Communications on Applied Nonlinear Analysis 

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A Fermatean Fuzzy Environment is used in Transportation Problems 

to Extend a New Score Function 

 
1Dr. M. Sangeetha, 2T. Mummoorthy 

1Professor, Department of Mathematics, Dr.N.G.P.Arts and Science College, Coimbatore, 

Tamil Nadu, India. m.sangeethaphd@gmail.com 

2Research Scholar, Department of Mathematics, Dr.N.G.P.Arts and Science College, Coimbatore, Tamil Nadu , India. 

mummoorthythangaraj@gmail.com 

 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: 

Since the economy and environment are in a state of instability. In a transportation 

difficulty, it is no longer possible to determine supply, demand, and transportation 

costs. This work aims to explore transportation, including supply, demand, and 

transportation costs expressed as FFNs. However, Pythagorean fuzzy numbers or 

generalized fuzzy numbers are used as the parameters in each of these 

recommendations. It is simpler to deal with uncertain data when making decisions 

because of the innovative idea of Fermatean fuzzy sets. For the first time, we have 

overcome the transportation problem with Fermatean fuzzy parameters. Our method 

was created to address the Fermatean fuzzy parameter transportation problem, and it 

was successfully handled by using a well-known method. The best result can then be 

obtained by performing arithmetic operations on Fermatean fuzzy numbers. 

Specifically, to illustrate the suggested approach, Our solution to a numerical instance 

and then presented and compared the results with the current literature. The approval 

of the effort along with its possibility future developments have been discussed. 

Keywords: Pythagorean fuzzy numbers, Fermatean fuzzy numbers, Score function, 

Accuracy function, Transportation problem, QM windows solver. 

 

1. Introduction 

In the dynamics of our fiercely competitive industry, figuring out the best strategy to create and 

deliver items to customers in the most efficient way, Among the many difficulties faced by the 

transport network's administration is adjusting to technological advancements, Increasing output, 

internationalization, shifting consumer expectations, market dynamics, and security concerns. 

However, given the current situation, addressing these issues becomes tiresome.A considerable 

structure is included in the transport system to handle these kinds of challenges and ensure that 

various types of items are delivered on time. 

A difficulty with transportation was initially developed by Hitchcock in (1941) [1]. The optimization 

requirement was proposed by Tjalling, C.Koopmans in (1949) [2]. The transportation algorithms was 

introduced by Dantzig, G. B. (1963) [3]. Transportation plays a major role in real-world scenarios. 

Korukoglu and Balli(2011) [4] presented an enhanced version of Vogel's approximation technique 

for problems related to transportation. The basic question in the transportation problem is to discover 

mailto:m.sangeethaphd@gmail.com
mailto:mummoorthythangaraj@gmail.com


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the least total transportation cost of a commodity to satisfy requests at destinations using available 

supplies at origins. In a traditional transportation problem, it is assumed that the values of supply and 

demands, as well as the costs of transportation, are precisely known. In a traditional transportation 

problem, it is assumed that the values of supply and demand, as well as the costs of transportation, 

are precisely known. Stated differently, the decision-makers are unable to precisely know the 

parameter data, judgment uncertainty, high information cost, or the data or outcome of a real-world 

transportation problem. Furthermore, these parameters may be more stable since imprecision might 

arise from a variety of factors, such as incomplete or inaccurate information, a lack of confidence in 

one's judgment, a high information cost, or the result of specific flexibility required by a source firm 

or demand marketplace. 

A fuzzy transportation problem naturally arises because the imprecision built into the parameters 

might not be of the probabilistic kind. This kind of uncertainty can be managed with fuzzy 

parameters. The notion of fuzzy sets, which was introduced by Zadech in (1965)[5], has been widely 

used in a variety of disciplines, including engineering, management, and economics, and is 

recognized as a useful instrument for resolving ambiguity and vagueness. Fuzzy sets theory has been 

developed over the last few decades by a large number of researchers using various techniques and 

unique innovations. Thereafter, Bellman,R.E., (1970)[6], introduced the concept of decision making 

problems involving uncertainty. Subsequently, Bellman,R.E (1970)[7], presented the notion of 

decision-making issues containing uncertainty. Fuzzy linear programming techniques are applied to 

the linear vector maximum problem by Zimmermann(1978) [8] in order to demonstrate the efficacy 

of these solutions. They examine the implications of combining different goal functions to achieve 

the optimal solution utilizing different approaches. Additionally, they offer insightful information on 

how well fuzzy linear programming works when solving multi-objective problems and give advice 

on which methods to use to reach the best possible compromise. 

Thereafter, S. Chanas et al. (1984)[9] a fuzzy supply and demand model with crisp cost was 

presented to handle the transportation problem. After that, a number of specialists developed the 

utilizing a multi-objective (MOTP) model or a single objective transportation problem, that considers 

different fuzzy contents. Chanas, S. & Kuchata, D.(1996)[10], investigated the TP's form with 

ambiguous parameters. 

After that, specialists from many fields created the transportation problem using the MOTP model, 

which is multi-objective and takes into account several fuzzy contexts or a single target TP. Tada & 

Ishii(1996)[11], Kaur & kumar(2011)[12], Considering the MOTP, Li & Lai (2000)[13] 

implemented into the MOTP a crude concession programming method, A multi-objective 

transportation facility localization problem was presented by Das, S.K., & Roy, S.K. (2019)[14]. To 

create an actual transportation network, Das, S.K., et al.(2020)[15], integrated a type-2 intuitionistic 

ambiguity in two dimensions. Ghosh et al.(2021)[16] created a MOSTP in the intuitionistic fuzzy 

context that was fixed. Midya et al.(2021)[17] created a multistage, fixed-charge MOSTP in a fuzzy, 

intuitionistic environment with a green supply network. Prabha(2021)[18], created a geometric mean 

approach to solve TP in fuzzy Pythagorean contexts. Kundu et al.(2014)[19], Singh & Yadav 

(2016)[20], Gupta & Anupum(2017)[21], Arora(2018)[22], and Hashmi et al.( 2019)[23] had 

contributed to inaccurate factors and concentrated on one objective TP, Ahmad & Adhami(2019)[24] 



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seen as non-linearity in MOTPs, resulting in nonlinear membership for each goal function within a 

system of neutrosophies. 

