k−1 (23)
and
¯b+
i
a1,1
(1 − ω∗
1 − · · · − ω∗
k−1),
¯b+
i
a1,2
ω∗
1, · · · ,
¯b+
i
a1,k
ω∗
k−1 (24)
which refers to the fuzzy constrained minimum of f(x), subject to the equality constraint given by
(15) and we denoted by
x∗
i =
¯b−
i
a1,1
(1−ω∗
1−· · ·−ω∗
k−1),
¯b−
i
a1,2
ω∗
1, · · · ,
¯b−
i
a1,k
ω∗
k−1,
¯b+
i
a1,1
(1−ω∗
1−· · ·−ω∗
k−1),
¯b+
i
a1,2
ω∗
1, · · · ,
¯b+
i
a1,k
ω∗
k−1
(25)
By the similar way, for equation given by (18), we have
x∗
j =
¯b−
j
a1,1
(1−ω∗
1−· · ·−ω∗
k−1),
¯b−
j
a1,2
ω∗
1, · · · ,
¯b−
j
a1,k
ω∗
k−1,
¯b+
j
a1,1
(1−ω∗
1−· · ·−ω∗
k−1),
¯b+
j
a1,2
ω∗
1, · · · ,
¯b+
j
a1,k
ω∗
k−1
(26)
which refers to the fuzzy constrained minimum of f(x), subject to the equality constraint given by
(16).
5. The Outline of Algorithm
The results shown in previous section can be used to obtain an algorithm for ﬁnding the
fuzzy optimal solution of QPP. The brief algorithm is as follows:
1. Compute ¯x(0)
, the unconstrained minimum of f(x) by using (12)

2. If ¯x(0)
satisfies all the constraints provided by the problem, then stop, ¯x(0)
becomes the
fuzzy optimal solution of the QPP. But if x(0)
/∈ F, then push all the indexes of the con-
straints violated by ¯x(0)
onto the set S, where S = {j|AT
j ¯x > bj, j ∈ {1, · · · , m}} for further
investigation.
3. Compute ¯x∗
j , the fuzzy constrained minimum of f(x) subject to equality constraint j where
j ∈ S. If ¯x∗
j ∈ F for one j ∈ S, then ¯x∗
j is the fuzzy optimal solution of QPP and stop.
Otherwise, search the fuzzy optimal solution of QPP which might be located on the equality
or intersection of two and more equality violated constraints by ¯x(0)
according to the method
explained in [1, 2, 10].
6. Numerical Results
An example has been illustrated to show the capability of the constraints exploration
method. Example taken from [16], where the objective function is to minimize the quadratic
function and the constraint functions consist of two linear functions. The first constraint have
an equality sign and the second constraint have an inequalities sign.
Minimized z = (x1 − 1)2
+ (x2 − 2)2
(27)
subject to
−x1 + x2 = 1 (28)
x1 + x2 ≤ 2 (29)
and
(x1, x2) ≥ (0, 0). (30)
The fuzzy QPP of the problem (27) - (30) with α ∈ [0, 1] can be written as
Minimized ¯Zα(¯x) = (¯x1 − ¯1)2
+ (¯x2 − ¯2)2
(31)
subject to
−¯x1 + ¯x2 = ¯1 (32)
¯x1 + ¯x2
¯2 (33)
with all variables are non-negative.
According to the algorithm in Section 5., the optimal solution of the fuzzy QPP is summa-
rized in 3 steps as follows:
Step 1 The determination of unconstrained minimum. For this problem we have
D =
2 0
0 2
, c = (−4, 2)T
, and q = 4, (34)
by using (12), we get
¯x(0)
=
α + 1
2
, −α + 2 ,
3α + 1
2
, −α + 3
.
Step 2 Test the feasibility of the ¯x(0)
. This can be done by substituting ¯x(0)
to both of the con-
straints. Clearly, ¯x(0)
violated constraint (33). Therefore, we are only focusing in finding the
optimal solution on the constraint which is given by equation (33).

