Arithmetic Operations of Generalized Triangular Picture Fuzzy Numbers with Applications

Subject Areas : Fuzzy Optimization and Modeling Journal

Mohammad Hasan ¹ , Abeda Sultana ² , Nirmal Mitra ³

1 - Department of Mathematics and Statistics, BUBT
2 - Department of Mathematics, Jahangirnagar University, Saver, Bangladesh
3 - Department of Mathematics and Statistics, Bangladesh University of Business and Technology, Dhaka, Bangladesh

Received: 2023-08-23 Accepted : 2023-11-14 Published : 2023-12-01

Keywords: Picture Fuzzy Set, Generalized Triangular Picture Fuzzy Number, Arithmetic Operations, Picture Fuzzy Linear Equations,

Abstract :

Picture fuzzy set is the generalization of intuitionistic fuzzy set as well as the fuzzy set considering the positive, neutral and negative membership functions of an element. In this article, we develop the arithmetic operations on generalized triangular picture fuzzy numbers by (α,γ,β)-cut method. Some related properties of them are explored. Finally, picture fuzzy linear equations are solved by using these arithmetic operations.

Full-Text:



E-ISNN: 2676-7007	Fuzzy Optimization and Modelling 2(2) (2021) 46-57
	Contents lists available at FOMJ     Fuzzy Optimization and Modelling     Journal homepage: http://fomj.qaemiau.ac.ir/

Paper Type: Research Paper

Arithmetic Operations of Generalized Triangular Picture Fuzzy Numbers with Applications

 

1.       Introduction

The concept of fuzzy set theory was introduced by L. A. Zadeh [33] in 1965 to handle uncertain data. Fuzzy set theory takes into account membership degree only. The non-membership degree is the direct complement of the membership degree. However, in many researches it is found that, this linguistic negation does not satisfy the logical negation always in the real life applications. Because, while selecting the membership degree for an object (element), there may be some kinds of hesitation while defining the membership function, as membership function may be Gaussian, triangular, exponential or any other membership functions. So, due to this hesitation, the non-membership degree is less than or equal to the complement of the membership degree. This is the reason why different results are obtained with different membership functions. To overcome this situation, after about two decades, in 1986, K.T. Atanassov [1] suggested the concept of intuitionistic fuzzy set, where the non-membership degree is not equal to the complement of the membership degree due to the fact that some kinds of hesitations or lack of knowledge is present while defining the membership function. So the intuitionistic fuzzy set theory is an important generalization of fuzzy set theory, where the membership degree and the non-membership degree separately in such a way that, sum of the two degrees must not exceed . It is observed that, fuzzy sets are intuitionistic fuzzy sets but converse is not necessarily correct.

Sometimes, fuzzy set and intuitionistic fuzzy set are found to be difficult to express the situations, when human thoughts involve more options like ‘yes’, ‘abstain’, ‘no’ and ‘refusal’. In this regards, Bui Cong Cuong and Vladik Kreinovich [8, 9] introduced picture fuzzy set in 2013, which is the direct extension of the fuzzy set and the intuitionistic fuzzy set by incorporating the concept of positive, negative and neutral membership degrees of an element. In picture fuzzy set, the authors have divided the hesitation margin,  of intuitionistic fuzzy set into two parts which are neutral membership degree and refusal membership degree. When both of the neutral and refusal membership degrees are zero, i.e., hesitation margin becomes zero, then the picture fuzzy set returns to intuitionistic fuzzy set.

Fuzzy set theory [33] and the intuitionistic fuzzy set theory [1] have been used for handling fuzzy decision-making problems for a long extent of time. But many researchers have shown interest in picture fuzzy set theory [8, 9] and applied it to the field of decision making. Picture fuzzy set manage the uncertain nature of human thoughts involving more options like yes, abstain, no and refusal terms.

In 1972, Chang and Zadeh [7] introduced the concept of fuzzy numbers and discussed its arithmetic operations. Later a host of researchers studied the concept of fuzzy numbers [3, 6, 12, 13, 17, 28, 31, 32, 34]. Islam et al. [16] discussed the comparison of classical method, extension principle and α-cuts and interval arithmetic method in solving system of fuzzy linear equations in 2019. The concept of intuitionistic fuzzy number was proposed by Burillo [4] in 1994. Mahapatra and Roy [20] presented triangular intuitionistic fuzzy number and used it for reliability evaluation. In 2010, they also defined trapezoidal intuitionistic fuzzy number [21] and discussed its arithmetic operations based on -cut method. Later numerous works have been done on intuitionistic fuzzy number and applied it in many branches of science and engineering (see [2, 5, 18, 19, 22, 23, 24, 25, 26, 27, 29, 30]. Dutta and Ganju [10] first defined extension principle for picture fuzzy sets and describe some arithmetic operations of picture fuzzy sets. P. Dutta. et al. [11] discussed the operations on picture fuzzy numbers and applied in Multi-criteria Group Decision Making Problems.

