417 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 

 
Generalized Heptagonal Membership Function for Fully Fuzzy  

Linear Fractional Programming Problems 

 

Israa H. Hasan1*   and Iden H. Al Kanani2   
1Department of Communication Engineering, University of Technology-Iraq. 

2Department of Mathematics, College of Science for Women University of Baghdad, Baghdad, Iraq. 

*Corresponding Author.  

 

 Received: 12 June 2023                         Accepted: 27 August 2023               Published: 20 January 2025 

 doi.org/10.30526/38.1.3596 

 
 Abstract   

         Identifying the optimum solution that satisfies the restrictions and maximizes or minimizes 

the objective function is the aim of fully fuzzy fractional programming (FFFLP). Due to the 

inclusion of both fuzzy parameters and fractional variables, this problem is difficult to solve. 

Several approaches, including linear programming, nonlinear programming, genetic algorithms, 

and computational intelligence algorithms, suggested handling FFFLP. For describing the 

uncertainty, vagueness, or imprecision of information in the real world, triangular and trapezoidal 

fuzzy numbers are frequently used. Indeed, it is not always practical to limit the membership 

function to a triangle or trapezoid. This paper submits a new type of Heptagonal fuzzy number 

and a novel ranking function technique based on generalized heptagonal membership functions 

suggested for ordering heptagonal fuzzy numbers. With the help of the algorithm of the simplex 

method for fully fuzzy simplex method, obtained the optimal fuzzy solution for FFFLP. 

Keywords: Fully Fuzzy Linear Fractional Programming, Heptagonal Fuzzy Number, Arithmetic 

Heptagonal Operations, Membership Function, Ranking function, Fully Fuzzy Simplex method. 

 

1. Introduction 

       Fuzzy problem solving relies heavily on the ranking function, which provides a methodical 

way to evaluate solutions when choosing an option. The fractional programming problem is a 

decision-making challenge involving the optimization of a constrained ratio of fuzzy 

programming problem Modern applications of the linear fractional programming problem found 

in a wide range of fields, including production planning, finance, healthcare, and other areas of 

engineering. 

Many papers use the features of fuzzy sets as a method to find the best solutions to fuzzy 

programming problems in which all variables are triangular or trapezoidal numbers (1,2). If a 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://doi.org/10.30526/38.1.3501
https://orcid.org/0000-0001-9289-0909
mailto:Israa.H.Hasn@uotechnology.edu.iq
https://orcid.org/0000-0002-6492-8196
mailto:idanha_math@csw.uobaghdad.edu.iq


IHJPAS. 2025, 38 (1) 
 

418 
 

fuzzy number has seven possible membership values; we say that it is heptagonal. The range of 

the heptagonal fuzzy number defined by a membership function that gives each value in the range 

a certain level of membership. The membership function for a heptagonal fuzzy number takes the 

shape of a triangle or trapezoid. (3-5) discussed a  form of fuzzy number named a Heptagonal 

fuzzy number and proposed a new version of the definition for the value and ambiguity of non-

normal fuzzy .For example , Namarta and others ; proposed a ranking method for heptagon fuzzy 

numbers based on area. The study involves the computation of the center of centroids for ranking 

heptagonal fuzzy numbers. 

Loganathan, T., (6, 7) proposed a method for solving FFFLP and all parameters and variables are 

triangular fuzzy numbers that preserves the fuzzy nature of the situation while maintaining their 

fuzzy characteristics. Using the close interval approximation of normalized heptagonal fuzzy 

numbers, which is one of the best interval approximations, Alharbi, M.G.; Khalifa, H.A. (8) 

attempted to solve the linear fractional programming problem with fully fuzzy normalized 

heptagonal fuzzy numbers. They transformed the original maximization (minimization) problem 

with an interval objective function into a multi-objective problem using order relations. 

 In order to rank triangular fuzzy numbers, Mitlif, R.J. (9,10) used a novel ranking function 

technique based on ordinary fuzzy numbers. The optimal solution found by first reducing the fuzzy 

fractional programming problem to a fractional programming problem and then solving it with the 

method. For solving FFLFP with trapezoidal fuzzy integers as the objective function and 

constraints, Gupta,D; Jain,P; Gupta,G.(11) presented a new trapezoidal fuzzy number ranking 

function, the proposed procedure is based on the simplex method and precise linear fractional 

programming. Triangular and trapezoidal fuzzy numbers and the values of the right-hand side 

represent the objective function and crisp numbers represents left-hand side constraints. A lots of 

researcher (12-18) submitted a fuzzy linear fractional programming problem and a new ranking 

function devised for converting the fuzzy linear fractional programming problem into an 

unambiguous one.  

