Solving Fuzzy Assignment Problem of Pythagorean Fuzzy 

Numbers Exploiting a Software Tool 

Dr. K. Senbagam1, Mrs. L. Saranya2 
1Assistant Professor, Department of Mathematics, PSG College of Arts & Science, Coimbatore 

2Research scholar, Department of Mathematics, PSG College of Arts & Science, Coimbatore 
saranya.logajaya@gmail.com 

Abstract: 

A fresh approach to computing assignment issues in the pythagorean fuzzy domain is the 
focus of this body of work. Pythagorean fuzzy numbers are utilised in this work for the sake 
of analysing working hours and working costs. For converting fuzzy sets integers into a 
crisp number, a new ranking process is used. Optimal solutions have to be found by using 
software techniques for minimization, maximization and unbalanced assignment problems. 
In the end, numerical examples are being solved by using the proposed technique. 

Keywords:  Maximization pythagorean fuzzy assignment problem , Fuzzy sets, Triangular 
pythagorean fuzzy numbers, Ranking function, Software tool 

1. Introduction

Operations research is an important field to finding new variety of solution in 
many real life problems. In OR, transportation problems are used to compute optimal value for 
goods shipping. The problem formulation is a subclass of the transportation problem. It refers 
to assign a work to the labour with the purpose of either minimizing the cost or reducing the 
number of hours into a single to one correspondence. Nowadays fuzzy sets are mainly used to 
all fields. Since many of the real life problems having only vagueness or imprecision values. 
An improvement on intuitionistic fuzzy sets, Pythagorean fuzzy sets take the precondition and 
wrapping legion regions and unwrap them. It is utilised for determining the best course of 
action in decision making, transportation, and business endeavours, among other applications. 
Fuzzy sets (FSs) have been crucial in various areas of contemporary culture. It is proposed to 
enhance the functionality of FSs and better express uncertain information. Information may be 
described in these sets and methods more thoroughly and precisely. Pythagorean fuzzy set 
(PFS), a new nonstandard fuzzy set, has just been developed. The PFS was created because we 
frequently come across these unique situations in practical decision-making processes, where 
an alternative may satisfy a decision-maker-provided criterion to a support (membership) 
degree and a nonmembership degree. 
For simplicity, the Pythagorean fuzzy numbers (PFNs) will be the PFS's constituent parts—
PFNs' fundamental operations, such as intersection, union, addition, and multiplication. The 
Pythagorean complement was the main focus of the study, and it was utilised to create PFS. 
Then, using these PFSs, he applied several aggregation techniques to situations involving 
multi-criteria decision-making (MCDM). They specifically demonstrated that the Pythagorean 
membership grades belong to the problematic number category known as -i numbers. Later, a 
straightforward and efficient Pythagorean fuzzy TOPSIS approach was presented for quickly 

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Article History: 

Received: 27-01-2024 

Revised: 12-04-2024 

Accepted: 28-04-2024 



addressing several MCDM issues. The research listed above, however, only considers how to 
handle PFSs by incorporating certain operations and aggregating techniques for discrete 
information.  
Different fuzzy sets can be used to communicate ambiguous and uncertain information in actual 
decision-making processes. However, we may run into instances where an option fulfils a condition 
supplied by the decision maker. As the basic building blocks of PFSs, Pythagorean fuzzy numbers 
(PFNs) are initially described in this study as change values when viewed as variables. 
                           Lotfi Zadeh (1965) pioneered the idea of fuzzy thinking in order to address 
issues relating to imprecision and vagueness. Atanassov (1986,1989) extended the concept of 
fuzzy sets with the fuzzy sets and discussed their many aspects. This was the first time the 
concept had been presented. Angelov made a suggestion for the optimization of an intuitively 
fuzzy environment in his paper. PP (1997). After that, he expanded the intuitive fuzzy set by 
adding new items and graphical representation (1999). The Hungarian technique, which had 
been suggested by Kuhn (1955), was utilised in order to resolve the assignment problem. 
Mahapatra and Roy (2009) conducted research on the reliability of fuzzy systems. The sorting 
function of triangular intuitionistic fuzzy numbers was analysed and explained by Nagoorgani 
and Ponnalagu (2012Yager (2013, 2014) provided an explanation of the concept of 
Pythagorean fuzzy sets, which are responsible for the complicated ambiguity and unreliability 
in a variety of domains. The distinguished attribute of the Pythagorean fuzzy method is to 
loosen up the restriction and which was more accessible than intuitionistic fuzzy sets. Yager 
(2014) presented a number of distinct kinds of aggregation methods for Pythagorean fuzzy 
subsets. Subsequently, in the instance of a multicriteria decision making problem, they 
described the problem in terms of Pythagorean degree of membership. Zhang and Xu (2014) 
presented a number of novel operational concepts of Pythagorean fuzzy sets and outlined their 
properties. The authors also discussed potential applications. In addition to this, they discussed 
a more comprehensive approach to the problems of decision-making based on many criteria, 
which can be resolved by using Pythagorean fuzzy sets. Luqman etal (2020) explained  
Pythagorean fuzzy risk matrices and calculate the risk priority idexes. An explanation of how 
to tackle the transportation problem in a pythagorean fuzzy environment was provided by 
Kumar etal (2019). To address the pythagorean fuzzy linear programming problem, 
Muhammad Akram, Inayat Ullah, and others (2021) came up with the idea of using non-
negative and unconstrained triangular pythagorean fuzzy numbers in conjunction with equality 
constraints. Pythagorean fuzzy modified distribution approach is designed to identify 
optimality. This method was developed by Priyanka Nagar, Pankaj Kumar Srivasta, and their 
colleagues (2021) in order to cope with optimisation of fuzzy species pythagorean 
transportation problem with conserved uncertainties. Using the application of the signature 
approach, Nagoor Gani and V.N. Mohamed Mohamed (2019) were able to solve the 
intuitionistic fuzzy assignment problem. In 2016, Ronald R. Yager provided an explanation of 
the content of pythagorean fuzzy subsets as well as the relationship between those and 
intuitionistic fuzzy subsets. Using Fermat's fuzzy numbers, Laxminarayan Sahoo (2021) was 
able to find a solution to the transportation problem. Ting and Yu Chen (2017) centred their 
attention on the multiple criterion decision analysis by making use of the IVPF details. After 
that, they went into specifics on the outshining decision making strategy for the proximity-
based assignment model. 