In this paper, we established the arithmetic operations on generalized triangular picture fuzzy numbers by cut method and some related properties of them are explored. At the end of this work, picture fuzzy linear equations are solved by using these arithmetic operations. 

The article is organized as follows: In section 2, some basic definitions are given which are essential to rest of the paper. In section 3, arithmetic operations of triangular picture fuzzy numbers are discussed. In section 4, solution of picture fuzzy linear equations is discussed. In section 5, conclusion remark is given.

2.       Preliminaries

In this section, some basic definitions are given which are essential to rest of the paper.

Definition 2.1 ([33]). Let be non-empty set. A fuzzy set  in  is given by

,

where.

Definition 2.2 ([1]). An intuitionistic fuzzy set  in  is given by

,

where and , with the condition .

The values  and  represent, respectively, the membership degree and non-membership degree of the element  to the set

For any intuitionistic fuzzy set  on the universal set, for

is called the hesitancy degree (or intuitionistic fuzzy index) of an element  in . It is the degree of indeterminacy membership of the element  whether belonging to  or not.

Obviously,  for any.

Particularly,  is always valid for any fuzzy set  on the universal set .

The set of all picture fuzzy sets in  will be denoted by .

Definition 2.3 ([8, 9]). A picture fuzzy set  on a universe of discourse  is of the form

,

where  is called the degree of positive membership of  in ,  is called the degree of neutral membership of  in  and  is called the degree of negative membership of  in , and where  and  satisfy the following condition:

.

Here  is called the degree of refusal membership of  in.

The set of all picture fuzzy sets in  will be denoted by.

Definition 2.4 ([8, 9]). Let, then the subset, equality, the union, intersection and complement are defined as follows:

i.                       iff and ;

ii.                       iff and ;

iii.                      ;

iv.                      ;

v.                      .

Definition 2.5. ([14]).  Let  be a picture fuzzy set on  and, then the -cut of  is given by

 

That is, ,  and  are ,  and - cut of positive membership, neutral membership and negative membership of a picture fuzzy set  respectively.

Definition 2.6.  Let  be a picture fuzzy set on. We now define a new picture fuzzy set denoted by, is defined by

,

where the positive membership, neutral membership  and negative membership  are as follows:

 

 

 

Theorem 2.7. (First decomposition theorem):

Let  be a non-empty set. For a picture fuzzy set  in ,

 

3.       Arithmetic Operations of Generalized Triangular Picture Fuzzy Numbers

In this section, the arithmetic operations of triangular picture fuzzy numbers are discussed. Throughout this section,  denotes the set of all real numbers.

Definition 3.1. ([15]). Let, with A generalized picture fuzzy number (GPFN)  is a special picture fuzzy set of real numbers  whose positive, neutral and negative membership functions ,  and  respectively satisfy the following conditions:

i.                      There exist at least three real numbers  and  such that  ,  and .

ii.                       and  are quasi concave and upper semi continuous on .

iii.                       is quasi convex and lower semi continuous on .

iv.                      The support of  is compact.

Definition 3.2.  A generalized triangular picture fuzzy number (GTPFN)

   is a special picture fuzzy set on  whose positive, neutral and negative membership functions are defined as follows:

 

A R T I C L E I N F O		A B S T R A C T In many practical situations of real life, we face some data which are more vague than exact. To give modelling these vague data a host of researchers have become involved and introduced numerous theories. Picture fuzzy set is one of these which is much capable to deal with these vague data. Picture fuzzy set is the generalization of intuitionistic fuzzy set as well as the fuzzy set considering the positive, neutral and negative membership degrees of an element. After the innovation of this concept, it has been widely studied and applied in many fields of real life situations specially in science and engineering.  In this article, we develop the arithmetic operations on generalized triangular picture fuzzy numbers by cut method. Some related properties of them are explored. Finally, picture fuzzy linear equations are solved by using these arithmetic operations.
Article history:
Keywords: Picture Fuzzy Set  Generalized Triangular Picture Fuzzy Number Arithmetic Operations Picture Fuzzy Linear Equations

Definition 3.3. A Generalized Triangular Picture Fuzzy Number   is called positive if  , for

Definition 3.4. A Generalized Triangular Picture Fuzzy Number   is called negative if  , for

Definition 3.5. A Generalized Triangular Picture Fuzzy Number   is called partial if at least one  , for

Definition 3.6. Let   be a GTPFN. Then the cut of  is a crisp subset of  which is defined as follows:

 

  

                                         ; .

Definition 3.7. Let   be a GTPFN. Then the cut of  is a crisp subset of  which is defined as follows:

 

       

                                              ; .

Definition 3.8. Let   be a GTPFN. Then the cut of  is a crisp subset of  which is defined as follows:

 

      

                                                            ; .

Definition 3.9. Let   be a GTPFN. Then thecut of  is given by

; , , ,.