Atanassov(1986)[25] introduced the idea of the Intuitionistic Fuzzy Set (IFS) that included both 

membership grade 𝜌 further to non-membership grade 𝜏 performs with a margin of hesitation 𝜋 in 

the way that follows 𝜌 + 𝜏 ≤ 1 and 𝜌 + 𝜏 +𝜂 = 1 However, there are some situations in which the 

sum of the membership grade and non-membership grade can be greater than one. Consequently, 

Senapati, T & Yager, R. R. (2013)[26] Pythagorean Fuzzy Set theory was recently introduced. (PFS), 

a progression of IFS. The following are some actual uses in Pythagorean fuzzy environments that 

have been suggested by different academics: Zhang & Xu (2014)[27] introduced a refined rank 

preference technique and created new creative PFS operating rules in 2014. Furthermore, Mohd & 

Abdullah (2017)[28] introduced the Pythagorean fuzzy analytic hierarchy method to establish the 

assessment criterion's weight. And after that, Kumar et al. ( 2019)[29] suggested models for TP 

optimization in a fuzzy Pythagorean setting. Jeyalakshmi et al.( 2021)[30] outlined the Monalisha 

technique for solving TP using PFS parameters. 

However, in practical implementations, there can be circumstances, like 0.9 and 0.6 supporting and 

opposing membership in the fuzzy sets, respectively. Since, It defies the requirements of the PFS and 

IFS limitations. Consequently, Senapati,T., & Yager, R. R. (2020)[31], Senapati,T., & Yager, R. R.( 

2020)[32] contrasted FFS with PFSs and IFSs and suggested the concept of FFS to handle these 

types of situations in decision-making. Senapati,T., & Yager, R. R.(2019)[33]. And after that, 

Laxminarayan Sahoo(2021)[34] conventional TP is first solved in FFTP, and the optimal solution is 

subsequently found by employing the QM Windows solver to solve the problem. Thus, 

Laxminarayan (2021)[35] established a three-point grading system for the FFS and applied the 

TOPSIS method to tackle a problem involving many criteria in a fuzzy Fermatean environment. 

Score functions for the defuzzification of FFS were the primary focus of this research study. An 

approach to transportation problems in Fermatean fuzzy surroundings is presented in this study. The 

suggested approach uses an orthopair fuzzy set to study the product's supply and demand as well as 

transportation costs. 〈 𝜌, 𝜏〉 that accomplishes that connection 0 ≤ 𝜌3 + 𝜏3 ≤ 1 and our method of 

converting the TP into crisp TP problems was to use score functions. The connection 0 ≤ 𝜌3 + 𝜏3 ≤ 

1 is considered because, regarding every TP parameter in the Fermatean fuzzy setting, Support for 

membership and opposition to membership degree add up to a cube that is either equal to or less than 

one. Next, the QM Window solver was used to solve the converted TP issue in order to determine the 

best solutions. The suggested approach is finally demonstrated by solving a numerical example, the 

computed results of which are presented and contrasted with the body of current literature. 

The remaining portions of the paper are arranged as follows: under Section 2 Given below are some 

fundamental definitions of Pythagorean and Fermatean fuzzy sets. In Section, the TP mathematical 

model is shown 3 and 4. The Part 6, The solution methodology that has been suggested is examined. 

In section, the outcomes of the fuzzy transportation problem are examined 7. The conclusions are 

presented in Section 8. 

2. Preliminaries 

This section provides some basic explanations of Fermatean fuzzy sets. 



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Table. 1. This section provides some basic explanations of Fermatean fuzzy sets. 
 

Authors and References Year Significance 

Lid.Xu[37] 1988 A fuzzy multi-objective programming technique for 

Yager,R.R.[38] 1988 In multi criteria decision making, order-weighted 

averaging aggregation operators 
Shyi-Ming Chen[39] 2012 Interval-valued intuitionistic fuzzy sets as the foundation 

for multi criteria fuzzy decision making 

Guiwu WEI et al.[40] 2013 The applications of fuzzy power aggregation operators to 

multiple attribute group decision-making 

Yager, R.R.[41] 2014 Multicriteria Decision-Making: Pythagorean 

Yingdong He et al.[42] 2014 Using intuitionistic fuzzy geometric interaction 

Yingdong He [42] 2015 Making decisions with generalised intuitionistic fuzzy 

power interaction averaging operators 

Harish Garg et al.[43] 2015 Multi-criteria decision making utilising entropy in a 

Harish Garg[44] 2016 Using Einstein's t-norm, generalised intuitionistic fuzzy 

interactive geometric interaction operators 

Shu-Ping Wan et al.[53] 2016 Using interval-valued fuzzy preference relations in group 

decision making:an intuitionistic fuzzy programming 

V.LakshmanaGomathi 2017 An intuition-based fuzzy multicriteria system based on 

Harish Garg[48] 2017 Utilizing generalized intuitionistic fuzzy soft sets and 

group-based approaches for decision-making 
Harish Garg[49] 2017 Pythagorean  fuzzy  aggregation  operators  based  on 

confidence levels and their application in decision- 

Harish Garg[50] 2018 Some methods for strategic decision-making problems 

with  immediate  probabilities  in  Pythagorean  fuzzy 
Feng Feng et al.[51] 2018 An Alternative Perspective on Multiattribute Decision- 

making techniques associated with generalized 

Abhishek Guleria & Rakesh 
Kumar Bajaj[52] 

2018 On Pythagorean fuzzy soft matrices, their operations, \\ 
and medicinal diagnosis applications 

Muhammad Sajjad Ali khan et 
al.[57] 