This paper proposes a new ranking function strategy based on the new generalized heptagonal 

membership function in order to solve an FFFLP by first transforming the FFFLP into a 

completely fuzzy linear problem. Finally, the optimal fuzzy solution can reached by fully fuzzy 

simplex method in which all the input variables are heptagonal fuzzy numbers. This paper 

constructed into eight sections. In section 2, the preliminary of the fuzzy set theory. Section 3 

proposes a generalized heptagonal fuzzy function, (𝜎-cut) function, and derived the proposed 

ranking function. In section 4, the fuzzy mathematical operations of heptagonal fuzzy numbers. 

Section 5 shows the mathematical model of FFFLP. The algorithm of the fully fuzzy simplex 

method is in section 6; a numerical example be given in section 7, and section 8 concludes the 

paper.  

 

 

 

 

 



IHJPAS. 2025, 38 (1) 
 

419 
 

2. Preliminary. 

Definition 2.1: (19)  

Let 𝑋 be a real set. The fuzzy set �̃� is defined by the membership function ℳ�̃�(𝑥), ℳ�̃�(𝑥): 𝒳→ 

[0, 1], is the degree of membership of 𝑥 ∈  𝒳 in the set �̃� and is denoted by  �̃�(𝑥) =

{(𝑥,ℳ�̃�(𝑥))| 𝑥 ∈  𝒳}.  

Definition 2.2: (19,20) 

A fuzzy number �̃� is a set whose membership function ℳ�̃�(𝑥) satisfies the following conditions: 

• �̃�   a normal fuzzy set if there exists at least one 𝑥0 in ℛ with ℳ�̃�(𝑥) = 1.  

• ℳ�̃�(𝑥) Piecewise continuous. 

• �̃�(𝑥) Convex if  ℳ�̃�(𝑥). [ℴ𝑥1 + (1 − ℴ) 𝑥2] ≥ Min ( ℳ�̃�(𝑥1), ℳ�̃� (𝑥2)}, 𝑥1, 𝑥2 ∈ 𝑋, ℴ ∈ [0, 

1]. 

Definition 2.3:  A Heptagonal Fuzzy Number (HEP) 

Is a fuzzy number that has seven membership values on the interval [0, 1] that determine the 

membership function of a heptagonal fuzzy number. The number of these dots stands for how much 

of a certain element belongs to the fuzzy set. 

The range of values for which the membership function is non-zero is called the support of a 

heptagonal fuzzy number. A heptagonal fuzzy number's center is the set of values where the 

membership function is one .The maximum value of the membership function defines the height of 

a heptagonal fuzzy number. The centroid of a heptagonal fuzzy number calculated by taking the 

weighted average of its seven membership values, with each value assigned a weight based on the 

degree to which it belongs to the heptagon. 

  

3 .Propose Generalized Heptagonal Membership Function. 

  Membership functions play a crucial role in finding solutions to fuzzy problems because of their 

capacity to accurately reflect the inherent vagueness and imprecision of real-world information. In 

this section, propose a nonlinear membership function ℳ𝒜𝐻𝐸�̃�
(𝑥)  of a Heptagonal fuzzy number.    

 𝒜𝐻𝐸�̃� = ( 𝛼1, 𝛼2, 𝛼3, 𝛼4, 𝛼5, 𝛼6, 𝛼7; 𝑘1, 𝑘2, 𝜔), Where  𝛼1 ≤ 𝛼2 ≤ 𝛼3 ≤ 𝛼4 ≤ 𝛼5 ≤ 𝛼6 ≤ 𝛼7 ∈

 ℛ, 𝑘1, 𝑘2 ∈ [0,1] and 0< 𝑘1 < 𝑘2 < 𝜔, 0 < 𝜔 ≤ 1: 



IHJPAS. 2025, 38 (1) 
 

420 
 

ℳ𝒜𝐻𝐸�̃�
(𝑥) =

{
 
 
 
 
 
 
 

 
 
 
 
 
 
 

0     𝑥 < 𝛼1

(𝑘1 (
𝑥 − 𝛼1
𝛼2 − 𝛼1

))1/3 𝛼1 ≤ 𝑥 < 𝛼2

 ( 𝑘1 + (𝑘2 − 𝑘1) (
𝑥 − 𝛼2
𝛼3 − 𝛼2

)) 1/3   𝛼2 ≤ 𝑥 < 𝛼3

(𝑘2 + (𝜔 − 𝑘2) (
𝑥 − 𝛼3
𝛼4 − 𝛼3

))1/3       𝛼3 ≤ 𝑥 < 𝛼4

(𝜔 + (𝑘2 − 𝜔) (
𝑥 − 𝛼4
𝛼5 − 𝛼4

)) 1/3       𝛼4 ≤ 𝑥 < 𝛼5

(𝑘2 − (𝑘2 − 𝑘1) (
𝑥 − 𝛼5
𝛼6 − 𝛼5

))1/3       𝛼5 ≤ 𝑥 < 𝛼6

(𝑘1 − 𝑘1 (
𝑥 − 𝛼6
𝛼7 − 𝛼6

))1/3

0

𝛼6 ≤ 𝑥 ≤ 𝛼7

𝑥 > 𝛼7

 