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                          In this paper, we proposed an assignment problem technique with triangular  
pythagorean fuzzy numbers. Then we explained a ranking function, arithmetic operations and 
expand a current method for computing optimum result of the pythagorean fuzzy assignment 
model for non-negative constraints in both balanced and unbalanced problems. Then apply the 
suggested techniques for computing real world problems.   
Pythagorean fuzzy numbers are a type of fuzzy numbers that extend the concept of fuzzy sets. 
They were introduced by Yager in 1996 as a way to represent uncertainty or vagueness in a 
more intuitive and expressive manner.  
 
In traditional fuzzy sets, membership degrees are assigned to elements of a set using a 
membership function. Pythagorean fuzzy numbers take a different approach by assigning a 
membership degree to each value within a range or interval. These membership degrees are 
determined based on the Pythagorean fuzzy membership function. 
 
A Pythagorean fuzzy number (PFN) is represented by a triple (L, M, R), where L, M, and R 
represent the left, middle, and right membership degrees, respectively. These membership 
degrees are assigned to the values within a specific interval, indicating the degree of 
membership of each value. 
 
The membership function for Pythagorean fuzzy numbers is defined as follows: 
- For values within the interval [L, M], the membership degree linearly increases from 0 to 1. 
- For values within the interval [M, R], the membership degree linearly decreases from 1 to 0. 
- Values outside the interval [L, R] have a membership degree of 0. 
 
The shape of the membership function resembles a triangular shape, where the middle point M 
is the peak of the membership function, and the left and right sides determine the linear increase 
and decrease of the membership degree. 
 
Pythagorean fuzzy numbers provide a more intuitive way to represent uncertainty, as they 
allow for the expression of both possibility and necessity. The middle value M represents the 
most typical or representative value, while the left and right values indicate the range of 
uncertainty or fuzziness around M. 
 
Pythagorean fuzzy numbers have found applications in decision-making, expert systems, 
pattern recognition, and other areas where uncertainty needs to be captured and analyzed. Their 
representation allows for a more detailed and nuanced representation of uncertainty compared 
to traditional fuzzy sets. 
 Here we ordered as, in part 2 give out some primary definitions. Part 3 explain a 
proposed method to solve triangular pythagorean fuzzy assignment problem with non negative 
constraints. Part 4 gives some numerical examples for both balanced and unbalanced 
assignment problems. Part 5 devoted the conclusion. 

 

Fuzzy Assignment Problem 

The fuzzy assignment problem is an extension of the classical assignment problem that 
incorporates uncertainty or fuzziness in the assignment process. In the classical assignment 
problem, a set of agents or workers needs to be assigned to a set of tasks, each with a certain 

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cost or benefit associated with it. The goal is to find the assignment that minimizes the total 
cost or maximizes the total benefit. 
In the fuzzy assignment problem, instead of having crisp costs or benefits associated with each 
assignment, there are fuzzy numbers or fuzzy costs. Fuzzy numbers represent uncertainty or 
vagueness in the values, allowing for a more flexible representation of preferences or criteria. 
The fuzzy assignment problem involves the following steps: 
Formulating the fuzzy assignment matrix: Create a matrix where the rows represent agents and 
the columns represent tasks. Each element of the matrix contains a fuzzy number representing 
the cost or benefit of assigning a particular agent to a particular task. The fuzzy numbers can 
be represented using membership functions or other fuzzy representation methods. 
Defining fuzzy assignment rules: Specify the rules or criteria for making the assignments based 
on the fuzzy numbers. These rules can be based on fuzzy logic, fuzzy reasoning, or other 
decision-making techniques. The rules consider the fuzzy numbers and their uncertainties to 
determine the best assignments. 
Solving the fuzzy assignment problem: Use fuzzy optimization techniques or fuzzy decision-
making approaches to find the optimal assignments that minimize the overall fuzzy cost or 
maximize the fuzzy benefit. This may involve aggregating the fuzzy numbers, performing 
fuzzy arithmetic operations, or applying fuzzy optimization algorithms to handle the 
uncertainties. 
Interpreting the results: Analyze and interpret the obtained assignments, considering the 
uncertainties and fuzziness in the problem. The assignments may be evaluated based on their 
overall satisfaction of the fuzzy assignment rules and the associated fuzzy costs or benefits. 
The fuzzy assignment problem allows for more realistic modeling of real-world situations 
where the costs or benefits of assignments are uncertain or imprecise. By incorporating 
fuzziness, it provides a more flexible and robust approach to decision-making and assignment 
optimization. 
The solution for a fuzzy assignment problem involves finding the optimal assignments that 
minimize the overall fuzzy cost or maximize the fuzzy benefit, taking into account the 
uncertainties and fuzziness in the problem. Here is a general approach for solving a fuzzy 
assignment problem: 
Formulate the fuzzy assignment problem: Define the problem by specifying the agents, tasks, 
and the fuzzy assignment matrix that represents the fuzzy costs or benefits associated with each 
assignment. 
Define the fuzzy assignment rules: Determine the criteria or rules for making the assignments 
based on the fuzzy numbers. These rules can be based on fuzzy logic, fuzzy reasoning, or other 
decision-making techniques. Consider the preferences, constraints, and objectives of the 
problem. 
Fuzzify the assignment matrix: Convert the fuzzy numbers in the assignment matrix into fuzzy 
sets using appropriate membership functions. This step involves assigning membership degrees 
to each fuzzy number based on the assigned fuzzy set.  
Apply fuzzy optimization techniques: Use fuzzy optimization approaches to find the optimal 
assignments that satisfy the fuzzy assignment rules. This may involve aggregating the fuzzy 
numbers, performing fuzzy arithmetic operations, or applying fuzzy optimization algorithms. 

