Fuzzy linear programming with the intuitionistic polygonal fuzzy numbers

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 14, No. 2, April 2024, pp. 2242∼2253
ISSN: 2088-8708, DOI: 10.11591/ijece.v14i2.pp2242-2253 ❒ 2242
Fuzzy linear programming with the intuitionistic polygonal
fuzzy numbers
Mahmoud H. Alrefaei, Marwa Z. Tuffaha
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
Article Info
Article history:
Received Jul 28, 2023
Revised Dec 18, 2023
Accepted Jan 5, 2024
Keywords:
Fuzzy numbers
Intuitionistic fully fuzzy linear
programming
Intuitionistic fuzzy numbers
Linear programming
Simplex method
ABSTRACT
In real world applications, data are subject to ambiguity due to several factors;
fuzzy sets and fuzzy numbers propose a great tool to model such ambiguity. In
case of hesitation, the complement of a membership value in fuzzy numbers can
be different from the non-membership value, in which case we can model us-
ing intuitionistic fuzzy numbers as they provide flexibility by defining both a
membership and a non-membership functions. In this article, we consider the
intuitionistic fuzzy linear programming problem with intuitionistic polygonal
fuzzy numbers, which is a generalization of the previous polygonal fuzzy num-
bers found in the literature. We present a modification of the simplex method
that can be used to solve any general intuitionistic fuzzy linear programming
problem after approximating the problem by an intuitionistic polygonal fuzzy
number with n edges. This method is given in a simple tableau formulation, and
then applied on numerical examples for clarity.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Mahmoud H. Alrefaei
Department of Mathematics and Statistics, Jordan University of Science and Technology
Irbid, Jordan
Email: alrefaei@just.edu.jo
1. INTRODUCTION
Linear programming (LP) is a widely used tool in operations research since many optimization prob-
lems can be expressed or simplified to a linear format. When full information about data and parameters are
known, traditional methods can be used for modelling and solving the problem. However, in many applications,
data are subject to vagueness or impreciseness; for example the prices may depend on the quantity or the day
in the year. In 1965, Zadeh [1] introduced the concept of fuzzy logic to formulate the vagueness of data. Fuzzy
numbers are represented by a membership function that represents the degree of membership of a number to a
given set. The shape of the membership function determines the type of fuzzy number such as the triangular,
trapezoidal, hexagonal and polygonal shapes. The fuzzy logic has been applied in many real life applications
and tens of journals has been established that deal with fuzzy logic and its applications. To give some, Singh
et al. [2] counted tens of thousands that deal with applications of fuzzy logic. Recent applications include data
acquisition can be found in Haddin et al. [3], in communication system [4], in grid connected PV inverter [5],
in Ambulance detection for smart traffic light applications [6] and many much more.
Bellman and Zadeh [7] are the first to introduce fuzzy environments in decision making, since then
many publications have suggested methods for solving linear programming in the fuzzy environment. In par-
ticular, fully fuzzy linear programming (FFLP) problems with triangular, trapezoidal and hexagonal fuzzy
numbers have been considered by several authors. For instance, Kumar and Kaur [8] considered the triangular
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Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 2243
FFLP and Das et al. [9] considered the trapezoidal FFLP. Recently, Tuffaha and Alrefaei [10] presented a
polygonal FFLP which is a generalization of all the above FFLP.
In many cases, the complement of a membership function on a fuzzy set does not mean a non-
membership, which happens in the case of hesitation. Therefore, the non-membership degree can be intro-
duced by another function called the non-membership function, which is the main character of intuitionistic
fuzzy numbers [11]. Solving optimization problems in intuitionistic fuzzy environment has been introduced by
Angelov [12] by maximizing the degree of membership and minimizing the degree of non-membership then
converting the problem to a crisp (unfuzzy) LP. Many solution methods have been presented for solving special
cases of IFLP such as the triangular and trapezoidal intuitionistic fuzzy linear programming (IFLP) problems.
Converting to easily-solvable crisp versions of the problem is common by using either ranking functions di-
rectly [13] or by dividing the problem based on (α, β)− cuts and then using a ranking function [14]. After the
conversion to crisp problems, the crisp simpler versions are ready to be solved by conventional methods such
as the simplex algorithm. Suresh et al. [15] suggested such a method, but Sidhu and Kumar [16] later showed
that the ranking function presented by the authors is incorrect; they fixed it and resolved the IFLP using the
corrected ranking function. On the other hand, a single step algorithm that directly solves the problem without
converting it to crisp was also suggested by Nagoorgani and Ponnalagu [17]. Discussion of duality of the IFLP
problem can also be found in [14], and a dual simplex method was proposed by Goli and Nasseri [18]. A fully
fuzzy IFLP with unconstrained LR-type of intuitionistic fuzzy numbers was introduced by Singh and Yadav
[19]. Their method resembles the Mehar’s method proposed by Kaur and Kumar [20] for solving fully fuzzy
linear programming where they use the (α, β)− cut and define a new product on the LR-type of intuitionistic
fuzzy numbers and use it to solve the problem. Recently, Malik et al. [21] present an approach for solving
a fully IFLP with unrestricted decision variables. A multi-objective IFLP for solving a closed loop supply
chain has been presented by Kousar et al. [22], while Singh and Yadav [23] solve a multi-objective IFLP using
various membership functions.
Most of the above articles are used for solving special cases of intuitionistic linear programming
problems with triangular, trapezoidal or hexagonal intuitionistic fuzzy numbers and so on. Moreover, most
of the above methods lack to provide general definitions of the binary operations between two intuitionistic
fuzzy numbers. Most of the privious binary operations were defined for specific cases such as triangular,
trapezoidal or hexagonal. Moreover, the product of two intuitionistic fuzzy numbers lack to preserve some
known properties such as the ranking of fuzzy numbers.
In this paper, we consider the intuitionistic fuzzy linear programming problem (IFLP) problem with
general intuitionistic fuzzy numbers. This intuitionistic fuzzy numbers can be approximated by an polygonal
intuitionistic fuzzy numbers with n edges (n−IPFN) and the simplex method is modified for solving this
problem using the binary operations presented by Alrefaei and Tuffaha [24]. Alrefaei and Tuffaha [24] provide
a generalized definition of the product between intuitionistic polygonal fuzzy number with n edges (n−IPFN)
that can preserve all the known properties such as the ranking of intuitionistic fuzzy numbers. These binary
operations can be used in all types of the known intuitionistic fuzzy numbers such as triangular, trapezoidal and
hexagonal fuzzy numbers because they are considered as special cases of n−IPFN.
This paper is organized as follows: section 2 previews some preliminaries about the intuitionistic
fuzzy numbers and give the definition of the polygonal intuitionistic fuzzy numbers. In section 3, we present the
intuitionistic fully fuzzy linear programming (IFFLP) with n−IPFN, in section 4, we present the intuitionistic
fuzzy simplex method and implement it into two numerical examples. Finally concluding remarks are presented
in section 5.
2. PRELIMINARY
We first give some definitions about the intuitionistic fuzzy sets and numbers and the definition of
intuitionistic polygonal fuzzy number (IPFN-n).
Definition 1. [11] An intuitionistic fuzzy set (IFS) ÃI
is a triple < X, µÃI , νÃI >, where X is any set and
µÃI , νÃI : X → [0, 1] are called the membership and non-membership functions, respectively. The comple-
mentary of this sum to 1 is called the degree of hesitation, πÃI (x) = 1 − µÃI (x) − νÃI (x) ∈ [0, 1].
Definition 2. [25] A real intuitionistic fuzzy number (IFN), ÃI
, is an intuitionistic fuzzy subset of R with a
membership µÃI and a non-membership νÃI functions, which can be described as:
Fuzzy linear programming with the intuitionistic polygonal fuzzy numbers (Mahmoud H. Alrefaei)