 

 

3.1. The (𝓸 − 𝒄𝒖𝒕) Function  

The ℴ − 𝑐𝑢𝑡 function is a mathematical function used in fuzzy logic to define a subset of a fuzzy 

set. It is a way of defining the degree of membership of an element in a fuzzy set .in this section 

constructs the (ℴ − 𝑐𝑢𝑡) function of a heptagonal fuzzy number as follows:  

𝒜𝐻𝐸𝑃𝓸
̃ =

{
 
 
 
 
 
 
 

 
 
 
 
 
 
 𝛼1 +

ℴ3

𝑘1
(𝛼2 − 𝛼1)           ℴ ∈ [0, 𝑘1]

𝛼2 + (
ℴ3 − 𝑘1
𝑘2 − 𝑘1

) (𝛼3 − 𝛼2) ℴ ∈ (𝑘1, 𝑘2]

𝛼3 + (
ℴ3 − 𝑘2
𝜔 − 𝑘2

)(𝛼4 − 𝛼3) ℴ ∈ (𝑘2, 𝜔]

𝛼4 + (
ℴ3 − 𝜔

𝑘2 − 𝜔
) (𝛼5 − 𝛼4) ℴ ∈ (𝑘2, 𝜔]

𝛼5 + (
ℴ3 − 𝑘2
𝑘1 − 𝑘2

) (𝛼6 − 𝛼5) ℴ ∈ (𝑘1, 𝑘2]

𝛼6 + (1 −
ℴ3

𝑘1
) (𝛼7 − 𝛼6) ℴ ∈ [0, 𝑘1]

 

(𝑖𝑛𝑓1𝒜𝐻𝐸𝑃𝓸
 ̃ , 𝑠𝑢𝑝3𝒜𝐻𝐸𝑃𝓸

̃ )= ([𝛼1 +
ℴ3

𝑘1
(𝛼2 − 𝛼1)], [𝛼6 + (1 −

ℴ3

𝑘1
) (𝛼7 − 𝛼6)]),ℴ ∈ [0, 𝑘1] 

(𝑖𝑛𝑓2𝒜𝐻𝐸𝑃𝓸
 ̃ , 𝑠𝑢𝑝2𝒜𝐻𝐸𝑃𝓸

̃ )= ([𝛼2 + (
ℴ3−𝑘1

𝑘2−𝑘1
) (𝛼3 − 𝛼2)] , [𝛼5 + (

ℴ3−𝑘2

𝑘1−𝑘2
) (𝛼6 − 𝛼5)]) , ℴ ∈

(𝑘1, 𝑘2]  

(𝑖𝑛𝑓3𝒜𝐻𝐸𝑃𝓸
 ̃ , 𝑠𝑢𝑝1𝒜𝐻𝐸𝑃𝓸

̃ )= ([𝛼3 + (
ℴ3−𝑘2

𝜔−𝑘2
)(𝛼4 − 𝛼3)],[ 𝛼4 + (

ℴ3−𝜔

𝑘2−𝜔
) (𝛼5 − 𝛼4)] , ℴ ∈

(𝑘2, 𝜔] 
 



IHJPAS. 2025, 38 (1) 
 

421 
 

3.2 Ranking Function 

Fuzzy problem solving depends significantly on ranking functions, which help evaluate and 

prioritize possible solutions. The ranking function allows alternatives to be evaluated according to 

their level of membership in a set or category, which is particularly useful in a fuzzy environment 

when there is uncertainty and imprecision in the data. This section proposed a novel ranking 

function depending on the new nonlinear heptagonal membership function as shown: 

Let  𝒜𝐻𝐸�̃� = ( 𝛼1, 𝛼2, 𝛼3, 𝛼4, 𝛼5, 𝛼6, 𝛼7; 𝑘1, 𝑘2, 𝜔  ) , depending on the following function: 

ℜ(𝒜𝐻𝐸�̃�) = (
1

2
) ∫ (𝑖𝑛𝑓𝑖𝒜𝐻𝐸𝑃𝓸

̃ +sup𝑗𝒜𝐻𝐸𝑃𝓸
̃ )𝑑ℴ  

𝜔

0
  𝑖 = 1,2,3     𝑗 = {

 𝑖 + 2 𝑖 = 1
𝑖 𝑖 = 2

𝑖 − 2 𝑖 = 3
 

Suppose that  ℜ(𝒜𝐻𝐸�̃�) =
1

2
∗ (𝐹1+ 𝐹2 + 𝐹3)                                                                              (1) 

Where;  

𝐹1 = ∫ ([𝛼1 + (
ℴ3

𝑘1
) (𝛼2 − 𝛼1)] +  [𝛼6 + (1 −

ℴ3

𝑘1
) (𝛼7 − 𝛼6)] )