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Defuzzify the solutions: Convert the fuzzy solutions into crisp assignments by applying a 
defuzzification process. This process involves summarizing the fuzzy information and 
obtaining a crisp result. 
Evaluate the solutions: Assess the quality of the obtained assignments based on the overall 
satisfaction of the fuzzy assignment rules and the associated fuzzy costs or benefits. Analyze 
the trade-offs between different assignments and consider the uncertainties and fuzziness in the 
problem. 
It's important to note that the specific solution approach may vary depending on the problem 
context, the formulation of the fuzzy assignment rules, and the available optimization 
techniques. Various fuzzy optimization methods, such as fuzzy linear programming, fuzzy goal 
programming, or evolutionary algorithms, can be employed to solve fuzzy assignment 
problems. 
 

Preliminaries 

Definition 1: [18 ] 

        Consider 𝒳 as a universal discourse and a pythagorean fuzzy set 𝔙 ∈ 𝒳 is defined as,     
     𝔙  = {(𝜌𝔙(𝔵), 𝜙𝔙 (𝔵)/𝔵 ∈ 𝒳 )} 
   Here membership function is denoted as 𝜌𝔙 and a non-membership function 
is denoted as 𝜙𝔙   respectively, where  
                       𝜌𝔙: 𝒳 → [0,1], 𝔵 ∈ 𝒳 → 𝜌𝔙(𝔵) ∈ [0,1] 
 
                      𝜙𝔙  : 𝒳 → [0,1],   𝔵 ∈ 𝒳 → 𝜙𝔙 (𝔵) ∈ [0,1] 
 
Such that 0 ≤ 𝜌𝔙

2(𝔵) + 𝜙𝔙 2(𝔵)≤ 1 ∀ 𝔵 ∈ 𝒳. Moreover,  ∀ 𝔵 ∈ 𝒳, 𝜋𝔙(𝔵) =
√1 − 𝜌𝔙2(𝔵) − 𝜙𝔙

2(𝔵) is known as a Pythagorean fuzzy index or degree of hesitancy of 𝔵 in 
𝔙. 
 In calculation purpose we consider, 𝛽 = (𝜌𝛽, 𝜙𝛽) is called a pythagoean fuzzy 
numbers (Zhang and Xu 2014 ), where 𝜌𝛽, 𝜙𝛽 ∈ [0,1], 𝜌𝛽2 + 𝜙𝛽2 ≤ 1 and 𝜋𝛽(𝔵) =

√1 − 𝜌𝛽2 − 𝜙𝛽
2 

 

Definition 2 [ 8 ] 

 

 A triangular pythagorean fuzzy number 𝔙=  {( 𝕖, 𝕗, 𝕘);𝓊,𝓋} is a pythagorean fuzzy 
set on 𝔑 , then their (𝜌𝔙) and (𝜙𝔙 ) it is defined as  

 𝜌𝔙(𝔵) = 

{
 
 

 
 

(𝔵−𝕖)𝓊

𝕗−𝕖
, 𝕖 ≤ 𝔵 ≤ 𝕗

𝓊, 𝔵 = 𝕗
(𝕘−𝔵)𝓊

𝕘−𝕗
, 𝕗 ≤ 𝔵 ≤ 𝕘

0, 𝔵 < 𝕖   𝑜𝑟  𝔵 > 𝕘

 

 

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 𝜙𝔙 (𝔵) = 

{
 
 

 
 

[𝕗−𝔵+𝓋(𝔵−𝕖)]

𝕗−𝕖
, 𝕖 ≤ 𝔵 ≤ 𝕗

𝓋, 𝔵 = 𝕗,
[𝔵 − 𝕗 + 𝓋(𝕘 − 𝕗), 𝕗 < 𝑥 ≤ 𝕘]

1, 𝔵 < 𝕖    𝑜𝑟   𝔵 > 𝕘 

                                                                 

 
Here, the merit of 𝓊 & 𝓋 defined the maximum and minimum degree of 𝜌𝔙 & 𝜙𝔙  properly, 
like 𝓊 ∈ [0,1] , 𝓋 ∈ [0,1] and 0≤ 𝓊2 +𝓋2 ≤ 1. 
  
Definition 3 

Considering two triangular Pythagorean fuzzy numbers  𝔅 = {( 𝕖, 𝕗, 𝕘);(ℯ ′, 𝕗, 𝕘′)}   𝔄 =
{(𝒽, 𝒾, 𝒿)(𝒽′, 𝒾, 𝒿′)} and  𝛿 be any real number . Then we defined,  
𝔅 ⊕  𝔄 = {(  𝕖 + 𝒽, 𝕗 + 𝒾, 𝕘 + 𝒿);(  𝕖′ + 𝒽′, 𝕗 + 𝒾, 𝕘′+𝒿′ )} 
- 𝔅 = {(- 𝕘, −𝕗, − 𝕖);(- 𝕘′, −𝕗, − 𝕖′)} 
 𝔅⊖  𝔄 =  {(  𝕖 − 𝒿, 𝕗 − 𝒾, 𝕘 − 𝒽);(  𝕖′ − 𝒿′, 𝕗 − 𝒾, 𝕘′-𝒽′)} 
 𝛿 𝔅= {( 𝛿 𝕖, 𝛿𝕗, 𝛿𝕘);( 𝛿 𝕖ℯ ′, 𝛿𝕗, 𝛿𝕘′)}, 𝛿 > 0 
            {( 𝛿𝕘, 𝛿𝕗, 𝛿 𝕖);( 𝛿𝕘′, 𝛿𝕗, 𝛿 𝕖′)}, 𝛿 < 0 
 

Definition 4 

                  𝔅={(𝕖, 𝕗, 𝕘);(ℯ ′, 𝕗, 𝕘′)} be a triangular Pythagorean non-negative fuzzy number,  
(𝔅 ≥ 0) if 𝕖′ ≥ 0. 
 

Definition 5 [11] 

                Ranking for triangular Pythagorean fuzzy number 𝔅= {( 𝕖, 𝕗, 𝕘);(ℯ ′, 𝕗, 𝕘′)} can be 
stated as,  

  𝔑(𝔅) = (𝕖+2𝕗+𝕘)+(𝕖
′+2𝕗+𝕘′)

8
 

Definition 6  

              Consider an assignment problem of assigning ‘m’workers to ‘m’ work and each 
worker having the capacity of working any work at various money / time period. Let 𝒞~𝑘𝑙 be 
a cost coefficient of triangular pythagorean fuzzy number of allocating the thl  work to the thk  
worker. Let the variable 𝕏~𝑘𝑙serve as the decision maker for the allocation of the assignment. 
Our objective is to reduce the overall cost of the minimization problem by distributing all of 
the labour to the workers who are available while keeping the net cost as low as possible. In 
maximization problem, we have to increase the profit. In unbalanced assignment problem, this 
condition holds but, rows and columns are not equal. 
   