2244 ❒ ISSN: 2088-8708
µ ˜
AI (x) =







µL
f
AI
(x) ; a < x ≤ b
1 ; b < x ≤ c
µR
f
AI
(x) ; c < x ≤ d
0 otherwise
, νÃ(x) =







νL
e
AI (x) ; a′
< x ≤ b′
0 ; b′
< x ≤ c′
νR
e
AI (x) ; c′
< x ≤ d′
1 otherwise
where a, b, c, d, a′
, b′
, c′
and d′
are real numbers, µL
f
AI
is increasing function, µR
f
AI
is decreasing function, νL
f
AI
is
decreasing function and νR
f
AI
is increasing function.
Now we have the definition of the intuitionistic polygonal fuzzy number.
Definition 3. [24] Let {(a0, a1, .., an; b0, b1, .., bn), (a′
0, a′
1, .., a′
n; b′
0, b′
1, .., b′
n)} be real numbers, an intuition-
istic polygonal fuzzy number (n−IPFN) is an IFN with membership and non-membership functions given by
(1) and (2) respectively.
fÃI (x) =







1
n [ x−ai
ai+1−ai
] + i
n ; ai ≤ x ≤ ai+1, i = 0, .., n − 1
1 ; an ≤ x ≤ b0
−1
n [ x−bi
bi+1−bi
] + n−i
n ; bi ≤ x ≤ bi+1, i = 0, .., n − 1
0 otherwise
(1)
and
gÃI (x) =









−1
n [
x−a′
i
a′
i+1−a′
i
] + n−i
n ; a′
i ≤ x ≤ a′
i+1, i = 0, .., n − 1
0 ; a′
n ≤ x ≤ b′
0
1
n [
x−b′
i
b′
i+1−b′
i
] + i
n ; b′
i ≤ x ≤ b′
i+1, i = 0, .., n − 1
1 otherwise
(2)
The set of all n−IPFN’s is denoted by IPn. An intuitionistic fuzzy number with 4n knots is repre-
sented as:
ÃI
= {(a0, a1, .., an; b0, b1, .., bn), (a′
0, a′
1, .., a′
n; b′
0, b′
1, .., b′
n)}
An example of a 2-IPFN is given in Figure 1.
Figure 1. An example of a 2-IPFN
Int J Elec & Comp Eng, Vol. 14, No. 2, April 2024: 2242-2253

Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 2245
We use the following ranking function for IPFN.
Definition 4. [24] Let ÃI
=< R, µÃI , νÃI > be an intuitionistic fuzzy number and let ÃI
= {(a0, a1, .., an;
b0, b1, .., bn), (a′
0, a′
1, .., a′
n; b′
0, b′
1, .., b′
n)}, then the ranking function is given by (3).
R(ÃI
) =
1
8n
[a0 + 2a1 + 2a2 + ... + 2an−1 + an + b0 + 2b1 + 2b2 + ... + 2bn−1 + bn
+ a′
0 + 2a′
1 + 2a′
2 + ... + 2a′
n−1 + a′
n + b′
0 + 2b′
1 + 2b′
2 + ... + 2b′
n−1 + b′
n] (3)
Let ÃI
and B̃I
∈ IPn be given by (4) and (5).
ÃI
= {(a0, a1, .., an; b0, b1, .., bn), (c0, c1, .., cn; d0, d1, .., dn)} (4)
B̃I
= {(a′
0, a′
1, .., a′
n; b′
0, b′
1, .., b′
n), (c′
0, c′
1, .., c′
n; d′
0, d′
1, .., d′
n), } (5)
then the relations between any two intuitionistic numbers is defined based on the ranking function as follows:
− ÃI
and B̃I
are called equivalent, denoted ÃI
≈ B̃I
, if R(ÃI
) = R(B̃I
),
− they are called equal, denoted ÃI
= B̃I
, if ai = a′
i, bi = b′
i, ci = c′
i and di = d′
i for all i = 0, 1, .., n,
− ÃI
⪰ B̃I
if R(ÃI
) ≥ R(B̃I
).
Let ÃI
and B̃I
∈ IPn be given in (4) and (5), then arithmetic operations on the IPFN are given as
follows;
Definition 5. [24] The addition of ÃI
and B̃I
is defined as follows:
ÃI
⊕ B̃I
= {(a0 + a′
0, a1 + a′
1, .., an + a′
n; b0 + b′
0, b1 + b′
1, .., bn + b′
n),
(c0 + c′
0, c1 + c′
1, .., cn + c′
n; d0 + d′
0, d1 + d′
1, .., dn + d′
n)}
The multiplication of intuitionistic number by a positive crisp number is to multiply each node by this num-
ber k̃AI
= {(ka0, ka1, .., kan; kb0, kb1, .., kbn), (kc0, kc1, .., kcn; kd0, kd1, .., kdn)}. However, if k is neg-
ative then k̃AI
= {(kan, kan−1, .., ka1; kbn, kbn−1, .., kb0), (kcn, kcn−1, .., kc0; kdn, kdd−1, .., kd0)}. The
subtraction of intuitionistic numbers ÃI
⊖ B̃I
is given by
ÃI
⊖ B̃I
= {(a0 − a′
n, a1 − a′
n−1, .., an − a′
0; b0 − b′
n, b1 − b′
n−1, .., bn − b′
0),
(c0 − c′
n, c1 − c′
n−1, .., cn − c′
0; d0 − d′
n, d1 − d′
n−1, .., dn − d′
0)}
Definition 6. [24] The product ÃI
⊗ B̃I
is defined as follows, assume that
ÃI
⊗ B̃I
= {(e0, e1, .., en; f0, f1.., fn), (g0, g1, .., gn; h0, h1.., hn)},
The values of are the solution of (6) – (8):
e0 + 2e1 + 2e2 + ... + 2en−1 + en + f0 + 2f1 + 2f2 + ... + 2fn−1 + fn + g0
+ 2g1 + 2g2 + ... + 2gn−1 + gn + h0 + 2h1 + 2h2 + ... + 2hn−1 + hn = I (6)
where
I =
1
8n
[(a0 + 2a1 + 2a2 + ... + 2an−1 + an + b0 + 2b1 + 2b2+
... + 2bn−1 + bn + c0 + 2c1 + 2c2 + ... + 2cn−1 + cn + c0+
2c1 + 2c2 + ... + 2cn−1 + cn) · (a′
0 + 2a′
1 + 2a′
2 + ... + 2a′
n−1+
a′
n + b′
0 + 2b′
1 + 2b′
2 + ... + 2b′
n−1 + b′
n + c′
0 + 2c′
1 + 2c′
2 + ...+
2c′
n−1 + c′
n + c′
0 + 2c′
1 + 2c′
2 + ... + 2c′
n−1 + c′
n)]. (7)

2246 ❒ ISSN: 2088-8708
For i = 1, 2, .., n we let:
ei − ei−1 = (ai − ai−1) + (a′
i − a′
i−1),
f0 − en = (b0 − an) + (b′
0 − a′
n),
fi − fi−1 = (bi − bi−1) + (b′
i − b′
i−1)
gi − gi−1 = (ci − ci−1) + (c′
i − c′
i−1),
h0 − gn = (d0 − cn) + (d′
0 − c′
n),
hi − hi−1 = (di − di−1) + (d′
i − d′
i−1)
f0 − h0 = b0 + b′
0 − d0 − d′
0 (8)
Note that (6) and (7) guarantee that the ranking of the product of two intuitionistic numbers is the product of
the ranking of the intuitionistic numbers: R(ÃI
⊗ B̃I
) = R(ÃI
)R(B̃I
).
3. THE INTUITIONISTIC FULLY FUZZY LINEAR PROGRAMMING (IFFLP) WITH N-IPFN
The standard form of a IFFLP problem with n−IPFN’s is given by (9):
min z̃I
= c̃I
x̃I
s.t. ÃI
x̃I
≈ b̃I
x̃I
⪰ 0̃I
(9)
where c̃I
= [c̃I
j ]1×l, x̃I
= [x̃I
j ]l×1, ÃI
= [ãI
ij]m×l, b̃I
= [b̃I
i ]m×1 are matrices with n−IPFN’s entries.
Moreover, b̃I
i ⪰ 0̃I
for all i = 1, .., m, and the matrix ÃI
is of rank m.
An IFFLP problem may not be in standard form because of some inequality constraints, unrestricted
variables or having an objective function to be maximized. In such cases, we can transform the problem into
the standard form as follows:
− An inequality of the form “ãI
i x̃I
⪯ b̃I
i ” can be transformed into an equality constraint by adding a non-
negative fuzzy unknown value s̃I
, called a slack intuitionistic fuzzy variable, such that the constraint be-
comes ãI
i x̃I
⊕ s̃I
≈ b̃I
i .
− If the inequality constraint is in the form “ãI
i x̃I
⪰ b̃I
i ”, then we may subtract a non-negative surplus
intuitionistic fuzzy variable p̃I
to transform the constraint into ãI
i x̃I
⊖ p̃I
≈ b̃I
i .
− An unrestricted variable x̃I
j can be replaced by two nonnegative variables, x̃I+
j and x̃I−
j , by putting x̃I
j =
x̃I+
j ⊖ x̃I−
j .
− Finally, if the objective function is to be maximized, then taking the additive inverse of the objective value
with the same values for the variables gives the solution of the original maximization problem.
3.1. Basic feasible solutions
After possibly rearranging the columns ˜
aj
I
of ÃI
, let ÃI
= [B̃I
ÑI
], where B̃I
is an m × m
invertible matrix consisting of m columns of ˜
aj
I
, and ÑI
is an m × (l − m) matrix with the rest of the
columns. Then the constrains can be written as:
ÃI
x̃I
≈ b̃I
[B̃I
ÑI
]x̃I
≈ b̃I
The variables vector can then be split as (10):
[B̃I
ÑI
]