2018 Multi-criteria group decision-making using Pythagorean 
fuzzy Einstein prioritized aggregation operators 

Senapati,T., Yager,R.R.[58] 2019 Applications of Fermatean Fuzzy WPM and a Few new 

Operations over Fermatean fuzzy numbers 

Harish Garg[44] 2019 Pythagorean  fuzzy  aggregation  operators  based  on 

neutrality operations and its applications 

Murat Kirisci[56] 2019 Comparing Intuitionistic Fuzzy Parametrized Fuzzy Soft 

Set and Riesz Summability in Medical decision-making 

Shu-ping Wan et al.[53] 2020 Using interval-valued Atanassov intuitionistic fuzzy 
programming, an approach for group decision making 

Shu-Ping Wan,  & Jiu-Ying 

Dong[54] 

2021 Hybrid multi-criteria group decision making using an 

interval-valued intuitionistic fuzzy mathematical 

Xiaolu Zhang, & Zeshul 
Xu[55] 

2021 Expanding TOPSIS for Using Pythagorean Fuzzy Sets in 
Multiple Criteria Decision Making 

Fang Zhou & Ting-Yu 

Chen[60] 

2021 A novel generalized distance measure paradigm and an 

extended Pythagorean fuzzy VIKOR approach with risk 



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Muhammad Akram et al.[61] 2022 Fermatean fuzzy soft expert knowledge for collective 

Muhammad Akram et al.[62] 2022 A novel framework for group decision-making using 2- 

tuple linguistic complicated q-rung image fuzzy sets as a 

Muhammad Akram & Zohra 

Niaz[63] 

2022 2-A Double Linguistic fermatean fuzzy decision-making 

method using COCOSO and CRITIC 

 

2.1. Definition 

A Fermatean fuzzy set on a universal basis X is represented as: 

𝐹 ̂ = {〈𝑥, 𝜌𝐹(̂𝑥), 〉, 𝜏𝐹(̂𝑥): 𝑥 ∈ 𝑋} 

Where 𝜌 (𝑥): 𝑋 → [0, 1] and , 𝜏 (𝑥): 𝑋 → [0, 1] with ( )  
3 ( ) 

3 
. 

𝐹  ̂ 𝐹^ 0 ≤ ( 𝜌 𝐹  ̂𝑥 ) + (𝜏𝐹 ̂ 𝑥 ) ≤ 1, ∀ 𝑥 ∈ 𝑋 

Further, in 𝐹 ,̂  the degree of membership and non-membership of 𝑥 ∈ 𝑋 are represented by 𝜌 𝐹 (̂ 𝑥)  and 

𝜏𝐹 (̂𝑥) .  A Fermatean fuzzy set 𝐹 ̂is degree of uncertainty is expressed as follows: 
 

 

3 3 
𝜗 𝐹 (̂ 𝑥)  = √1 − (𝜌𝐹(̂𝑥)) 

3 

− (𝜏𝐹(̂𝑥)) 

Then set 𝐹 ̂ = {〈𝑥, 𝜌𝐹 (̂𝑥) , 〉, 𝜏𝐹 (̂𝑥): 𝑥 ∈ 𝑋} is indicated as 𝐹 ̂ = 〈 𝜌 𝐹  ̂, 𝜏 𝐹  ̂〉 due to explicitness. 

2.2. Definition 

A few fundamental ideas regarding FFSs are briefly discussed in this section. Let us consider three 

FFS 𝐹 ̂ = 〈 𝜌 𝐹  ̂, 𝜏 𝐹  ̂〉, 𝐹 1̂  = 〈𝜌𝐹 1̂  , 𝜏 𝐹 1̂  
〉 and 𝐹 2̂  = 〈𝜌𝐹 2̂  , 𝜏 𝐹 2̂  

〉, regarding universal set X and 𝜂 > 0. These 

are the definitions of the elementary operations on the FFS. 

 ̂  ̂ 3 3 3 3 3 

1. Addition: 𝐹1 ⊕ 𝐹2 = ⟨ √(𝜌𝐹 1̂) + (𝜏𝐹 2̂) − (𝜌𝐹 1̂  ) ( 𝜏𝐹 2̂  ) , 𝜏 𝐹 1̂  𝜏 𝐹 2̂  ⟩ 

 ̂  ̂ 3 3 3 3 3 

2. Multiplication: 𝐹1 ⊗ 𝐹2 = ⟨𝜌𝐹 1̂  
𝜌 𝐹 2̂  

, √(𝜏𝐹1̂  
) + (𝜏𝐹 2̂  

) − (𝜏𝐹 1̂  
) (𝜏𝐹 2̂  

) ⟩ 

 ̂ 3 3 η η 

3. Scalar Multiplication: η⊙ 𝐹 = ⟨ √1 − (1 − 𝜌𝐹^) , ( 𝜏 𝐹 )̂  ⟩ 

η̂ η 3 3 η 

4. Exponent: 𝐹 = ⟨ ( 𝜏 𝐹 )̂  , √1 − (1 − 𝜌𝐹^) ⟩ 

5. Union: 𝐹 1̂  ∪ 𝐹 2̂  = ⟨ 𝑀𝑎𝑥(𝜌𝐹1̂ 
, 𝜌𝐹 2̂  

), 𝑀𝑖𝑛(𝜏𝐹1̂ 
, 𝜏𝐹2̂

)⟩ 

6. Intersection: 𝐹 1̂  ∩ 𝐹 2̂  = ⟨ 𝑀𝑖𝑛(𝜌𝐹1̂ , 𝜌𝐹 2̂  ), 𝑀𝑎𝑥(𝜏𝐹1̂ , 𝜏𝐹 2̂  )⟩ 

7. Component: 𝐹 �̂�  = ⟨ 𝜌 𝐹  ̂ , 𝜏 𝐹  ̂⟩ 
1 2 

Example 1. 