Let the variable 𝕏~𝑘𝑙serve as the decision maker for the allocation of the assignment. Our 
objective is to reduce the overall cost of the minimization problem by distributing all of the 
labour to the workers who are available while keeping the net cost as low as possible. 
                  Mathematical formulation of Pythagorean fuzzy assignment problem as,  

Maximize/Minimize 𝒵~= ∑ ∑ ℂ~𝑘𝑙𝕏
~
𝑘𝑙

𝑚
𝑙=1

𝑚
𝑘=1            →(1) 

                Subject to         ∑ 𝕏~𝑘𝑙
𝑚
𝑙=1 =1    for k = 1, 2 ...m 

                              ∑ 𝕏~𝑘𝑙
𝑚
𝑘=1 =1    for l = 1, 2 ... m          →(2) 

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                      𝕏~𝑘𝑙= 0 (or) 1    
3   Methodology 

                  The proposed method is helped to compute the optimum result of the assignment 
problem under Pythagorean fuzzy environment. 

1. Construct the table of costs associated with the assignment based on the supplied  
problem. 

       2. Determine whether the issue that has been provided is balanced or unbalanced.       
       3. If the situation that has been presented to you is imbalanced, then you should introduce    
          a dummy row or column with entries that have no cost. 
      4. Consider ℂ~𝑘𝑙= {(𝔷𝑘𝑙,   1′ 𝔷𝑘𝑙,

2′  𝔷𝑘𝑙 
3′ ) ; ( 𝔷𝑘𝑙,   1′′ 𝔷𝑘𝑙,

2′  𝔷𝑘𝑙,
3′′)} 

                         𝕏~𝑘𝑙= {(𝔭𝑘𝑙′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′ ) (𝔭𝑘𝑙′′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙′′ )} in (1).  

             Where  ℂ~𝑘𝑙 , 𝕏~𝑘𝑙 are all triangular Pythagorean fuzzy numbers . 
           Then problem (1) becomes, 
 
Maximize/Minimize              𝒵~= ∑ ∑ {(𝔷𝑘𝑙,   

1′ 𝔷𝑘𝑙,
2′  𝔷𝑘𝑙,

3′ ); ( 𝔷𝑘𝑙,   
1′′ 𝔷𝑘𝑙,

2′ 𝔷𝑘𝑙,
3′′)}   

m
l=1

𝑚
𝑘=1

{(𝔭𝑘𝑙′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′ )(𝔭𝑘𝑙′′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙′′ )} 

Subject to 
             ∑ {(𝔭𝑘𝑙

′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′ )(𝔭𝑘𝑙
′′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙
′′ )}𝑚

𝑙=1  = 1 for k = 1,2,...m 
             ∑ {(𝔭𝑘𝑙

′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′ )(𝔭𝑘𝑙
′′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙
′′ )}𝑚

𝑘=1  = 1 for l = 1,2,... m           
 
      5. By using ranking method, then the problem becomes, 
    Maximize/Minimize            ℝ  (𝒵)~=ℝ(∑ ∑ {(𝔷𝑘𝑙,   

1′ 𝔷𝑘𝑙,
2′  𝔷𝑘𝑙,

3′ ); ( 𝔷𝑘𝑙,   
1′′ 𝔷𝑘𝑙,

2′ 𝔷𝑘𝑙,
3′′)}   

m
l=1

𝑚
𝑘=1

{(𝔭𝑘𝑙′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′ )(𝔭𝑘𝑙′′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙′′ )}) 

             ∑ {(𝔭𝑘𝑙
′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙
′ )(𝔭𝑘𝑙

′′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′′ )}𝑚
𝑙=1  = 1 for k = 1,2,...m 

             ∑ {(𝔭𝑘𝑙
′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙
′ )(𝔭𝑘𝑙

′′ , 𝔮𝑘𝑙
′ , 𝔯𝑘𝑙

′′ )}𝑚
𝑘=1  = 1 for l = 1,2,... m           

 
      5. With the help of lingo software technique, solve the crisp problem we get the values 
            𝕏~𝑘𝑙= {(𝔭𝑘𝑙′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙
′ ) (𝔭𝑘𝑙′′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙′′ )}, for k = 1,2,...m  and l = 1,2,... m           
 
      6.  Substitute the above values in    
           ∑ ∑ {(𝔷𝑘𝑙,   

1′ 𝔷𝑘𝑙,
2′  𝔷𝑘𝑙,

3′ ); ( 𝔷𝑘𝑙,   
1′′ 𝔷𝑘𝑙,

2′ 𝔷𝑘𝑙,
3′′)}  

m
l=1

𝑚
𝑘=1 {(𝔭𝑘𝑙′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙
′ )(𝔭𝑘𝑙′′ , 𝔮𝑘𝑙

′ , 𝔯𝑘𝑙′′ )},  
           we can find the optimal solution and the allocation of the Pythagorean fuzzy  
           assignment problem. 
 

Numerical Examples 

4.1. Minimization Pythagorean Fuzzy Assignment Problem 

             A two wheeler dealer wishes to put 2 machines into 2 works. The mechanics 
have truly exceptional types of competencies and they showcase one of kind degrees 
of affectivity from one job to another. The supplier has estimated the wide variety of man-
hours that would be required for every job-man combination. From the 
following desk locate the greatest venture that will end result in minimal man-hours needed. 
 