2789 ISSN: 1693-6930
Step 3 By choosing 2 points on (33), usually the intersection of the line (33) with the coordinates
axis is chosen. It gives the constrained minimum with respect to (33) as an active constraint
and we denoted as ¯x∗
26 where
¯x∗
26 =
1
2
8α3
− 37α2
+ 106α − 73
α2 + 2α + 1
, −
1
2
4α3
− 21α2
+ 34α − 21
α2 + 2α + 1
,
(α − 2)(2α2
− 13α + 5)
(α + 1)2
, −
(α − 2)(3α − 5)(2α − 5)
(α + 1)2
.
By using the feasibility test in Step 2, clearly the constrained minimum, ¯x∗
26 satisﬁes equation (32)
and (33) or ¯x∗
26 ∈ F. Then the optimal solution for the example is ¯x∗
26 = ¯x∗
.
Now, at α = 1
¯x∗
= (
1
2
,
3
2
)
as an optimal solution of the problem, and agreed with the solution that given by [16].
7. Conclusion
A quadratic programming problems (QPP) is an optimization problem where the objective
function is quadratic function and the constraints are linear functions. Many engineering problems
can be represented as QPP, such as sensor network localization, principle component analysis
and optimal power ﬂow. That is, some performance metric is being optimized with subject to
design limits. In this paper we solve the quadratic programming problem by using α-cuts and
constraints exploration approach. The fuzzy solution are characterized by fuzzy numbers, through
the use of the concept of violation constraints by the fuzzy unconstrained and the optimal solution.
By this approach, the fuzzy optimal solution of quadratic programming problem which is occurring
in the real life situation can be easily obtained.
Acknowledgement.
The authors would like to thank Center for Research and Innovation Management (CRIM),
Center of Excellence, research grant PJP/2017/FKEKK/HI10/S01532 and Universiti Teknikal Malaysia
Melaka (UTeM) for their encouragement and help in sponsoring this research.
References
[1] Ismail Bin Mohd and Yosza bin Dasril. Constraint exploration method for quadratic programming problem,
Journal of Applied Mathematics and Computation. 2000; 112:161-170.
[2] Ismail Bin Mohd dan Yosza. Cross-Product direction exploration approach for solving the quadratic pro-
gramming problems. Journal ff Ultra Scientist of Physical Sciences. 2000; 12(2):155-162.
[3] Bose S, Dennise FG, Chandy KM, and Low SH. Solving Quadratically Constrained Quadratic Programs
on Acyclic Graph with Application to Optimal Power Flow. IEEE Xplore: Proceeding Annual Conference
of Information Sciences and System (CISS), New York. 2014: 19-21.
[4] Antoniou A, and Wu-Sheng L. Practical Optimization, Algorithms and Engineering Applications. New
York: Springer. 2007.
[5] Zadeh LA, Fuzzy sets. Journal of Information and Control. 1965; 8:338-353.
[6] Bellman RE, and Zadeh LA. Decision making in a fuzzy environment. Journal of Management Science.
1970; 17:141-164.
[7] Tanaka H, Okuda T, and Asai K. On fuzzy mathematical programming. Journal of Cybernetics and Sys-
tem. 1973; 3:37-46.
[8] Zimmerman HJ. Fuzzy programming and linear programming with several objective functions. Journal of
Fuzzy Sets and System. 1978; 1:45-55.
[9] E. Ammar and H. A Khalifah, Fuzzy portfolio optimization a quadratic programming approach, Journal of
Chaos, Solitons and Fractals, 2003; 18:1045-1054

[10] Ismail Bin Mohd and Yosza. A note on the violated constraints approach for solving the quadratic pro-
gramming problem. Journal of Ultra Scientist of Physical Sciences. 1998; 10(2):141-148
[11] Kaufmann A, and Gupta MM. Introduction to fuzzy arithmetic: Theory and Applications, Van Nostrand
Reinhold. New York. 1985.
[12] Liou TS, and Wang MJ. Ranking fuzzy numbers with integral value. Journal of Fuzzy Sets and Systems.
1992; 50:247-255.
[13] Buckley JJ, Esfandiar E, and Thomas F. Fuzzy mathematics in economics and engineering. Physica-
Verlag. 2002.
[14] Zohany Q, Zhang, and Wany G. Fuzzy random variable-valued exponential function, logarithmic func-
tion and power function. Journal of Fuzzy Sets and Systems, 1998; 99:311-324.
[15] Maleki H.R, Tata M, and Mashinchi M. Linear programming with fuzzy variables. Journal of Fuzzy Sets
and Systems. 2000; 109:1026-1092.
[16] Wismer DA, and Chattergy R. Introduction to Nonlinear Optimization, Noth-Holland Pub. Company.
1978.

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