Let 𝐹 ̂= ⟨ 0.6, 0.3⟩, 𝐹 1̂  = ⟨0.7, 0.6⟩ , 𝐹 2̂  = ⟨0.3 , 0.8⟩be three FFS and η=2 be a scalar then:- 

1. Addition: 𝐹 1̂   ⊕ 𝐹 2̂  = ⟨0.7,0.6⟩ + ⟨0.3,0.8⟩ = ⟨0.711,0.110⟩ (By definition 2.2) 



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𝐹 

So, 
1+𝜌 −𝜏 

𝐹^ 𝐹 

2. Multiplication: 𝐹 1̂  ⊗ 𝐹 2̂  = ⟨0.7,0.6⟩ + ⟨0.3,0.8⟩ = ⟨0.21,0.6282⟩ (By definition 2.2) 
 

3. Scalar Multiplication: η⊙ 𝐹 ̂= 2 ⊙ ⟨ 0.6,0.3⟩ = ⟨ 0.7276,0.09⟩ (By definition 

2.2) 
 

 

4. Exponent: 𝐹 η̂  = ⟨ (0.6)2, 3√1 − (1 − 0.33)2⟩ = ⟨ 0.36,0.3⟩ 
 

(By 

 

definition 

2.2)   

5. Union: 𝐹 1̂  ∪ 𝐹 2̂  = ⟨ 𝑀𝑎𝑥(0.7,0.3), 𝑀𝑖𝑛(0.6,0.8)⟩ = ⟨0.7,0.8⟩ (By definition 2.2) 

6. Intersection: 𝐹 1̂  ∩ 𝐹 2̂  = ⟨ 𝑀𝑖𝑛(0.7,0.3), 𝑀𝑎𝑥(0.6,0.8)⟩ = ⟨0.3,0.6⟩ (By definition 2.2) 

7. Component: 𝐹 �̂�  = ⟨0.6,0.3⟩𝐶 = ⟨0.3,0.6⟩ (By definition 2.2) 

3. Fermatean fuzzy score function 

Examining FFS 𝐹 ̂= 〈𝜌 ̂ , 𝜏 ̂ 〉, subsequently score function of 𝐹 ̂suggested as 𝑆 ( 𝐹 )̂  = (𝜌3 − 𝜏3 ). 
𝐹 𝐹 𝐹 𝐹 𝐹 

In this instance. Certain score functions have been defined by us 𝑆𝐹(𝐹 )̂  ∈ [−1,1]. Nevertheless, we 

created a few score functions 𝑆𝐹( 𝐹 ̂ ). 𝑆𝐹( 𝐹 ̂) ∈ [0,1] that are listed below: 
(i). Type1  ̂ 1 

(1 + 𝜌3 3) 
𝑆1𝐹(𝐹) =  − 𝜏 

2 

(ii). 
Type 2  ̂  =  1 (1 + 2𝜌

3 
− 𝜏

3) 𝑆2𝐹(𝐹) 
3 𝐹 𝐹 

(iii). Type 3  ̂  = 1  𝜌2 − 𝜏2)| | 
𝑆3𝐹(𝐹) (1 + 

2 𝐹 𝐹  𝜌𝐹  − 𝜏𝐹  

3. 1. Fermatean fuzzy set Score function 

Examining an FFS 𝐹 ̂= 〈 𝜌 𝐹  ̂, 𝜏 𝐹  ̂〉. Next, the procedure that follows 𝐹 ̂provides an 𝑆𝐹(𝐹 )̂  explanation 

and representation of the score function. 

𝑆 ( 𝐹 )̂  = (𝜌3 − 𝜏3) 
𝐹 𝐹 𝐹 

Property 2.1. Consider an FFS 𝐹 ̂= 〈 𝜌 𝐹  ̂, 𝜏 𝐹  ̂〉 then 𝑆𝐹(𝐹 )̂  ∈ [0,1] then 𝑆1𝐹(𝐹)̂ ∈ [0,1], 𝑆2𝐹(𝐹)̂ ∈ 

[0,1] and 𝑆3𝐹(𝐹 )̂ ∈ [0,1]. 

Proof. For any Fermatean fuzzy set 𝜌3 ≥0 and 𝜏3 ≤ 1 
𝐹 𝐹 

⇒1 − 𝜏3 ≥ 0. 

Hence ,1 + 𝜌3 − 𝜏3 ≥ 0 and certainly 𝑆 (𝐹 )̂ ≥ 0. 
𝐹 𝐹 1𝐹 

Again, 𝜌3 + 𝜏3 ≤ 1 
𝐹 𝐹 

⇒1 + 𝜌3 + 𝜏3 ≤ 2 
𝐹 𝐹 

⇒1 + 𝜌3 − 𝜏3 ≤ 2 as 𝜏3 ≥ 0 
𝐹 𝐹 𝐹 

3 3 
𝐹 𝐹 ≤ 1 and 𝑆 (𝐹) ≤ 1. 
2 𝐹 



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𝑆  ̂

For type 1 score function, hence 𝑆1 𝐹(𝐹 )̂  ∈ [0,1], 𝑆2 𝐹(𝐹 )̂  ∈ [0,1] and 𝑆3 𝐹(𝐹 )̂  ∈ [0,1]. Especially, if 

𝐹 ̂ = ⟨0, 1⟩, then 𝑆1𝐹(𝐹 )̂  = 0, 𝑆2𝐹(𝐹 )̂  = 0, 𝑆3𝐹(𝐹 )̂  = 0. Again if if 𝐹 ̂ = ⟨0, 1⟩, then 𝑆1𝐹(𝐹 )̂  = 1, 

𝑆2𝐹(𝐹 )̂ = 1, 𝑆3𝐹(𝐹 )̂ = 1. 

3.2. The FFS’s Accuracy Function. 

Let us take an FFS. 𝑆𝐹(𝐹 )̂  and indicated in the following aspect. 