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 𝒥1 𝒥2 
ℳ1 (3,5,9)(1,5,11) (5,7,10)(3,7,12) 
ℳ2 (4,7,11)(2,7,13) (5,8,10)(3,8,12) 

                                
                  Where the time entries are in triangular Pythagorean fuzzy number  
Solution 

 

1      Let   𝒳~
11= {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )} 
                𝒳~

12 = {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )} 

                       𝒳~
21 = {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}                 →    (𝐼) 
                      𝒳~

22= {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} 
Then  the above problem can be written as,  
Minimize 𝒵~ =     {(3, 5, 9) (1, 5, 11)} {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )} + 
         {(5, 7, 10) (3, 7, 12)} {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ 
 
                             {(4, 7, 11) (2, 7, 13)} {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )} + 
                             {(5, 8, 10) (3, 8, 12)} {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )} 

Subject to 
            {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ {(𝔭12′ , 𝔮12
′ , 𝔯12

′ ) (𝔭12′′ , 𝔮12
′ , 𝔯′12′ )}=1 

            {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}+ {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )}= 1 

            {(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )}+ {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}= 1 
            {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} = 1 
 
Where   
{(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )},{(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )},{(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}, 
{(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ ) are all ≥ 0. 

 
2    Minimize 𝒵~= {(3𝔭11′ , 5𝔮11′ , 9𝔯11′ ) (1𝔭11′′ , 5𝔮11′ , 11𝔯11′′ )} +       
                                {(5𝔭12′ , 7𝔮12′ , 10𝔯12′ ) (3𝔭12′′ , 7𝔮12′ , 12𝔯′12′ )}+ 
                                {(4𝔭21′ , 7𝔮21′ , 11𝔯21′ ) (2𝔭21′′ , 7𝔮21′ , 13𝔯21′′ )}+ 

                                {(5𝔭22 ′ , 8𝔮22′ , 10𝔯22′ ) (3𝔭22′′ , 8𝔮22′ , 12𝔯22′′ )}  
Subject to 

{(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )}+ {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}=1 
            {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}+ {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )}= 1 
            {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}= 1 

            {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )}+ {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )} = 1 

 
3   Applying the ranking function, 
 Minimize 𝒵~=  ℝ {(3𝔭11′ +5𝔭12′ + 4𝔭21

′ + 5𝔭22
′ ), (5𝔮11′ +7𝔮12′ + 7𝔮21

′ + 8𝔮22
′ ), 

                                           (9𝔯11′ +10𝔯12′ + 11𝔯21
′ + 10𝔯22

′ );  
                                           (1𝔭′11′ + 3𝔭′12

′ + 2𝔭′21
′ + 3𝔭′22

′ ), (5𝔮11
′ + 7𝔮12

′ + 7𝔮21
′ + 8𝔮22

′ ),  

Communications on Applied Nonlinear Analysis 
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                                           (11𝔯′11′ + 12𝔯′12
′ + 13𝔯′21

′ + 12𝔯′22
′ )}                               

Subject to 
            {(𝔭11′ + 𝔭12

′ , 𝔮11
′ + 𝔮12 

′ , 𝔯11
′ + 𝔯12

′ ) ; (𝔭′11′ + 𝔭12 
′′ , 𝔮11

′ + 𝔮12
′ , 𝔯11′′ + 𝔯′12′ )} = 1 

            {(𝔭21′ + 𝔭22
′ , 𝔮21

′ + 𝔮22
′ , 𝔯21

′ + 𝔯22
′ ) ; (𝔭21′′ + 𝔭′22′ , 𝔮21′ + 𝔮22

′ , 𝔯′21′ + 𝔯′22
′ )} = 1 

            {(𝔭11′ + 𝔭21
′ , 𝔮11

′ + 𝔮21
′ , 𝔯11

′ + 𝔯21
′ ) ; (𝔭′11′ + 𝔭21

′′ , 𝔮11
′ + 𝔮21

′ ,  𝔯′11′ + 𝔯21
′′ )}= 1 

            {(𝔭12′ + 𝔭22
′ , 𝔮12

′ + 𝔮22
′ , 𝔯12

′ + 𝔯22
′ ) ; (𝔭′12′ + 𝔭′22

′ , 𝔮12
′ + 𝔮22

′ ,  𝔯′12′ +𝔯′22′ )} = 1 
 
4     By using arithmetic operations, then the problem converts into crisp assignment problem, 
Minimize 𝒵~= 
3𝔭11
′ +5𝔭12

′ +4𝔭21
′ +5𝔭22

′ +1𝔭11
′′ +3𝔭12

′′ +2𝔭21
′′ +3𝔭22

′′ +20𝔮11
′ +28𝔮12

′ +28𝔮21
′ +32𝔮22

′ +9𝔯11
′ +10𝔯12

′ +11𝔯21
′ +10𝔯22

′ +11𝔯11
′′ +12𝔯12

′′ +13𝔯21
′′ +12𝔯22

′′

8
 

 Subject to 
𝔭11
′ + 𝔭12

′ = 1          𝔮11
′ +, 𝔮12

′ =1          𝔯11′ + 𝔯12
′ = 1          𝔭′11′ + 𝔭12

′′ = 1        𝔯11′′ + 𝔯′12′ = 1 
𝔭21
′ + 𝔭22

′ = 1        𝔮21
′ + 𝔮22

′ = 1         𝔯11′ + 𝔯12
′ = 1          𝔭21′′ + 𝔭′22′ = 1        𝔯′21′ + 𝔯′22

′ = 1 
𝔭11
′ + 𝔭21

′ = 1        𝔮11′ + 𝔮21
′ = 1         𝔯11′ + 𝔯21

′ =1          𝔭′11′ + 𝔭21
′′ = 1        𝔯′11′ + 𝔯21

′′ = 1 
𝔭12
′ + 𝔭22

′ = 1         𝔮12′ + 𝔮22
′ = 1        𝔯12′ + 𝔯22

′ = 1         𝔭′12′ + 𝔭′22
′ = 1       𝔯′12′ +𝔯′22′ = 1 

     
5. By using software computations we get the following values, 
 

𝔭11
′ = 1 𝔮11

′ = 1 𝔯11
′ = 1 𝔭11

′′ = 1   𝔯′11
′ = 1 

𝔭12
′ = 0 𝔮12

′ = 0 𝔯12
′ = 0  𝔭′12

′ = 0  𝔯′12
′ = 0 

𝔭21
′  = 0  𝔮21

′ = 0 𝔯21
′ = 0 𝔭′21

′ = 0  𝔯′21
′ = 0 

𝔭22
′    = 1 𝔮22

′ = 1 𝔯22
′ = 1 𝔭′22

′ = 1  𝔯′22
′ = 1 

 
6     Substitute the above values in (I)   
       we get  𝒳~

11= {(1,1,1)(1,1,1)}, 𝒳~
12 = {(0,0,0)(0,0,0)} , 𝒳~

21 = {(0,0,0)(0,0,0)} ,                                          
                      𝒳~

22 = {(1,1,1)(1,1,1)}    →    (𝐼𝐼) 
7    Substitute  (𝐼𝐼) in the objective function, we get the optimum solution of the    
       Pythagorean fuzzy assignment problem is {(8,13,19);(4,13,23)}       
         and  the job allocations are,  ℳ1 → 𝒥1,  ℳ2 → 𝒥2 
 