𝑘1

0

 𝑑ℴ 

 ∴   𝐹1   =
1

4
[ 𝑘1

3(𝛼2 − 𝛼1 + 𝛼6 − 𝛼7) + 𝑘1(4𝛼1 + 4𝛼7)]                                                           (2) 

𝐹2 = ∫ ([𝛼2 + (
ℴ3 − 𝑘1
𝑘2 − 𝑘1

) (𝛼3 − 𝛼2)]
𝑘2

𝑘1

+ [𝛼5 + (
ℴ3 − 𝑘2
𝑘1 − 𝑘2

) (𝛼6 − 𝛼5)]) 𝑑ℴ 

     =
 𝑘1

3

4
 (𝛼3 − 𝛼2+𝛼5 − 𝑎6) ++ 

𝑘1
2𝑘2 

4
 (𝛼3 − 𝛼2+𝛼5 − 𝛼6) ++ 

𝑘2
2𝑘1

4
(𝛼3 − 𝛼2+𝛼5 −

𝛼6)−𝑘1(𝛼3+𝛼5) + 
𝑘2

3

4
(𝛼3 − 𝛼2+𝛼5 − 𝛼6) +𝑘2 (𝛼2 + 𝛼6)                                                (3)  

𝐹3 = ∫ ([𝛼3 + (
ℴ3 − 𝑘2
𝜔 − 𝑘2

) (𝛼4 − 𝛼3)]
𝜔

𝑘2

+ [𝛼4 + (
ℴ3 − 𝜔

𝑘2 − 𝜔
) (𝛼5 − 𝛼4)])𝑑ℴ 

     =
𝑘2

3

4
(2𝛼4 − 𝛼3 − 𝛼5)+

𝑘2
2𝜔

4
(2𝛼4 − 𝛼3 − 𝛼5)+ 

𝑘2𝜔
2

4
(2𝛼4 − 𝛼3 − 𝛼5)−2𝛼4𝑘2 +

𝜔3

4
(2𝛼4 −

𝛼3 − 𝛼5)+ 𝜔(𝛼3 + 𝛼5).                                                                                                        (4) 

Now substitute equations (2), (3), (4) in eq. (1): 

∴  𝕽(𝓐𝑯𝑬�̃�) =
1

8
[𝑘1

3
(𝛼3 − 𝛼1 + 𝛼5 – 𝛼7) + 𝑘1

2𝑘2(𝛼3 – 𝛼2+𝛼5 − 𝑎6) +

 𝑘2
2𝑘1 (𝛼3 – 𝛼2+𝛼5 − 𝑎6) +

𝑘1

4
 (𝛼1 – 𝛼3– 𝛼5  + 𝛼7) + 𝑘2

3(2𝛼4  − 𝛼2 − 𝑎6) +

 𝑘2
2𝜔(2𝛼4 − 𝛼3 − 𝛼5) + 𝑘2𝜔

2 (2𝛼4 − 𝛼3 − 𝛼5) + 𝑘2(4𝛼2 −  8𝛼4 + 4𝑎6) + 𝜔
3(2𝛼4 −

𝛼3 − 𝛼5) + 𝜔(4𝛼3  + 4𝛼5)] 

 0 <  𝑘1 < 𝑘2 < 𝜔 , 0 < 𝜔 ≤ 1 

4. Fuzzy Mathematical Operations of Generalized Heptagonal Fuzzy Numbers  (3,4) 

       Let 𝒜𝐻𝐸�̃�  𝑎𝑛𝑑   ℬ𝐻𝐸�̃� be two arbitrary generalized Heptagonal fuzzy numbers, such that 

𝒜𝐻𝐸�̃� = (𝛼1, 𝛼2, 𝛼3, … , 𝛼7; 𝑘1, 𝑙1, 𝜔1) , ℬ𝐻𝐸�̃� = (𝑏1, 𝑏2, 𝑏3, … , 𝑏7; 𝑘2, 𝑙2, 𝜔2) 

 Define the Addition, Subtraction, and multiplication operations [10] as follows: 

• 𝒜𝐻𝐸�̃�⨁ℬ𝐻𝐸�̃� = (𝛼1 + 𝑏1, 𝛼2 + 𝑏2, 𝛼3 + 𝑏3, … , 𝛼7 + 𝑏7;min(𝑘1, 𝑘2) ,min(𝑙1, 𝑙2) ,min (𝜔1, 𝜔2)) 
• 𝒜𝐻𝐸�̃�⊖ℬ𝐻𝐸�̃� = 

(𝛼1 − 𝑏7, 𝛼2 − 𝑏6, 𝛼3 − 𝑏5, … , 𝛼6 − 𝑏2, 𝛼7 − 𝑏1;min(𝑘1, 𝑘2) ,min(𝑙1, 𝑙2) ,min (𝜔1, 𝜔2)) 