x̃I
B
x̃I
N

≈ b̃I
B̃I
x̃I
B ⊕ ÑI
x̃I
N ≈ b̃I
x̃I
B ⊕ B̃I(−1)
ÑI
x̃I
N ≈ B̃I(−1)
b̃I
(10)
Int J Elec Comp Eng, Vol. 14, No. 2, April 2024: 2242-2253

Int J Elec Comp Eng ISSN: 2088-8708 ❒ 2247
One solution is x̃I
=

x̃I
B
x̃I
N

=

B̃I(−1)
b̃I
0̃I

, which is called a basic solution. B̃I
is called the basis, and the
components of x̃I
B are called the basic variables. If x̃I
B ⪰ 0̃, then x̃I
is called a basic feasible solution (b.f.s.).
4. THE INTUITIONISTIC FUZZY SIMPLEX METHOD
Assume problem (9) has a basic feasible solution x̃′I
=

x̃I
B
x̃I
N

=

B̃I(−1)
b̃I
0̃I

, whose objective value
is given by:
˜
zI
0 = c̃I

B̃I(−1)
b̃I
0̃I

= [c̃I
B c̃I
N ]

B̃I(−1)
b̃I
0̃I

= c̃I
BB̃I(−1)
b̃I
(11)
The objective function in augmented form is:
z̃I
=

c̃I
B c̃I
N

x̃I
B
x̃I
N

= c̃I
Bx̃I
B ⊕ c̃I
N x̃I
N (12)
From (10), we have (13):
x̃I
B ≈ B̃I(−1)
b̃I
⊖ B̃I(−1)
ÑI
x̃I
N (13)
Substituting (13) in (12) and simplifying:
z̃I
⊕ (c̃I
BB̃I(−1)
ÑI
⊖ c̃I
N )x̃I
N ≈ c̃I
BB̃I(−1)
b̃I
Let z̃I
N = c̃I
BB̃I(−1)
ÑI
, then:
z̃I
⊕ (z̃I
N ⊖ c̃I
N )x̃I
N ≈ c̃I
BB̃I(−1)
b̃I
(14)
From (10) and (14), and putting b̃
I
= B̃I(−1)
b̃I
, the current b.f.s. can be represented in the tableau form as in
Table 1:
Table 1. A tableau form for the simplex method
x̃I
B x̃I
N RHS
z̃I 0̃I z̃I
N ⊖ c̃I
N c̃I
BB̃I(−1)
b̃I
x̃I
B
˜
II B̃I(−1)
ÑI b̃
I
Note that Row 0 (the second row in the table) represents the objective value of the current solution
z̃I
= c̃I
BB̃I(−1)
b̃I
since x̃I
N = 0. Row 1 (the third row in the table is in fact m rows represent the values of the
basic variables x̃I
B = b̃
I
.)
Without loss of generality, we assume the absence of degeneracy, i.e. we assume that b̃
I
≻ 0. The
case of degeneracy, where b̃
I
has zero values is known to cause some problems and needs a special discussion
that will be studied later.
Let J be the current set of indices of the non-basic variables, then z̃I
j ⊖c̃I
j where j ∈ J are the elements
of z̃I
N ⊖ c̃I
N . Now, from (14) we have:
z̃I
≈ z̃I
0 ⊖
X
j∈J
(z̃I
j ⊖ c̃I
j )x̃I
j (15)
Note that if z̃I
j ⊖ c̃I
j ≺ 0 for all j ∈ J, then increasing the value of any nonbasic variable of the x̃I
j ’s increases
the value of the objective function z̃I
. Therefore, the current solution cannot be improved anymore, and it is
optimal. On the other hand, if z̃I
j ⊖ c̃I
j ⪯ 0 for all j ∈ J, and z̃I
k ⊖ c̃I
k ≈ 0 for some k ∈ J, then increasing the
value of x̃I
k does not affect the objective value, which means that we have alternative optimal solutions with the

2248 ❒ ISSN: 2088-8708
same objective value. However, such case is not treated differently than the previous case in this paper. In other
words, even if we have alternative optimal solutions, we will take only one of them into consideration. Finally,
if there exists z̃I
k ⊖ c̃I
k ≻ 0 for some k ∈ J, then increasing the ranking value of x̃I
j obviously decreases the
objective value, which means that the current solution is not optimal and there is a better one. Thus, x̃I
k needs
to enter the basis, and another variable needs to leave it.
Determining the leaving variable can be done as follows:
Let ỹI
j for j ∈ J be the columns of the matrix B̃I(−1)
ÑI
, i.e. ỹI
j = B̃I(−1)
ãI
j = [ỹI
ij]. Since the nonbasic
variables other that x̃I
k will stay nonbasic with zero value, then from (13) and (15) the problem can be written
as:
min z̃I
≈ z̃I
0 ⊖ (z̃I
k ⊖ c̃I
k)x̃I
k
s.t.