4.2. Maximization Pythagorean Fuzzy Assignment Problem 

 A software technology consultancy wants to assign the project work to two team 
leader. They are receiving the quotation from the team leader for various working cost. 
Company wants to increase their revenue. How can they assign the work to the team for 
increasing their profit? The working costs are given in the following table. 
 

 Team 1 Team 2 
Project 1 (4,6,8) (1,6,11) (4,7,9) (2,7,14) 
Project 2 (3,4,6) (1,4,8) (5,8,9) (3,8,13) 

 
  Where the cost entries are in triangular Pythagorean fuzzy number.  

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 31 No. 2 (2024)

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Solution 

1 Let       𝒳~
11= {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )} 
                𝒳~

12 = {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )} 

                       𝒳~
21 = {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}                 →    (𝐼) 
                      𝒳~

22= {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} 
 
Then the above problem can be written as,  
Maximize 𝒵~=     {(4, 6, 8) (1, 6, 11)} {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ 
         {(4, 7, 9) (2, 7, 14)} {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ 
                              {(3,4, 6) (1,4,8)} {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}+ 
                              {(5, 8, 9) (3, 8,13)} {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )} 

Subject to 
            {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )} = 1 

            {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}+ {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )}= 1 

            {(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )}+ {(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}= 1 
            {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} = 1 
 
Where   
{(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )},{(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )},{(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}, 
{(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ ) are all ≥ 0. 

 
2     Maximize 𝒵~= {(4𝔭11′ , 6𝔮11′ , 8𝔯11′ ) (1𝔭11′′ , 6𝔮11′ , 11𝔯11′′ )} +       
                                {(4𝔭12′ , 7𝔮12′ , 9𝔯12′ ) (2𝔭12′′ , 7𝔮12′ , 14𝔯12′ )}+ 
                                 {(3𝔭21′ , 4𝔮21′ , 6𝔯21′ ) (1𝔭21′′ , 4𝔮21′ , 8𝔯21′′ )}+ 

                                {(5𝔭22′ , 8𝔮22′ , 9𝔯22′ ) (3𝔭22′′ , 8𝔮22′ , 13𝔯22′′ )}  
Subject to 
            {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )}=1 

            {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}+ {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )}= 1 

            {(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )}+{(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}= 1 
            {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} = 1 
3    Applying the ranking function, 
 Maximize 𝒵~=  ℝ {(4𝔭11′ +4𝔭12′ + 3𝔭21

′ + 5𝔭22
′ ), (6𝔮11′ +7𝔮12′ + 4𝔮21

′ + 8𝔮22
′ ), 

                                        (8𝔯11′ +9𝔯12′ + 6𝔯21
′ + 9𝔯22

′ ) ;  
                                        (1𝔭′11′ + 2𝔭′12

′ + 1𝔭′21
′ + 3𝔭′22

′ ), ), (6𝔮11′ +7𝔮12′ + 4𝔮21
′ + 8𝔮22

′ ),    
                                        (11𝔯′11′ + 14𝔯′12

′ + 8𝔯′21
′ + 13𝔯′22

′ )}                                   
Subject to 
            {(𝔭11′ + 𝔭12

′ , 𝔮11
′ +, 𝔮12

′ , 𝔯11
′ + 𝔯12

′ ) ; (𝔭′11′ + 𝔭12
′′ , 𝔮11

′ +, 𝔮12
′ , 𝔯11′′ + 𝔯′12′ )} = 1 

            {(𝔭21′ + 𝔭22
′ , 𝔮21

′ + 𝔮22
′ , 𝔯21

′ + 𝔯22
′ ) ; (𝔭21′′ + 𝔭′22′ , 𝔮21′ + 𝔮22

′ , 𝔯′21′ + 𝔯′22
′ )} = 1 

            {(𝔭11′ + 𝔭21
′ , 𝔮11

′ + 𝔮21
′ , 𝔯11

′ + 𝔯21
′ ) ; (𝔭′11′ + 𝔭21

′′ , 𝔮11
′ + 𝔮21

′ ,  𝔯′11′ + 𝔯21
′′ )}= 1 

            {(𝔭12′ + 𝔭22
′ , 𝔮12

′ + 𝔮22
′ , 𝔯12

′ + 𝔯22
′ ) ; (𝔭′12′ + 𝔭′22

′ , 𝔮12
′ + 𝔮22

′ ,  𝔯′12′ +𝔯′22′ )} = 1 
 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 31 No. 2 (2024)