•  𝒜𝐻𝐸�̃�⨂ℬ𝐻𝐸�̃�  = (𝛼1 ∗ 𝑏1, 𝛼2 ∗ 𝑏2, … , , 𝛼7 ∗ 𝑏7; min(𝑘1, 𝑘2) , min(𝑙1, 𝑙2) , min (𝜔1, 𝜔2))  



IHJPAS. 2025, 38 (1) 
 

422 
 

•   ℷ⨂𝒜𝐻𝐸�̃�         =  (ℷ𝛼1, ℷ𝛼2, ℷ𝛼3, … , ℷ𝛼7; 𝑘1, 𝑙1, 𝜔1)                          𝑖𝑓  ℷ > 0 

                                = (ℷ𝛼7, ℷ𝛼6, ℷ𝛼5, ℷ𝛼4, ℷ𝛼3, ℷ𝛼2, ℷ𝛼1; 𝑘1, 𝑙1, 𝜔1)       𝑖𝑓  ℷ < 0 

 

5. Fully Fuzzy Linear Fractional Programming Problem (FFLFPP) (9),(20) 

      Consider the following FFLFP problem having m fuzzy constraints and n fuzzy variables: 

𝑀𝑎𝑥 �̃� ≅
�̃�𝑇�̃�⨁�̃�

�̃�𝑇�̃�⨁�̃�
=

�̃�(𝑥)

�̃�(𝑥)
  

𝑆. 𝑡𝑜 

                �̃�⨂�̃� ≤=≥ �̃�  ,                 �̃� ≥ 0̃ 

where,  

�̃�𝑇 = (�̃�𝑗)1∗𝑛  , �̃�
𝑇=(�̃�𝑗)1∗𝑛 ,  �̃� = (�̃�𝑗)𝑛∗1,  𝛽, 𝛿 ∈ heptagonal fuzzy numbers. 

�̃� = (�̃�𝑖𝑗)𝑚∗𝑛 , �̃�=(�̃�𝑖)𝑚∗1. 

�̃�𝑇 , �̃�𝑇, �̃�, �̃�, �̃� are heptagonal fuzzy numbers       ∀ 1 ≤ 𝑗 ≤ 𝑛,    1 ≤ 𝑖 ≤ 𝑚 

 

6. The Algorithm of Fully Fuzzy Simplex (FFS) Method.  

Step 1: Use the development complementary method (21,22) to convert the FFLFPP into FFLPP.  

Step 2: Adding fuzzy slack variables �̃�𝑖, 𝑖 = 1, 2, … ,𝑚 to convert all the inequalities of the 

constraints into equations. 

Step 3: Construct the fully fuzzy simplex tableau as shown in Table 1.: 
 

Table 1. FFS Method Tableau 

Basic var. �̃�1 �̃�2 …  �̃�𝑛 �̃�1 �̃�2 … �̃�𝑛 R.H.S ℜ(𝑅.𝐻. 𝑆) 

(�̃� ⊖ �̃�𝑗) −�̃�1 −�̃�2 … − �̃�𝑛 (0,0,…,0) (0,0,…,0) … (0,0,…,0) 𝛽𝑛 ℜ(𝛽𝑛) 

�̃�𝟏 �̃�11 �̃�12 … �̃�1𝑛 (0,0,...,1) (0,0,…,0) … (0,0,…,0) �̃�1 ℜ(�̃�1) 

�̃�𝟐 �̃�21 �̃�22 … �̃�2𝑛 (0,0,…,0) (0,0,…,1) … (0,0,…,0) �̃�2 ℜ(�̃�2) 

⋮ ⋮ ⋮ … ⋮ ⋮ … ⋮ ⋮ ⋮ 

�̃�𝒎 �̃�𝑚1 �̃�𝑚2 … �̃�𝑚𝑛 (0,0,…,0) (0,0,…,0) … (0,0,…,1) �̃�𝑚 ℜ(�̃�𝑚) 
 

Step 4: In the maximization problem, select the most negative value of ℜ(�̃� ⊖ �̃�𝑗) as a fuzzy 

entering variable, and if the problem is minimum, the most positive value of the ℜ(�̃� ⊖ �̃�𝑗)  

represents the entering fuzzy variable. 

Step 5: Determine the fuzzy variable that leaves the basic solution by: 

𝜃 = min {
𝕽(�̃�𝑖)

𝕽(𝑡ℎ𝑒 𝑐𝑜𝑙𝑢𝑚𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒  )
} .   𝑖 = 1,2… ,𝑚 

Step 6: Use the arithmetic operations of the heptagonal fuzzy number to get the new fuzzy tableau 

(new iteration) of a fully fuzzy simplex table. 

Step 7: Repeat the steps even to get the optimal solution. 