x̃I
B1
x̃I
B2
...
x̃I
Bm



 ≈





b̃
I
1
b̃
I
2
...
b̃
I
m





⊖




ỹI
1k
ỹI
2k
...
ỹI
mk



 ⊗ x̃I
k
x̃I
k ⪰ 0, x̃I
Bi
⪰ 0 ∀i = 1, .., m
where x̃I
Bi
are the elements of x̃I
B (i.e. the basic variables). The constraints can be written as:
x̃I
Bi
≈ b̃
I
i ⊖ ỹI
ik ⊗ x̃I
k ∀i = 1, .., m
For each i, we note that:
− If ỹI
ik ⪯ 0, then x̃I
Bi
increases or does not get affected as x̃I
k increases, so x̃I
Bi
continues to be nonnegative.
− If ỹI
ik ≻ 0, then x̃I
Bi
will decrease as x̃I
k increases.
In order to maintain the nonnegativity of the variables, x̃I
k is increased until the first point at which some basic
variable x̃I
Br
drops to zero. In fact, we can increase x̃I
k until:
x̃I
k =
b̃
I
r
ỹI
rk
= min
(
b̃
I
i
ỹI
ik
; ỹI
ik ≻ 0, i = 1, .., m
)
(16)
and then, x̃I
Br
is the variable that leaves the basis and we call it the blocking variable, and (16) is called
the minimum ratio. In fact, the only purpose of finding the minimum ratio (16) is to determine the blocking
variable. Therefore, we can use the ranking function to facilitate the calculations, and the following ranked
minimum ratio is enough to achieve the purpose:
R[b̃
I
r]
R[ỹI
rk]
= min
(
R[b̃
I
i ]
R[ỹI
ik]
; ỹI
ik ≻ 0, i = 1, .., m
)
(17)
In the tableau format, we can change the basis using the elementary row operations, which are known
to maintain an equivalent problem, such that x̃I
k enters the basis and x̃I
Br
leaves it. One possible case still needs
to be discussed, that is when ỹI
k ⪯ 0, i.e. the ranking values of all its elements are less than or equivalent to
zero. In this case, there is no blocking variable, and the value of x̃I
k can be increased indefinitely giving always
a better objective value without violating any of the constraints. Thus, the problem is unbounded.
Trying to update the solution gives:
x̃I
B = b̃
I
⊖ (ỹI
k ⊗ x̃I
k)
x̃I
N = ẽI
k ⊗ x̃I
k
where ẽI
k is a (l − m)-vector with zero entries, except 1 in the k-th position. Then the new solution becomes:

x̃I
B
x̃I
N

≈

B̃I(−1)
b̃
I
0̃I
#
⊕

−ỹI
k
ẽI
k

⊗ x̃I
k
We call the vector ˜
dI
=

−ỹI
k
ẽI
k

the direction of unboundedness, which satisfies:

− ˜
dI
⪰ 0 clearly.
− c̃I ˜
dI
≺ 0 since: c̃I ˜
dI
= [c̃I
B c̃I
N ] ˜
dI
= −c̃I
BỹI
k ⊕ c̃I
k = −z̃I
k ⊕ c̃I
k ≺ 0
− ÃI ˜
dI
≈ 0 since:
ÃI
⊗ ˜
dI
= [B̃I
ÑI
] ⊗