𝑆 ( 𝐹 )̂  = (𝜌3 + 𝜏3) 
𝐹 𝐹 𝐹 

3.3. The Ranking function of Fermatean fuzzy sets. 

Let us examine these two FFSs. 𝐹 1̂  = ⟨𝜌𝐹 1̂  , 𝜏 𝐹 1̂  ⟩ and 𝐹 2̂  = ⟨𝜌𝐹 2̂  , 𝜏 𝐹 2̂  ⟩, the ranking rules of 𝐹 1̂  and 𝐹 2̂  

are given an interpretation in the following the manner: 

𝑆∗( 𝐹 1̂) ≥ 𝑆∗( 𝐹 2̂) then 𝑄∗( 𝐹 1̂) > 𝑄∗( 𝐹 2̂) iff 𝐹 1̂  > 𝐹 2̂  
𝐹 𝐹 𝐹 𝐹 

𝑆∗( 𝐹 1̂) ≤ 𝑆∗( 𝐹 2̂) then 𝑄∗( 𝐹 1̂) > 𝑄∗( 𝐹 2̂) iff 𝐹 1̂  < 𝐹 2̂  
𝐹 𝐹 𝐹 𝐹 

𝑆∗( 𝐹 1̂) = 𝑆∗( 𝐹 2̂) then 𝑄∗( 𝐹 1̂) < 𝑄∗( 𝐹 2̂) iff 𝐹 1̂  = 𝐹 2̂  
𝐹 𝐹 𝐹 𝐹 

1. Example. 

Let 𝐹 1̂  = ⟨0.7,0.6 ⟩ and 𝐹 2̂  = ⟨0.51 ,0.4 ⟩ be two FFSs, we have the subsequent procedure, utilizing the 
type 1 score function  ∗  ̂  3 3 . 

 

𝑆∗( 𝐹 ̂ ) = 

𝑆𝐹( 𝐹1) = 
2 

(1 + 𝜌𝐹^ − 𝜏𝐹 )̂  
1 

(1 + 0.73 − 0.63) = 0.5635 
 

𝐹 1 2 
∗  ̂ 1 

3 3 

𝑆𝐹( 𝐹2) = 
2 

(1 + 0.5 − 0.4 ) = 0.5305 

𝑆∗( 𝐹 1̂) ≥ 𝑆∗( 𝐹 2̂) ⇒ 𝐹 1̂  > 𝐹 2̂  
𝐹 𝐹 

2. Example. 

Let 𝐹 1̂  = ⟨0.7,0.5 ⟩ and 𝐹 2̂  = ⟨0.9,0.3 ⟩ be two FFSs, we have the subsequent procedure, utilizing the 
type 2 score function ( 𝐹 ) = 1(1 + 2𝜌 − 𝜏 . 

∗ 3 3) 
𝐹 2 3 

𝐹 𝐹 

𝑆∗( 𝐹 ̂ ) = 
1 

(1 + 2(0.7)3 − 0.63) = 0.3630 
 

𝐹 1 3 
∗  ̂ 1 

3 3 

𝑆𝐹( 𝐹2) = 
2 

(1 + 2(0.8) − 0.7 ) = 0.5603 

𝑆∗( 𝐹 1̂) ≤ 𝑆∗( 𝐹 2̂) ⇒ 𝐹 1̂  < 𝐹 2̂  
𝐹 𝐹 

4. Mathematical formulation. 

Analyze a TP extensive u source points and v incurable points, where 𝑏𝑗 > 0 units are needed by the 

𝑗𝑡ℎincurable point and 𝑎𝑖 > 0 units are transmitted by the 𝑖𝑡ℎ source point. A unit shipping cost, 

𝑐𝑖𝑗, exists for each link (𝑖, 𝑗) from source point 𝑖𝑡ℎ to incurable j. Finding the cost of the coefficient 

of solution for sending the given data is the work, which must be completed in order to meet 



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0 0 

𝑖𝑗 𝑖𝑗 

specifications and reduce overall transport costs. 𝑥𝑖𝑗 indicates the number of groups that need to be 

transferred from source i to incurable j. The mathematical expression for the standard TP is as 

follows: 

 

 

 

Subject to: 

𝑢 𝑣 

Minimize 𝑍0 = ∑ ∑ 𝐶𝑖𝑗𝑋𝑖𝑗 

𝑖=1 𝑗=1 

 
(1) 

𝑣 
𝑗=1 

𝑣 
𝑗=1 

𝑋𝑖𝑗 

𝑋𝑖𝑗 

≤ 𝑎𝑖, for i = 1,2, . . . , u 

≤ 𝑏𝑗, for i = 1,2, . . . , v 

𝑥𝑖𝑗 ≥ 0, ∀ 𝑖, 𝑗 

When 𝑎𝑖, 𝑏𝑗, 𝑐𝑖𝑗 or even when Fermatean fuzzy parameters are accepted for any of them separately. 

According, FFTP is ordinary TP in (1). With 0 ≤ (𝜌)3 + (𝜏)3 ≤ 1, the FFS have the form ⟨𝜌, 𝜏⟩. The 

Fermatean fuzzy transportation problem can be formulated in this manner: 

𝑢 𝑣 

 

 
Subject to : 

Min ⟨𝜌𝑥^0 , 𝜏𝑥 0̂⟩ = ∑ ∑⟨𝜌𝑧0̂, 𝜏𝑧 0⟩ 
𝑖=1 𝑗=1 

(2) 

∑𝑣 𝑥𝑖𝑗 ≤ ⟨𝜌𝑎  ̂, 𝜏𝑎  ̂⟩, for 𝑖 = 1, 2, . . . , 𝑢 
𝑗=1 𝑖 𝑖 

 

 