Step 8: The optimal fuzzy solution of the maximum problem is reached, when (  ℜ(�̃� ⊖ �̃�𝑗) ≥

0 ), and at the minimum problem ( ℜ(�̃� ⊖ �̃�𝑗) ≤ 0 ).  
 

7. Numerical Results 

    The following fractional linear programming problem paper (23):  

 𝑀𝑎𝑥 𝜔 =
4̃�̃�1+6̃�̃�2−2̃

6̃�̃�1+9̃�̃�2+3̃
 



IHJPAS. 2025, 38 (1) 
 

423 
 

𝑠. 𝑡          �̃�1+3̃�̃�2 ≤ 5̃ 

               2̃�̃�1+�̃�2 ≤ 2̃ 

               �̃�1, �̃�2 ≥ 0 

Now, take the above example with all the variables are heptagonal fuzzy numbers: 

 

𝑀𝑎𝑥 �̃� =
(−2,0,2,4,6,8,10;𝜔1)⨂�̃�1 ⨁(0,2,4,6,8,10,12;𝜔1)⨂�̃�2⊖(−4,−2,0,2,4,6,8;𝜔1)

(0,2,4,6,8,10,12;𝜔2)⨂�̃�1 ⨁(3,5,7,9,11,13,15;𝜔2)⨂�̃�2⨁(−3,−1,1,3,5,7,9;𝜔2)
     

      𝑠. 𝑡 

(−5,−3,−1,1,3,5,7; 𝜔1)⨂�̃�1⨁(−3,−1,1,3,5,7,9;𝜔1)⨂�̃�2 ≤ (−1,1,3,5,7,9,11; 𝜔1)  

(−4,−2,0,2,4,6,8;𝜔2)⨂�̃�1⨁(−5,−3,−1,1,3,5,7;𝜔2)⨂�̃�2 ≤ (−4,−2,0,2,4,6,8; 𝜔2)  

                                  �̃�1, �̃�2 ≥ 0  ,  𝜔1, 𝜔2 are the weighted of fuzzy numbers. 

 

Applying the proposed algorithm, the first step uses the development complementary method to 

convert the problem to (FFLPP) problem. 

Let  𝑀𝑎𝑥 �̃� =
𝑀𝑖𝑛 𝑤1

𝑀𝑎𝑥 𝑤2
             𝑤ℎ𝑒𝑟𝑒,    

  𝑀𝑖𝑛 𝑤1 = (−2,0,2,4,6,8,10; 𝜔1)⨂�̃�1 ⨁(0,2,4,6,8,10,12;𝜔1)⨂�̃�2⊖ (−4,−2,0,2,4,6,8; 𝜔1) 

 𝑀𝑎𝑥 𝑤2 = (0,2,4,6,8,10,12;𝜔2)⨂�̃�1 ⨁(3,5,7,9,11,13; 𝜔2)⨂�̃�2⨁(−3,−1,1,3,5,7,9;𝜔2) 

 

Since  𝑀𝑎𝑥 𝑤1 = 𝑀𝑖𝑛(−𝑤1)  

 ∴ 𝑀𝑎𝑥  𝑤1 =⊖ [(−2,0,2,4,6,8,10; 𝜔1)⨂�̃�1 ⨁(0,2,4,6,8,10,12;𝜔1)⨂�̃�2⊖

(−4,−2,0,2,4,6,8; 𝜔1)] 

 = [(−10,−8,−6,−4,−2,0,2;𝜔1)⨂�̃�1⨁(−12,−10,−8,−6,−4,−2,0; 𝜔1)⨂�̃�2 

⨁(−4,−2,0,2,4,6,8; 𝜔1] 

The new form of the fuzzy objective function is: 𝑀𝑎𝑥 𝜔′ =  𝑀𝑎𝑥 𝑤1 +  𝑀𝑎𝑥 𝑤2 

 

𝑀𝑎𝑥 𝜔′=[(−10,−8,−6,−4, −2,0,2;𝜔1)⨂�̃�1⨁(−12,−10,−8, −6,−4,−2,0; 𝜔1)⨂�̃�2 

⨁(−4,−2,0,2,4,6,8;𝜔1)⨁ [(0,2,4,6,8,10,12; 𝜔2)⨂�̃�1 ⨁(3,5,7,9,11,13;𝜔2)⨂�̃�2 

⨁(−3,−1,1,3,5,7,9;𝜔2)  

 

Then the new problem becomes as follows: 

 𝑀𝑎𝑥 𝜔′ = (−10,−6,−2,2,6,10,14;min ( 𝜔1, 𝜔2)) ⨂�̃�1 

 ⨁(−9,−5,−1,3,7,11,15;min( 𝜔1, 𝜔2))⨂�̃�2⨁(−7,−3,1,5,9,13,17;min ( 𝜔1, 𝜔2)) 