https://internationalpubls.com 151



4     By using arithmetic operations, the problem converts into crisp assignment problem, 
Maximize 𝒵~= 
4𝔭11
′ +4𝔭12

′ +3𝔭21
′ +5𝔭22

′ +1𝔭11
′′ +2𝔭12

′′ +1𝔭21
′′ +3𝔭22

′′ +24𝔮11
′ +28𝔮12

′ +16𝔮21
′ +32𝔮22

′ +8𝔯11
′ +9𝔯12

′ +6𝔯21
′ +9𝔯22

′ +11𝔯11
′′ +14𝔯12

′′ +8𝔯21
′′ +13𝔯22

′′

8
 

 Subject to 
𝔭11
′ + 𝔭12

′ = 1          𝔮11
′ +, 𝔮12

′ =1          𝔯11′ + 𝔯12
′ = 1       𝔭′11′ + 𝔭12

′′ = 1          𝔯11′′ + 𝔯′12′ = 1 
𝔭21
′ + 𝔭22

′ = 1        𝔮21
′ + 𝔮22

′ = 1         𝔯11′ + 𝔯12
′ = 1       𝔭21′′ + 𝔭′22′ = 1        𝔯′21′ + 𝔯′22

′ = 1 
𝔭11
′ + 𝔭21

′ = 1        𝔮11′ + 𝔮21
′ = 1         𝔯11′ + 𝔯21

′ =1         𝔭′11′ + 𝔭21
′′ = 1       𝔯′11′ + 𝔯21

′′ = 1 
𝔭12
′ + 𝔭22

′ = 1         𝔮12′ + 𝔮22
′ = 1        𝔯12′ + 𝔯22

′ = 1        𝔭′12′ + 𝔭′22
′ = 1      𝔯′12′ +𝔯′22′ = 1 

 
5        By using software computations we get the following values, 

𝔭11
′ = 1 𝔮11

′ = 1 𝔯11
′ = 1 𝔭11

′′ = 1   𝔯′11
′ = 1 

𝔭12
′ = 0 𝔮12

′ = 0 𝔯12
′ = 0  𝔭′12

′ = 0  𝔯′12
′ = 0 

𝔭21
′  = 0  𝔮21

′ = 0 𝔯21
′ = 0 𝔭′21

′ = 0  𝔯′21
′ = 0 

𝔭22
′    = 1 𝔮22

′ = 1 𝔯22
′ = 1 𝔭′22

′ = 1  𝔯′22
′ = 1 

 
6     Substitute the above values in  (I)   
       we get  𝒳~

11= {(1,1,1)(1,1,1)}, 𝒳~
12 = {(0,0,0)(0,0,0)} , 𝒳~

21 = {(0,0,0)(0,0,0)} ,                                          
                      𝒳~

22 = {(1,1,1)(1,1,1)}      →    (𝐼𝐼) 
7    Substitute (𝐼𝐼)   in the objective function, we get the optimum solution of the  
      Pythagorean fuzzy assignment problem is {(9,14,17);(4,14,24)}               
       and  the job allocations are,     𝑝𝑟𝑜𝑗𝑒𝑐𝑡 1 → 𝑇𝑒𝑎𝑚 1, 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 2 → 𝑇𝑒𝑎𝑚 2 
 
4.3 Unbalanced Pythagorean Fuzzy Assignment Problem 

    Calculate the optimum solution of the Pythagorean fuzzy assignment problem from     
    the following table values. 
 

 𝒫1 𝒫2 
𝒲 (2,4,8)(1,4,10) (2,3,5)(1,3,7) 

 
Solution 

  The problem that has been presented is an example of an unbalanced 
Pythagorean fuzzy assignment problem. Add dummy row/column with zero cost entries, for 
converting it into the balanced problem. Hence the problem becomes, 
 

 𝒫1 𝒫2 
𝒲1 (2,4,8)(1,4,10) (2,3,5)(1,3,7) 
𝒲2 (0,0,0)(0,0,0) (0,0,0)(0,0,0) 

 
1 Let       𝒳~

11= {(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )} 

                𝒳~
12 = {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )} 
                       𝒳~

21 = {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}                 →    (𝐼) 

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                      𝒳~
22= {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )} 

Then, modify the above problem as follows, 
Minimize  𝒵~ =   {(2, 4, 8) (1,4,10)} {(𝔭11′ , 𝔮11′ , 𝔯11′ )(𝔭11′′ , 𝔮11′ , 𝔯11′′ )}+ 
        { (2,3,5) (1,3,7)} {(𝔭12′ , 𝔮12′ , 𝔯12′ )(𝔭12′′ , 𝔮12′ , 𝔯′12′ )}+ 
                             {(0, 0, 0) (0,0,0)} {(𝔭21′ , 𝔮21′ , 𝔯21′ )(𝔭21′′ , 𝔮21′ , 𝔯21′′ )}+  
                             {(0, 0, 0) (0,0,0)} {(𝔭22′ , 𝔮22′ , 𝔯22′ )(𝔭′22′ , 𝔮22′ , 𝔯′22′ )} 
Subject to 
            {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )}=1 

            {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}+ {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )}= 1 

            {(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )}+{(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}= 1 
            {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} = 1 
 
Where   
{(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )},{(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )},{(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}, 
{(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ ) are all ≥ 0. 

 
2        Minimize  𝒵~= {(2𝔭11′ , 4𝔮11′ , 8𝔯11′ ) (1𝔭11′′ ,4𝔮11′ , 10𝔯′11′ )} +  

           {(2𝔭12′ , , 3𝔮12′ , 5𝔯12′ ) (1𝔭12′′ ,  3𝔮12′ , 7𝔯12′′ )}+ 
           {(0𝔭21′ , 0𝔮21′ , 0𝔯21′ ) (0𝔭21′′ , 0𝔮21′ , 0𝔯21′′ ) +  

                                   {(0𝔭22′ , 0𝔮22′ , 0𝔯22′ ) (0𝔭22′′ , 0𝔮22′ , 0𝔯22′′ )}  
Subject to 
            {(𝔭11′ , 𝔮11

′ , 𝔯11
′ )(𝔭11′′ , 𝔮11

′ , 𝔯11′′ )}+ {(𝔭12′ , 𝔮12
′ , 𝔯12

′ )(𝔭12′′ , 𝔮12
′ , 𝔯′12′ )}=1 

            {(𝔭21′ , 𝔮21
′ , 𝔯21

′ )(𝔭21′′ , 𝔮21
′ , 𝔯21′′ )}+ {(𝔭22′ , 𝔮22

′ , 𝔯22
′ )(𝔭′22

′ , 𝔮22
′ , 𝔯′22′ )}= 1 

            {(𝔭11′ , 𝔮11
′ , 𝔯11

′ )(𝔭11′′ , 𝔮11
′ , 𝔯11′′ )}+{(𝔭21′ , 𝔮21

′ , 𝔯21
′ )(𝔭21′′ , 𝔮21

′ , 𝔯21′′ )}= 1 
            {(𝔭12′ , 𝔮12

′ , 𝔯12
′ )(𝔭12′′ , 𝔮12

′ , 𝔯′12′ )}+ {(𝔭22′ , 𝔮22
′ , 𝔯22

′ )(𝔭′22
′ , 𝔮22

′ , 𝔯′22′ )} = 1 
 
3 Applying ranking function, 
 Minimize 𝒵~= ℝ {{(2𝔭11′ +2𝔭12′ + 0𝔭21

′ + 0𝔭22
′ ), (4𝔮11′ +3𝔮12′ + 0𝔮21

′ + 0𝔮22
′ ),   

                                             (8𝔯11′ +5𝔯12′ + 0𝔯21
′ + 0𝔯22

′ ) ;  
                                        (1𝔭11′′ + 1𝔭12′′ + 0𝔭21′′ + 0𝔭22′′ ), ), (4𝔮11′ +3𝔮12′ + 0𝔮21