∑𝑣 𝑥𝑖𝑗 ≤ ⟨𝜌  ̂ , 𝜏  ̂ ⟩, for 𝑗 = 1, 2, . . . , 𝑣 
𝑖=1 𝑏𝑖 𝑏𝑖 

Where 0 ≤ (𝜌𝑧^ )3 + (𝜏𝑧^ )3 ≤1, 

0 ≤ (𝜌𝑎^𝑖 
)3 + (𝜏𝑎^𝑖 

)3 ≤1, i=1,2,…,u 

0 ≤ (𝜌
𝑏^𝑗 

)3 + (𝜏
𝑏^𝑗 

)3 ≤1, j=1,2,…,v 

𝑥𝑖𝑗 ≥ 0, and 0 ≤ (𝜌𝑐^ )3 + (𝜏𝑐^ )3 ≤1, 

i=1,2,…,u j=1,2,…,v 

The computational model of a TP in Fermatean fuzzy surroundings,or FFTP, is now problem(2). It 
should be noted that the FFTP is considered balanced, or imbalanced FFTP, if  ∑𝑢  ⊕ ⟨𝜌𝑎^ , 𝜏𝑎^ ⟩ = 

𝑖=1 𝑖 𝑖 
∑𝑢 ⊕ ⟨𝜌𝑎^ , 𝜏𝑎^ ⟩. Fermatean additives are used to describe the structure ∑⊕ as an integration. 

𝑖=1 𝑖 𝑖 

4.1. The proposed method algorithm for solving FFTP. 

In various fuzzy circumstances, the literature now in publication offers a range of methods for 

expressive both the optimal and initial feasible solutions for the transportation problem (TP). 

Intuitionistic fuzzy, Pythagorean fuzzy etc. The linear programming methodology, Vogel's 

approximation method, row, column, and matrix minima, as well as the north-west corner. Moreover, 

there are several algorithmic evolution approaches that aim to address the same issue. All the 

∑ 

∑ 



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𝑗 = 1,2,3,4 

parameters in a Fermatean fuzzy environment are represented as FFSs, but no researcher has 

considered the TP inside this setting. Consequently, we solved the TP in the Fermatean fuzzy 

environs in this study using the well-known QM-Solver. The suggested algorithm involves the 

following steps: 

1). As well as Fermatean fuzzy supply and demand, ascertain the score function value for each of 

these situations. 

2). Determine the entire supply and total demand to confirm the transportation issue. 

3). If supply and demand are equal throughout, go to step 7; if not, go to the next step. This suggests 

a balanced approach to the transportation issue. 

4). To improve the balance, add a dummy variable in step 7 if the supply and demand are not equal. 

5). The transportation issue should be expressed as a linear programming problem (LPP). 

6). Use the QM for Windows solver to discover the best solution to the balanced transportation 

problem. 

Table 2: Enter data concerning supply, demand, and transportation costs. 
 

 H1 H2 H3 H4 Supply 
Z1 ⟨0.1,0.8⟩ ⟨0.3,0.7⟩ ⟨0.2,0.7⟩ ⟨0.3, 0.7⟩ ⟨0.8,0.2⟩ 

Z2 ⟨0.2,0.8⟩ ⟨0.1,0.9⟩ ⟨0.7,0.1⟩ ⟨0.8, 0.1⟩ ⟨0.7,0.2⟩ 

Z3 ⟨0.6,0.3⟩ ⟨0.4,0.6⟩ ⟨0.7,0.2⟩ ⟨0.8,0.2⟩ ⟨0.9,0.1⟩ 

Demand ⟨0.3, 0.8⟩ ⟨0.6,0.4⟩ ⟨0.8, 0.2⟩ ⟨0.6,0.5⟩  

 

5. The analysis and findings with a numerical example. 

We provide an appropriate example in this part to demonstrate our suggested method for solving the 

problem. Examine an FFTP where Table 4 provides all of the parameters. ⟨𝜌𝑎^𝑖, 𝜏𝑎^𝑖⟩. supplies i = 

1,2,3 requests for each j=1,2,3,4, ⟨𝜌 𝑏 �̂�
,  𝜏𝑏 �̂�

⟩ ,  and expenses ⟨𝜌𝑐^𝑖𝑗 , 𝜏𝑐𝑖𝑗 ⟩. We regard i = 1,2,3: j = 

1,2,3,4 to be FFNs. In this case, there are four destinations: H1, H2, H3, and H4 and three origins: Z1, 

Z2 and Z3. The following LPP(3), for which Table 4 contains all of the input data, was solved before 

we could solve the FFTP. 

Minimize SqF⟨𝜌𝑧^ , 𝜏𝑧^ ⟩ ∑3  ∑4  SqF⟨𝜌𝑐^ , 𝜏𝑐 ⟩⊙𝑥𝑖𝑗 (3) 
0 0 𝑖=1 𝑗=1 𝑖𝑗 𝑖𝑗 

Subject to 

∑4 𝑥𝑖𝑗 ≤ SqF (⟨𝜌𝑎  ̂ , 𝜏𝑎  ̂⟩), 𝑖 = 1,2,3, 
𝑗=1 𝑖 𝑖 

 

Where 0 ≤ (𝜌𝑧^0 )3 + (𝜏𝑧^0 )3 ≤1, 

0 ≤ (𝜌𝑎^𝑖)3 + (𝜏𝑎^𝑖)3 ≤ i = 1,2,3 



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𝑖𝑗 𝑖𝑗 

0 0 

𝑖 𝑖 

0 0 

𝑖 𝑖 

3 

∑ 𝑥𝑖𝑗 ≥ S1F (⟨𝜌𝑏 𝑖
,  𝜏𝑏 𝑖

⟩),  𝑗 = 1,2,3,4 
𝑖=1 

3 

∑ 𝑥𝑖𝑗 ≥ S2F (⟨𝜌𝑏 𝑖
,  𝜏𝑏𝑖

⟩), 𝑗 = 1,2,3,4 
𝑖=1 

0 ≤ (𝜌  )3 + (𝜏  )3 ≤ j = 1,2,3,4 
𝑏^𝑗 

𝑥𝑖𝑗 ≥ 0, and 0 ≤ (𝜌𝑐^ )3 + (𝜏𝑐^ )3 ≤1, 

𝑏^𝑗 

i =1,2,3,  j=1,2,3,4 

LPP(4), LPP(5) and LPP(6) are our three LPPs, using group 1(q=1), and group 2(q=2), and 

group(q=3) score functions and Table 3 in LPP(3). Theses are mentioned as follows: 