𝑠. 𝑡 

(−5,−3,−1,1,3,5,7; 𝜔1)⨂�̃�1⨁(−3,−1,1,3,5,7,9;𝜔1)⨂�̃�2 ≤ (−1,1,3,5,7,9,11; 𝜔1)  

(−4,−2,0,2,4,6,8;𝜔2)⨂�̃�1⨁(−5,−3, −1,1,3,5,7;𝜔2)⨂�̃�2 ≤ (−4,−2,0,2,4,6,8; 𝜔2)  

                                  �̃�1, �̃�2 ≥ 0 

The second step, is to convert the problem to the standard form by adding the fuzzy slack variables, 

𝑀𝑎𝑥 𝜔 ′̃ = (−10, −6,−2,2,6,10,14;min ( 𝜔1, 𝜔2)) ⨂�̃�1  

⨁(−9,−5,−1,3,7,11,15;min( 𝜔1, 𝜔2))⨂�̃�2 

 ⨁0̃⨂�̃�1⨁0̃⨂�̃�2⨁(−7,−3,1,5,9,13,17;min ( 𝜔1, 𝜔2)) 



IHJPAS. 2025, 38 (1) 
 

424 
 

𝑠. 𝑡  

(−5,−3,−1,1,3,5,7; 𝜔1)⨂�̃�1⨁(−3,−1,1,3,5,7,9; 𝜔1)⨂�̃�2⨁1̃⨂�̃�1⨁ ≤ (−1,1,3,5,7,9,11; 𝜔1)  

(−4,−2,0,2,4,6,8;𝜔2)⨂�̃�1⨁(−5,−3, −1,1,3,5,7;𝜔2)⨂�̃�2⨁1̃⨂�̃�2 ≤ (−4,−2,0,2,4,6,8; 𝜔2)  

                                               �̃�1, �̃�2, �̃�1, �̃�2 ≥ 0̃ 

Suppose that,  𝜔1 = 𝜔2 = 1, The first tableau of the fully fuzzy simplex is shown in Table 2 below: 

 Table 2. Primary Table of the FFS Method         

 B.V 𝒙𝟏 𝒙𝟐  �̃�𝟏 �̃�𝟐 𝑹.𝑯. 𝑺 𝜽 

𝑨𝟎  𝝎′̃⊖ �̃�𝒋 

 
(
−14,−10,−6,
−2,2,6,10; 1

) 
(−15,−11,−7, 

−3,1,5,9; 1) 
(0̃;1) (0̃;1) 

(−7,−3,1, 

5,9,13,17; 1) 
 

𝑨𝟏  �̃�1 
(−5,−3,−1, 

1,3,5,7; 1) 

(−3,−1,1, 

3,5,7,9; 1) 
(1̃;1) (0̃;1) 

(−1,1,3, 

5,7,9,11; 1) 
5/3 

𝑨𝟐  �̃�𝟐 

 

(−4,−2,0, 

2,4,6,8; 1) 

(−5,−3,−1, 

1,3,5,7; 1) 
(0̃;1) (1̃;1) 

(−4,−2,0, 

2,4,6,8; 1) 
2/1 

  

By the ranking, function techniques,  

The entering variable = 𝑚𝑖𝑛{ℜ( 𝜔 ′̃⊖ �̃�1),ℜ( 𝜔 ′̃⊖ �̃�2)} = 𝑚𝑖𝑛{−2,−3} = −3  

The leaving variable depends on 𝜃  

 

𝜃 = 𝑚𝑖𝑛 {
ℜ(−1,1,3,5,7,9,11; 1)

ℜ(−5,−3,−1,1,3,5,7; 1)
,
ℜ(−4,−2,0,2,4,6,8; 1)

ℜ(−5,−3,−1,1,3,5,7; 1)
} = min {

5

3
,
2

1
} =

5

3
 

 

Therefore, the entering variable is �̃�2 and the leaving variable is  �̃�1. 

The pivot element of heptagonal fuzzy numbers is: (−5,−3, −1,1,3,5,7; 1)  

Now, use the operation of the heptagonal fuzzy numbers to find the new table as follows: 

The pivot row 𝐴1
′ = (

1

3
)⨂𝐴1, and the other rows: 𝐴0

′ = 𝐴0⨁𝐴1 , 

 𝐴2
′ = (−1⨂𝐴1

′)⨁𝐴2 

The new iteration of the fully fuzzy simplex method, as shown in Table 3: 

 

Table 3. New Iteration for the FFS method 

       B.V 𝒙𝟏 𝒙𝟐  �̃�𝟏 �̃�𝟐 𝑹.𝑯. 𝑺 

𝑨𝟎
′ 

 

 𝝎′̃⊖ �̃�𝒋 
(
−19,−13,−7,−1,

5,11,17; 1
)  (

−18,−12,−6,0,
6,12,18; 1

) 
(0,0, 

…,1;1 ) 