′ + 0𝔮22
′ ),    

                                        (10𝔯11′′ + 7𝔯12′′ + 0𝔯′21′ + 0𝔯22
′′ )}}                                    

Subject to 
            {(𝔭11′ + 𝔭12

′ , 𝔮11
′ +, 𝔮12

′ , 𝔯11
′ + 𝔯12

′ ) ; (𝔭′11′ + 𝔭12
′′ , 𝔮11

′ +, 𝔮12
′ , 𝔯11′′ + 𝔯′12′ )} = 1 

            {(𝔭21′ + 𝔭22
′ , 𝔮21

′ + 𝔮22
′ , 𝔯21

′ + 𝔯22
′ ) ; (𝔭21′′ + 𝔭′22′ , 𝔮21′ + 𝔮22

′ , 𝔯′21′ + 𝔯′22
′ )} = 1 

            {(𝔭11′ + 𝔭21
′ , 𝔮11

′ + 𝔮21
′ , 𝔯11

′ + 𝔯21
′ ) ; (𝔭′11′ + 𝔭21

′′ , 𝔮11
′ + 𝔮21

′ ,  𝔯′11′ + 𝔯21
′′ )}= 1 

            {(𝔭12′ + 𝔭22
′ , 𝔮12

′ + 𝔮22
′ , 𝔯12

′ + 𝔯22
′ ) ; (𝔭′12′ + 𝔭′22

′ , 𝔮12
′ + 𝔮22

′ ,  𝔯′12′ +𝔯′22′ )} = 1 
 
4      By using arithmetic operations, then the problem converts into crisp assignment 
problem, 
Minimize 𝒵~ = 
2𝔭11
′ +2𝔭12

′ +0𝔭21
′ +0𝔭22

′ +1𝔭11
′′ +1𝔭12

′′ +0𝔭21
′′ +0𝔭22

′′ +16𝔮11
′ +12𝔮12

′ +0𝔮21
′ +0𝔮22

′ +8𝔯11
′ +5𝔯12

′ +0𝔯21
′ +0𝔯22

′ +10𝔯11
′′ +7𝔯12

′′ +0𝔯21
′′ +0𝔯22

′′

8
 

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 Subject to 
𝔭11
′ + 𝔭12

′ = 1          𝔮11
′ +, 𝔮12

′ =1          𝔯11′ + 𝔯12
′ = 1       𝔭′11′ + 𝔭12

′′ = 1         𝔯11′′ + 𝔯′12′ = 1 
𝔭21
′ + 𝔭22

′ = 1        𝔮21
′ + 𝔮22

′ = 1         𝔯11′ + 𝔯12
′ = 1       𝔭21′′ + 𝔭′22′ = 1        𝔯′21′ + 𝔯′22

′ = 1 
𝔭11
′ + 𝔭21

′ = 1        𝔮11′ + 𝔮21
′ = 1         𝔯11′ + 𝔯21

′ =1         𝔭′11′ + 𝔭21
′′ = 1       𝔯′11′ + 𝔯21

′′ = 1 
𝔭12
′ + 𝔭22

′ = 1         𝔮12′ + 𝔮22
′ = 1        𝔯12′ + 𝔯22

′ = 1        𝔭′12′ + 𝔭′22
′ = 1      𝔯′12′ +𝔯′22′ = 1 

 
5    By using software computations we get the following values 
 

𝔭11
′ = 0 𝔮11

′ = 0 𝔯11
′ = 0 𝔭11

′′ = 0   𝔯′11
′ = 0 

𝔭12
′ = 1 𝔮12

′ = 1 𝔯12
′ = 1  𝔭′12

′ = 1  𝔯′12
′ = 1 

𝔭21
′  = 1  𝔮21

′ = 1 𝔯21
′ = 1 𝔭′21

′ = 1  𝔯′21
′ = 1 

𝔭22
′    = 0 𝔮22

′ = 0 𝔯22
′ = 0 𝔭′22

′ = 0  𝔯′22
′ = 0 

 
6       Putting the above values in (I) 
         we get   𝒳~

11= {(0,0,0)(0,0,0)},  𝒳~
12 = {(1,1,1)(1,1,1)} ,  𝒳~

21 = {(1,1,1)(1,1,1)} ,                                           
                         𝒳~

22 = {(0,0,0)(0,0,0)}       →    (𝐼𝐼) 
 
7     Substitute   (𝐼)  into the objective function , we get the optimum solution of the  
       pythagorean fuzzy assignment problem is {(2,3,5);(1,3,7)}       
       and  the job allocations are, 𝒲1 → 𝒫2, 𝒲2 → 𝑁𝑜 𝑗𝑜𝑏 
 
 5   Conclusion 

                      In an intuitionistic fuzzy sets some of the conditions are give up, it will form 
Pythagorean fuzzy sets.  It will cover more areas comparing with intuitionistic fuzzy sets. 
Hence it gives an extra opportunity to improving the knowledge, creating it feasible to narrate 
numerous actual complication in a extra fabulous path. We have solved Pythagorean fuzzy task 
problem primarily based on triangular Pythagorean fuzzy numbers. We have introduced some 
functions of the proposed approach to actual world problem. The benefits of the current find 
out about are, (i) it is very effortless to practice in selection making and (ii) It solves the 
problem of the task when all of the variables and parameters are positive non-negative 
triangular Pythagorean integers. (iii) consider giant quantity of data’s arise, we can without 
difficulty calculate the most fulfilling cost by using this software.  In future we are designing 
to prolong our exertion to (i) Interval Pythagorean fuzzy assignment problem (ii) Fractional 
pythagoean fuzzy assignment problem. 
 

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