Minimize SqF⟨𝜌𝑧  ̂ , 𝜏𝑧  ̂ ⟩ ∑3 ∑4 SqF⟨𝜌𝑐^ , 𝜏𝑐 ⟩⊙𝑥𝑖𝑗 (4) 
0 0 𝑖=1 𝑗=1 𝑖𝑗 𝑖𝑗 

Subject to 

∑4 𝑥𝑖𝑗 ≤ S1F (⟨𝜌𝑎  ̂ , 𝜏𝑎  ̂⟩), 𝑖 = 1,2,3, 
𝑗=1 𝑖 𝑖 

 

Where 0 ≤ (𝜌𝑧^ )3 + (𝜏𝑧^ )3 ≤1, 

0 ≤ (𝜌𝑎^ )3 + (𝜏𝑎^ )3 ≤ i = 1,2,3 

0 ≤ (𝜌  )3 + (𝜏  )3 ≤ j = 1,2,3,4 

 
𝑥𝑖𝑗 ≥ 0, i=1,2,3, j=1,2,3,4 

𝑏 𝑗 𝑏^𝑗 

 
Minimize SqF⟨𝜌𝑧  ̂ , 𝜏𝑧  ̂ ⟩ ∑3 ∑4 SqF⟨𝜌𝑐^ , 𝜏𝑐 ⟩⊙𝑥𝑖𝑗 (5) 

0 0 𝑖=1 𝑗=1 𝑖𝑗 𝑖𝑗 

Subject to 

∑4 𝑥𝑖𝑗 ≤ S2F (⟨𝜌𝑎  ̂, 𝜏 𝑎  ̂⟩), 𝑖 = 1,2,3, 
𝑗=1 𝑖 𝑖 

 

Where 0 ≤ (𝜌𝑧^ )3 + (𝜏𝑧^ )3 ≤1, 

0 ≤ (𝜌𝑎^ )3 + (𝜏𝑎^ )3 ≤ i = 1,2,3 

0 ≤ (𝜌  )3 + (𝜏  )3 ≤ j = 1,2,3,4 

 
𝑥𝑖𝑗 ≥ 0, i=1,2,3, j=1,2,3,4 

𝑏 𝑗 𝑏^𝑗 

Minimize SqF⟨𝜌𝑧  ̂ , 𝜏𝑧  ̂ ⟩ ∑3 ∑4 SqF⟨𝜌𝑐^ , 𝜏𝑐 ⟩⊙𝑥𝑖𝑗 (6) 
0 0 𝑖=1 𝑗=1 𝑖𝑗 𝑖𝑗 

Subject to 

∑4 𝑥𝑖𝑗 ≤ S3F (⟨𝜌𝑎  ̂ , 𝜏𝑎  ̂⟩), 𝑖 = 1,2,3, 
𝑗=1 𝑖 𝑖 



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0 0 

𝑖 𝑖 

 

 

Where 0 ≤ (𝜌𝑧^ )3 + (𝜏𝑧^ )3 ≤1, 

0 ≤ (𝜌𝑎^ )3 + (𝜏𝑎^ )3 ≤ i = 1,2,3 

0 ≤ (𝜌  )3 + (𝜏  )3 ≤ j = 1,2,3,4 

 
𝑥𝑖𝑗 ≥ 0, i=1,2,3, j=1,2,3,4 

𝑏 𝑗 𝑏^𝑗 

Table 3: Enter data concerning supply, demand, and transportation costs. 
 

 H1 H2 H3 H4 Supply 

Z1 0.1633 0.2370 0.2243 0.2370 0.7520 
Z2 0.1680 0.0910 0.5616 0.6743 0.6675 
Z3 0.5945 0.4240 0.6675 0.7520 0.8640 

Demand 0.2575 0.5760 0.7520 0.5455  

 

Using the recommended algorithm, the type-1 FFTP has the following as its optimal solution: x13 = 

0.2065, x14=0.5455, x21=0.2575, x22=0.410, x32=0.166, x33=0.5455. Consequently, The type-1 FFTP 

has a minimum expense of 0.6907. 

Table 4: Enter data concerning supply, demand, and transportation costs. 
 

 H1 H2 H3 H4 Supply 

Z1 0.1633 0.2370 0.2213 0.2370 0.6720 
Z2 0.1680 0.0910 0.5616 0.6743 0.5593 
Z3 0.4683 0.6880 0.5593 0.6720 0.1523 

Demand 0.1806 0.4560 0.6720 0.4356  

 

Using the recommended algorithm, the type-1 FFTP has the following as its optimal solution: x13 = 

0.597, x14 = 0.075, x21 = 0.1033, x22 = 0.456, x31 = 0.0773, x33 = 0.075. Consequently, The type-2 

FFTP has a minimum expense of 0. 31983. 

Table 5: Enter data concerning supply, demand, and transportation costs. 
 

 H1 H2 H3 H4 Supply 

Z1 0.1295 0.1200 0.1375 0.1200 0.4800 
Z2 0.1200 0.1000 0.4440 0.5705 0.3625 
Z3 0.1905 0.0800 0.3625 0.4800 0.7200 

Demand 0.1125 0.1200 0.4800 0.0555  

 

Using the recommended algorithm, the type-3 FFTP has the following as its optimal solution: x13 = 

0.4245, x14 = 0.555, x21 = 0.1125, x32 = 0.120, x33 = 0, Consequently, The type-3 FFTP has a 

minimum expense of 0. 2887. 