−ỹI
k
ẽI
k

= (B̃I
⊗ (−ỹI
k)) ⊕ (ÑI
⊗ ẽI
k)
= (B̃I
⊗ (−B̃I(−1)
⊗ ãI
k)) ⊕ ãI
k ≈ (−˜
II
⊗ ãI
k) ⊕ ãI
k ≈ −ãI
k ⊕ ãI
k ≈ 0
In fact, having a vector ˜
dI
satisfying these three properties is a necessary and sufficient condition for
the unboundedness of the problem. To show this, let ˜
dI
be a l-vector satisfying: ÃI ˜
dI
≈ 0, c̃I ˜
dI
≺ 0 and
˜
dI
⪰ 0, and suppose that the problem is bounded, i.e. it has an optimal solution x̃I∗
with the optimal objective
value z̃I∗
= c̃I
x̃I∗
. Now let x̃I′
= x̃I∗
⊕ ˜
dI
this maintains the nonnegativity of the variables since ˜
dI
⪰ 0 and
it is a feasible solution since:
ÃI
x̃I′
= ÃI
⊗ (x̃I∗
⊕ ˜
dI
) = ÃI
x̃I∗
⊕ ÃI ˜
dI
≈ ÃI
x̃I∗
≈ b̃I
since x̃I∗
is feasible. On the other hand, the objective value at x̃I′
is:
z̃I′
= c̃I
⊗ (x̃I∗
⊕ ˜
dI
) = c̃I
x̃I∗
⊕ c̃I ˜
dI
≺ c̃I
x̃I∗
= z̃I∗
which means the x̃I′
gives a better objective value and is a better solution, which contradicts with the assump-
tion that x̃I∗
is the optimal solution.
4.1. Summarizing the modified simplex algorithm
To summarize, assuming a minimization problem, the simplex algorithm proceeds as follows:
Step 1: Let z̃I
k ⊖ c̃I
k = max{z̃I
j ⊖ c̃I
j : j ∈ J}. If z̃I
k ⊖ c̃I
k ⪯ 0̃I
, then stop; the current solution is optimal.
Otherwise, proceed to the next step.
Step 2: If ỹI
k ⪯ 0̃I
, then stop; the solution is unbounded. Otherwise, proceed.
Step 3: Let
R[b̃
I
r]
R[ỹI
rk]
= min{
R[b̃
I
i ]
R[ỹI
ik]
; ỹI
ik ≻ 0, i = 1, .., m}, then x̃I
k enters the basis and x̃I
Br
leaves it, so pivot
at ỹI
rk as follows:
− Multiply row r by the multiplicative inverse of ỹI
rk.
− Update the other rows using the following elementary row operations: R′
0 = (−(z̃I
k ⊖ c̃I
k) ⊗ Rr) ⊕ R0,
R′
i = (−ỹI
ik ⊗ Rr) ⊕ Ri, i ∈ {1, .., m}{r}, where R0 is row zero (i.e. the objective row), and Ri is row
number i in the tableau after the objective row.
Then, update the set of indices of the non-basic variables J, and go to Step 1.
Remark 1. Note that the choice of the pivoting location depends on the ranking values, and the pivoting
steps consist of elementary row operations which depend on the arithmetic operations on the n−IPFN that
preserve the ranking values. This results in the following important property: Suppose that we replace each
intuitionistic fuzzy number in problem (9) by its ranking value, and replace the intuitionistic fuzzy variables
with crisp (unfuzzy) ones. Then we get the crisp linear programming problem:
min z = cx
s.t. Ax = b
x ≥ 0 (18)
We call this problem the ranked linear programming (RLP) problem. It is clear that the values of the variables
in the optimal solution of problem (18) equal the ranking values of the fuzzy variables in the optimal solution
of problem (9). Moreover, the steps of solving the original problem are equivalent to the steps of solving the
corresponding RLP problem in number and order, which means that the fuzzy simplex method proposed in this
paper terminates in a finite number of iterations.

2250 ❒ ISSN: 2088-8708
5. NUMERICAL EXAMPLES
The following example illustrates the above.
Example 1. Consider the following IFFLP problem:
min {(1, 2; 4, 5), (−1, 2; 5, 6)} ⊗ x̃I
1 ⊕ {(−6, −2.5; −1, 1), (−7.5, −4; 1, 3)} ⊗ x̃I
2
s.t. {(−4, −2; −0.5, 2), (−5, −3.5; 1, 4)} ⊗ x̃I
1 ⊕ {(2, 3; 3, 5), (0, 2; 3, 6)} ⊗ x̃I
2 ⪯ {(4, 5; 6, 9), (2, 4; 8, 10)}
{(1, 3; 5, 6), (0, 3; 6, 8)} ⊗ x̃I
1 ⊕ {(−1, 1; 3, 6), (−3, 0; 4, 6)} ⊗ x̃I
2 ⪯ {(4, 6; 9, 10), (1, 5; 10, 11)}
x̃I
1, x̃I
2 ⪰ 0̃I
(19)
Adding the slack variables, Table 2 gives the first simplex tableau:
Table 2. The first tableau for Example 1
x̃I
1 x̃I
2 ỹI
1 ỹI
2 RHS
z̃I
{(−5, −4; −2, −1), (−6, −5; −2, 1)} {(−1, 1; 2.5, 6), (−3, −1; 4, 7.5)} 0 0 0
RV −3 2
ỹI
1 {(−4, −2; −0.5, 2), (−5, −3.5; 1, 4)} {(2, 3; 3, 5), (0, 2; 3, 6)} 1 0 {(4, 5; 6, 9), (2, 4; 8, 10)}
RV −1 ▷3◁ 6
ỹI
2 {(1, 3; 5, 6), (0, 3; 6, 8)} {(−1, 1; 3, 6), (−3, 0; 4, 6)} 0 1 {(4, 6; 9, 10), (1, 5; 10, 11)}
RV 4 2 7
where the ranking value (RV) of each intuitionistic fuzzy number is written below it.
z̃I
k ⊖ c̃I
k = max{{(−5, −4; −2, −1), (−6, −5; −2, 1)}, {(−1, 1; 2.5, 6), (−3, −1; 4, 7.5)}), 0, 0}
= {(−1, 1; 2.5, 6), (−3, −1; 4, 7.5)} = z̃I
2 ⊖ c̃I
2 ≻ 0̃,
thus the current solution is not optimal and the non basic variable z̃I
2 enters the basis. Using the ranked mini-
mum ratio test (17), we find:
R[b̃
I
r]
R[ỹI
rk]
= min{
6
3
,
7
2
} =
R[b̃
I
1]
R[ỹI
12]
= 2
Therefore, ỹI
1 leaves the basis and we pivot at {(2, 3; 3, 5), (0, 2; 3, 6)} by performing the elementary row
operations:
R1 ←{(2, 3; 3, 5), (0, 2; 3, 6)}−1
⊗ R1
R0 ← − {(−1, 1; 2.5, 6), (−3, −1; 4, 7.5)} ⊗ R1 ⊕ R0
R2 ← − {(−1, 1; 3, 6), (−3, 0; 4, 6)}) ⊗ R1 ⊕ R2
This gives the second simplex tableau in Table 3:
Table 3. The final tableau for Example 1
x̃1 x̃2 ỹ1 ỹ2 RHS
z̃ {(− 40
3 , − 29
6 ; 1
6 , 20
3 ), {(−10, − 3
2 ; 3
2 , 10), {(− 20
3 , − 7
6 ; 1
3 , 10
3 ), {(−6, −0.5; 1, 4), {(−12, − 11
2 ; −3, 3),
(− 107
6 , − 53
6 ; 14
3 , 44
3 )} (− 33
2 , −6; 6, 33
2 )} (− 55
6 , − 8
3 ; 10
3 , 22
3 )} (−8.5, −2; 4, 8)} (− 33
2 ; −8, 2, 8)}
RV − 7
3 0 − 2
3 0 −4
x̃2 {(− 16
3 , − 4
3 ; 1
6 , 11
3 ), {(−2, 1; 1, 4), {(− 5
3 , 1
3 ; 1
3 , 4
3 ), {(−2, 0; 0, 1), {(−2, 1; 2, 6),
(− 22
3 , − 17
6 ; 8
3 , 23
3 )} (−5, 0; 2, 7)} (− 8
3 , 1
3 ; 4
3 , 10
3 )} (−3, 0; 1, 3)} (−5, 0; 5, 9)}
RV − 1
3 1 1
3 0 2
ỹ2 {(− 22
3 , 5
3 ; 43
6 , 41
3 ), {(−10, −2; 2, 10), {(− 20
3 , − 5
3 ; 1
3 , 10
3 ), {(−5, 0; 2, 5), {(−8, 0; 6, 13),
(− 31
3 , − 5
6 ; 35
3 , 65
3 )} (−15, −5; 5, 15)} (− 23
3 , − 8
3 ; 7
3 , 22
3 )} (−6, −1; 4, 9)} (−14, −3; 11, 19)}
RV 14
3 0 − 2
3 1 3
z̃k ⊖c̃k = 0, thus the solution is optimal. The optimal solution for the problem is: x̃∗
1 = 0, x̃∗
2 = {(−2, 1; 2, 6),
(−5, 0; 5, 9)} with the fuzzy objective value z̃∗
= {(−12, −11
2 ; −3, 3), (−33
2 ; −8, 2, 8)}. Now, if we solve the
RLP problem for problem (19), which is:

min 3x1 − 2x2
s.t. −x1 + 3x2 ≤ 6
4x1 + 2x2 ≤ 7
x1, x2 ≥ 0
we find that the optimal solution is: x∗
1 = 0, x∗
2 = 2 with the optimal objective value z∗
= −4. As expected,
we have x∗
1 = R(x̃∗
1), x∗
2 = R(x̃∗
2) and z∗
= R(z̃∗
).
The next example is an unbounded IFFLP with 3−IPFN’s.
Example 2. Consider the following problem:
max {(−3, −1, 1, 2; 2, 3, 5, 7), (−4, −1.5, 0.5, 2; 3, 3.5, 5, 8)} ⊗ x̃I
1⊕
{(−10, −7, −6, −4; −2, 0, 1, 4), (−12, −9, −6, −4.5; −1, 1, 2, 5.5)} ⊗ x̃I
2
s.t. {(−5, −4, −3, −2; −1, 2, 2.5, 3), (−8, −5, −3, −2; 0, 1, 3, 4)} ⊗ x̃I
1⊕
{(−4, −3, −1, 0; 2, 3, 5, 6), (−7, −5, −2, 0; 3, 5, 6, 8)} ⊗ x̃I
2 ⪯
{(−2, 0, 1, 2; 2, 3, 4, 6), (−6, −2, −1, 1; 3, 5, 7, 8)}
{(−4, −3, −2, −1; −1, 0, 1, 2), (−10, −5, −3, −2; 1, 2, 3, 5)} ⊗ x̃I
1⊕
{(−3, −1, 1, 2; 2, 3, 5, 7), (−6, −3, −2, 0; 4, 5, 7, 12)} ⊗ x̃I
2 ⪰
{(−6, −5, −4, −2; −2, 0, 1, 2), (−18, −13, −8, −5; −1, 5, 9, 14)}
x̃I
1, x̃I
2 ⪰ 0̃I
(20)
Note that this is a maximization problem, so we multiply row 1 by −1 to convert to a minimization
problem and when we get an optimal solution, we multiply the objective value by −1 to get the actual optimal
value for the maximization problem. Note also that the second constraint is a ⪰ type inequality, so we multiply
by −1. Now, we add slack variables for both constraints to get the the first simplex tableau in Table 4. Since
z̃I
k ⊖ c̃I
k = (−3, −1, 1, 2; 2, 3, 5, 7), (−4, −1.5, 0.5, 2; 3, 3.5, 5, 8)} = z̃I
1 ⊖ c̃I
1 ≻ 0̃, the optimal solution is not
reached. Using the minimum ratio test, the next iteration is by pivoting at {(−2, −1, 0, 1; 1, 2, 3, 4), (−5, −3
, −2, −1; 2, 3, 5, 10)} in the last row, which gives the tableau in Table 5. It is clear that the variable x̃2 tries to
enter the basis. However,
ỹ2 =

ỹ32
ỹ12

=

{(−11, −8, −4, −2; 0, 2, 6, 9), (−19, −12, −7, −4; 3, 7, 9, 14)}
{(−7, −5, −3, −2; −2, −1, 1, 3), (−12, −7, −5, −4; 0, 2, 3, 6)}

⪯ 0̃
Therefore, the problem is unbounded with the direction of unboundedness:
˜
d =




−ỹ12
1
−ỹ32
0



 =




{(−3, −1, 1, 2; 2, 3, 5, 7), (−6, −3, −2, 0; 4, 5, 7, 12)}
{(1, 1, 1, 1; 1, 1, 1, 1), (1, 1, 1, 1; 1, 1, 1, 1)}
{(−9, −6, −2, 0, 2, 4, 8, 11), (−14, −9, −7, −3; 4, 7, 12, 19)}
{(0, 0, 0, 0; 0, 0, 0, 0), (0, 0, 0, 0; 0, 0, 0, 0)}