(0,… 

,0;1) 
(
−7.3, −2.6,2,6.6,
11.3,16,20.6; 1

) 

𝑨𝟏
′ 

 �̃�𝟐 

 
(
−1.6, −1,−0.3,0.3,

1,1.6,2.3; 1
) (

−1,−0.3,0.3,1,
1.6,2.3,3; 1

) (
0,0,
… ,0.3

) (
0,

… ,0; 1
) (

−0.3,0.3,1,1.6,
2.3,3,3.6; 1

) 

𝑨𝟐
′ 

 �̃�𝟐 

 
(
−6.3, −3.6, −1,1.6,

4.3,7,9.6; 1
) (

−8,−5.3, −2.6,0,
2.7,5.3,8; 1

) (
−0.3,0,
0,… ,0; 1)

) (
0,0,
… ,1; 1

) (
−7.6,−5,−2.3,0.3

, 3,5.6,8.3; 1
) 

  

Since, ℜ(�̃� ⊖ �̃�1) < 0  ,then repeated the steps to find the optimal solution as follows in Table 4. 

 

 

 



IHJPAS. 2025, 38 (1) 
 

425 
 

 

Table 4.  The Optimal Table Solution for the FFS method 

B.V 𝒙𝟏 𝒙𝟐  �̃�𝟏 �̃�𝟐 𝑹.𝑯. 𝑺 

 𝝎′̃

⊖ �̃�𝒋 
(
−22.7, −15.1, −7.6, 0
, 6.5,15.2,22.7; 1

)  (
−22.8, −15.1, −7.5, 0
, 7.6,15.18, 22.8 ; 1

) 
(-0.2,0,0,0  

0,0,0.2;1 ) 
(0,…,0.6;1) (

−14.9, −7.6, −0.3,6.9,
14.3,21.6,28.9; 1

) 

 �̃�𝟐 (
−3.5, −2.4, −1.13,0,

1.2,2.3,3.5; 1
) (

−2.6, −1.36, −0.2,1,
2.1,3.3,4.6; 1; 1

) (
0,0, … ,
0.06; 1

) (−0.2, … ,0; 1) (
−1.9, −0.8,0.4,1.5,

2.7,4,5.1; 1
) 

 �̃�𝟏 

 
(
−3.7, −2.1, −0.6, 1,
2.5, 4.2,5.7; 1

) (
−4.8, −3.1, −1.5,0,
1.6,3.18,4.8; 1

) (
−0.2,0,0,
… ,0; 1

) (0,0, … ,0.6; 1) (
−4.5, −3.0, −1.38, 0.18,

1.8, 3.36, 4.98; 1
) 

 

Finally,  ℜ( �̃�𝟏) = 0.204 , ℜ( �̃�𝟐) = 1.5  and ℜ(�̃� ⊖ �̃�𝑗) ≥ 0 .So , the optimal solution is reached. 

𝑀𝑎𝑥 �̃� =

(−2,0,2,4,6,8,10;1)⨂(
−4.5,−3.0,−1.38,0.18,

1.8,3.36,4.98;1
) ⨁(0,2,4,6,8,10,12;1)

⨂(
−1.9,−0.8,0.4,1.5,
2.7,4,5.1;10;1

)⊖(−4,−2,0,2,4,6,8;1)

(0,2,4,6,8,10,12;1)⨂(
−4.5,−3.0,−1.38,0.18,

1.8,3.36,4.98;1
) ⨁(3,5,7,9,11,13,15;1)

⨂(
−1.9,−0.8,0.4,1.5,
2.7,4,5.1;10;1

)⨁(−3,−1,1,3,5,7,9;1)

 

 

𝑀𝑎𝑥 �̃�=
ℜ(  1,−7.6,−5.16,7.72,32.4,68.88,115;1)

ℜ( −8.7,−11,−1.72,17.58,49.1,92.6,145.2;1)
=
37.5252

47.7252
=0.7863. 

While, the crisp optimal solution of the problem is   �̃�1 = 0.2,  �̃�2 = 1.6,  𝜔 = 0.5. 

 

8. Conclusion   

The paper proposes a novel ranking function for heptagonal fuzzy numbers based on the 

proposed generalized heptagonal membership function. The algorithm of the fully fuzzy simplex 

method with the help of the proposed ranking function is suitable for finding the optimal fuzzy 

solution of a fully fuzzy fractional linear programming problem. Through a numerical example, it 

proved that the optimal solution obtained for a fully fuzzy fractional linear programming problem 

using the arithmetic operations of heptagonal fuzzy numbers is more efficient according to the crisp 

solution of the problem. 

   

Acknowledgment 

We would like to express our deep appreciation to the reviewers of this work and the publishers 

of the "Ibn Al-Haitham for pure and Science Journal" for this splendid opportunity. 

 

Conflict of Interest 

     The authors declare that they have no conflicts of interest. 

 

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