3 

∑ 𝑥𝑖𝑗 ≥ S3F (⟨𝜌𝑏
�̂�
,  𝜏𝑏

�̂�
⟩),  𝑗 = 1,2,3,4 

𝑖=1 



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Table 6: The LPP(4), LPP(5), and LPP(6) numerical outcomes 
 

Method Model Score function Optimal Solutions Minimum Cost 

  

LPP(4) 

1 
𝑆1𝐹(𝐹)̂ = (1 + 𝜌3 − 𝜏3) 

2 𝐹 𝐹 
X13=0.2065,  

𝑆1𝐹 (𝐹 )̂  =0.6907  X14=0.5455, 

X21=0.2575, 

Our X22=0.410, 

proposed X32=0.166, 

Method X33=0.5455, 

  

LPP(5) 

1 
𝑆2𝐹(𝐹)̂ = (1 + 2𝜌3 − 𝜏3) 

3 𝐹 𝐹 

X13=0.0.597,  

𝑆2𝐹 (𝐹 )̂  =0.2887  X14=0.075, 

X21=0.1033, 

 X22=0.456, 

 X31=0.0773, 

 X33=0.075, 

  

LPP(6) 

1 
𝑆 ( 𝐹 ) = (1 + 𝜌2 

 

3𝐹 2 𝐹 

− 𝜏2)|𝜌 ̂ − 𝜏 ̂ | 
𝐹 𝐹 𝐹 

X13=0.4245,  

𝑆3𝐹 (𝐹 )̂  =0.1082  X14=0.555, 

X21=0.1125, 

 X22=0.120, 

 X31=0.120, 

 X33=0.555, 

 

Table: 7 Comparison of numerical results for our results and proposed results by L. Shaoo 
 

Method Model Score function Optimal Solutions Minimum Cost 

  

LPP(4) 

1 
𝑆1𝐹(𝐹)̂ = (1 + 𝜌3 − 𝜏3) 

2 𝐹 𝐹 

X13=0.2065,  

𝑆1𝐹 (𝐹 )̂  =0.6907  X14=0.5455, 

X21=0.2575, 

Our X22=0.410, 

proposed X32=0.166, 

Method X33=0.5455, 

  

LPP(5) 

1 
𝑆2𝐹(𝐹)̂ = (1 + 2𝜌3 − 𝜏3) 

3 𝐹 𝐹 

X13=0.0.597,  

𝑆2𝐹 (𝐹 )̂  =0.2887  X14=0.075, 

X21=0.1033, 

 X22=0.456, 

 X31=0.0773, 

 X33=0.075, 

  

LPP(6) 

1 
𝑆3𝐹(𝐹)̂ = (1 + 𝜌2 

2 𝐹 

− 𝜏2)|𝜌 ̂ 
𝐹 𝐹 

− 𝜏 𝐹 |̂  

X13=0.4245,  

𝑆3𝐹 (𝐹 )̂  =0.1082  X14=0.555, 

X21=0.1125, 

 X22=0.120, 

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   X33=0.555,  

  

LPP(4) 

1 
𝑆1𝐹(𝐹)̂ = (1 + 𝜌3 − 𝜏3) 

2 𝐹 𝐹 

X13=0.2065,  

𝑆1𝐹 (𝐹 )̂  =0.6907  X14=0.5455, 

 X21=0.2065, 

Proposed X22=0.5455, 

method by X32=0.756, 

L. Sahoo X33=0.108, 

  

LPP(5) 

1 
𝑆2𝐹(𝐹)̂ = (1 + 2𝜌3 − 𝜏3) 

3 𝐹 𝐹 

X12=0.0198,  

𝑆2𝐹 (𝐹 )̂  =0.2887  X14=0.473, 

X21=0.361, 

 X22=0.395, 

 X33=0.6743, 

 X34=0.108, 

 LPP(6) 1 
𝑆3𝐹(𝐹)̂ = (1 + 𝜌2 

2 𝐹 

− 𝜏2)|𝜌 ̂ 
𝐹 𝐹 

− 𝜏 𝐹 |̂  

X14=0.4245, 

X21=0.555, 

𝑆3𝐹 (𝐹 )̂  =0.1082 

 X22=0.1125, 

 X33=0.5705, 

 

The numerical outcomes of solving LPP using the QM window solver are shown in Table 6. Table 7 

presents the comparison results. The Fermatean fuzzy cost of 0.6907, 0.2887, and 0.1082 is visible. 

In a Fermatean fuzzy environment, the score function thus provides the ideal solution to a fuzzy 

transportation problem. Additionally, Table 7 shows that the scoring function values are contrasted 

with Laxminarayan Sahoo. 

 

 

Figure. 1 Comparison results for our results and Proposed results by L. Shaoo 

Graph 1 additionally demonstrates that our suggested method produces remarkable results for three 

different forms of FFTP in a Fermatean fuzzy scenario when compared to Laxminarayan Sahoo's 

strategy. Consequently, we claimed that our method of addressing the transportation problem in an 



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ambiguous environment is unique. In addition, the Fermatean fuzzy transportation problem is what 

we have for our numerical experiment. It implies that our recommended strategy is a fresh approach 

to handling uncertainty in Fermatean fuzzy circumstances. 

6. Conclusion. 

The parameters relating to the transportation problem involve Fermatean fuzzy sets. The 

transportation issue, where costs, supply, and needs are not well defined in real-world situations, 

frequently dominates decision-making. Fermatean fuzzy parameters provide a more practical 

solution than accurate parameters in transportation problems since uncertainty is a common 

occurrence in real-life situations. Computing results show that Fermatean fuzzy sets are more 

appropriate than PFSs and can handle higher levels of uncertainty. It is now possible to ensure that 

the score function value for any Fermatean fuzzy sets falls inside the unit interval by presenting a 

new definition of the score function for Fermatean fuzzy sets. The suggested approach covered here 

is incredibly straightforward, easy to use, and effective for addressing practical decision-making 

issues in fields including operations, engineering, and management science. Future research and 

additional decision-making topics involving Fermatean fuzzy parameters. 

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