Table 4. The first tableau for Example 2
x̃1 x̃2 x̃3 x̃4 RHS
z̃
{(−3, −1, 1, 2; 2, 3, 5, 7),
(−4, −1.5, 0.5, 2; 3, 3.5, 5, 8)}
{(−10, −7, −6, −4; −2, 0, 1, 4),
(−12, −9, −6, −4.5; −1, 1, 2, 5.5)}
0 0 0
RV 2 −3
x̃3
{(−5, −4, −3, −2; −1, 2, 2.5, 3),
(−8, −5, −3, −2; 0, 1, 3, 4)}
{(−4, −3, −1, 0; 2, 3, 5, 6),
(−7, −5, −2, 0; 3, 5, 6, 8)}
1 0
{(−2, 0, 1, 2; 2, 3, 4, 6),
(−6, −2, −1, 1; 3, 5, 7, 8)}
RV −1 1 2
x̃4
{(−2, −1, 0, 1; 1, 2, 3, 4),
(−5, −3, −2, −1; 2, 3, 5, 10)}
{(−7, −5, −3, −2; −2, −1, 1, 3),
(−12, −7, −5, −4; 0, 2, 3, 6)}
0 1
{(−2, −1, 0, 2; 2, 4, 5, 6),
(−14, −9, −5, 1; 5, 8, 13, 18)}
RV ▷1◁ −2 2

2252 ❒ ISSN: 2088-8708
Table 5. The final tableau for Example 2
x̃1 x̃2 x̃3 x̃4 RHS
z̃
{(−11, −7, −3, 0;
0, 3, 7, 11),
(−24, −11.5, −5.5, −2;
5, 7.5, 11, 18)}
{(−16, −9, −4, 0;
2, 6, 11, 18),
(−24, −13, −6.5, −3.5;
5, 10.5, 14.5, 23.5)}
{(−5, −3, −1, 0;
0, 1, 3, 5),
(−6, −3, −1.5, −1;
0, 1.5, 3.5, 6)}
{(−7, −5, −3, −2;
−2, −1, 1, 3),
(−8, −5, −3.5, −3;
−2, −0.5, 1.5, 4)}
{(−13, −10, −7, −4;
−4, −1, 2, 5),
(−26, −18, −12.5, −6;
−1., 3.5, 10.5, 18)}
RV 0 1 0 −2 −4
x̃3
{(−7, −5, −3, −1;
−2, 2, 3.5, 5),
(−13, −8, −5, −3;
2, 4, 8, 14)}
{(−11, −8, −4, −2;
0, 2, 6, 9),
(−19, −12, −7, −4;
3, 7, 9, 14)}
1 1
{(−4, −2, 0, 4;
4, 8, 10, 12),
(−20, −11, −6, 2;
8, 13, 20, 26)}
RV 0 −1 4
x̃1
{(−2, −1, 0, 1;
1, 2, 3, 4),
(−5, −3, −2, −1;
2, 3, 5, 10)}
{(−7, −5, −3, −2;
−2, −1, 1, 3),
(−12, −7, −5, −4;
0, 2, 3, 6)}
0 1
{(−2, −1, 0, 2;
2, 4, 5, 6),
(−14, −9, −5, 1;
5, 8, 13, 18)}
RV 1 −2 2
6. CONCLUSION
We have considered the intuitionistic fuzzy linear programming (IFLP) with intuitionistic polygonal
fuzzy numbers (IPFN). Most of the previous work in the literature are special cases of the IPFN such as
triangular or rectangular IFN’s and most of them convert the problem into a crisp and then solve with the
traditional methods. The polygonal IFN’s considered in this paper is a generalization of these existing IFN’s.
We have discussed how to modify the simplex method to solve IFLP with IPN problems without converting
it to crisp. We showed how to use the simplex method in the tableau format and discuss the cases of the final
solution based on the optimality conditions. We implement the modified simplex method to solve two examples
one has exact solution and the other one has a unbounded solutions.
ACKNOWLEDGEMENT
This work was funded by the Deanship of Scientific Research in Jordan University of Science and
Technology under research project number 20180486.
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BIOGRAPHIES OF AUTHORS
Mahmoud H. Alrefaei is a professor of operations research at Jordan University of
Science and Technology (JUST), Jordan. He received his Ph.D. in mathematics and industrial
engineering from the University of Wisconsin-Madison, USA in 1997. He was a research assis-
tant at Georgia Institute of Science and Technology from 1995 to 1997. He joined the Math-
ematics and Statistics Department at JUST in 1997, and joined Qatar University from 2006 to
2012. His research interest includes simulated annealing; multi-objective simulation; supply chain
management; stochastic optimization; statistical selection and ordinal optimization. He has pub-
lished more than 60 publications in international reputable journals. Further info on his homepage:
http://www.just.edu.jo/eportfolio/Pages/Default.aspx?email=alrefaei. He can be contacted at email:
alrefaei@just.edu.jo.
Marwa Tuffaha received her M.Sc. degree in applied mathematics from Jordan University
of Science and Technology and is currently a Ph.D candidate in Applied Mathematics at the Univer-
sity of Western Ontario. Her research interests are in the field of mathematical biology, in specific,
population genetics. Her research interests includes databases programming languages, algorithms
logic and foundations of mathematics, applied mathematics and analysis. She can be contacted at
email: mtuffaha@uwo.ca.

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