 

Definition 3.10. Let  and  be two GTPFNs and their corresponding cuts be

 

 

for any   and , then the four basic arithmetic operations such as addition, subtraction, multiplication and division are defined as follows:

Addition:

 

                        .

Subtraction:

  

                             .

Multiplication:

 

                         , where

,

,

 

Division:

 

, where,

 ,

 ,

where

Example 3.11. Let   and  be two positive triangular picture fuzzy numbers where the positive , neutral and negative membership functions are as follows:

 , ,

 

and

 , ,

 

The corresponding cut of the above triangular picture fuzzy numbers  and  are as follows:

 

 

Now,

 

 

 

                         

 

                       

Now the positive, neutral and negative membership functions of resulting GTPFNs are

 , ,

  .

 , ,

  .

 

  ,

 

 

  ,

 

Proposition 3.12. Addition of two generalized triangular picture fuzzy numbers is also a generalized triangular picture fuzzy number.

Proof:  Let  and  be two GTPFNs . The Addition of two GTPFNs based on cut method is

 , where

 

 

, and ,  , , , . 

Now,

 

 

 

.

Let .

Now

 

 

.

Let

Now,

          

                          ; if .

Therefore,  is an increasing function.

Also

 

Again,

 

 

 

 

 

 

Let

Now,

 ; if

Therefore,  is an decreasing function.

Also

 

So the positive membership function of  is

 

Hence the positive membership function preserved the triangular shape.

In similar manner, we can also prove that the neutral membership of  is

 

So the neutral membership function preserved the triangular shape.

Now, for the negative membership function

  

 

 

 .

Let

Now

 

 

 

 

Let

Now,

           

 ; if

Therefore,  is an decreasing function.

Also, .

          

Again,

 

 

 

 

.

Let

Now,

 ; if .

Therefore,  is an increasing function.

Also, .

           .

So the negative membership function of  is

 

Hence the negative membership function preserved the triangular shape.

Therefore, the addition of two GTPFNs is also a GTPFN.

Proposition 3.13. Subtraction of two generalized triangular picture fuzzy numbers is also a generalized triangular picture fuzzy number.

Proof : Let  and  be two GTPFNs . The subtraction of two GTPFNs based on cut method is

 , where

 

 

, and ,  , , , . 

Now,

 

 

 

 

 .

Let .

Now,

 

 

 

Let

Now,

 ; if .

Therefore,  is an increasing function.

Also .

.

Again,

 

 

 

 

 

 

Let

Now,

; if

Therefore,  is an decreasing function.

Also .

.

So the positive membership function of  is

 

So the positive membership function preserved the triangular shape.

In similar manner, we can also prove that the neutral membership of  is

 

So the neutral membership function preserved the triangular shape.

Now, for the negative membership function

 

 

 

 

 

 

Let

Now,

 

 

 

 

 

.

Let .

Now,

 ; if .

Therefore,  is a decreasing function.

Also   .

          . 

Again,

 

 

 

 

.

Let .

Now,

 

 ; if

Therefore,  is an increasing function.

Also, .

 

 

So the negative membership function of  is

 

So the negative membership function preserved the triangular shape.

Therefore, the subtraction of two GTPFNs is also a GTPFN.

4.       Solution of Picture Fuzzy Linear Equations

In this section, the solution of picture fuzzy linear equations of the forms  and  , where  and  are GTPFNs are discussed.

4.1 Solution of picture fuzzy linear equations of the form

Let  and  be two positive GTPFNs Then

                                              (1)

is a picture fuzzy linear equation

Let . Then  is not a solution of the equation (1).

For any ,  and  let

        ,

         and

        .

Then the equation (1) has a solution iff

 (i),

and  for every ,  and  and

 (ii)  ,  and  implies

,

, and

.

The property (i) ensures that the interval equation

                                             (2)

has a solution which is  . Property (ii) ensures that the solutions of the interval equations for  and  are nested; that is, if  ,  and  then . If a solution  exists ,  and  and property (ii) is satisfied, then the solution of  of the picture fuzzy linear equation is given by

 

Example 4.2. Let  and  in our equation be the following generalized triangular shape picture fuzzy numbers

 

, ,

,

 

Fig. 2: GTPFN .

 ,

 .

 

Fig. 3: GTPFN .

Then the corresponding  cuts of  and  are 

 and

.

It is easy to verify that,

,

,

.

Or,

,

,

.

Consequently,

 

It is easy to check that,  ,  and  implies

 for each pair  ,   and .

Therefore, the solution of the picture fuzzy equation is

     implies

 

, , .

 

 

Fig. 4: Solution  of the equation 1.

4.3 Solution of picture fuzzy linear equations of the form

Let  and  be two positive GTPFNs Then

                                              (3)

is a picture fuzzy linear equation

Let . Then  is not a solution of the equation (3).

For any ,  and  let

        ,

         and

        , then we obtained the interval equation

                            (4)

The equation (3) can be solved the interval equation (4) for ,  and .

Then the equation (3) has a solution iff

(i) (

 

And  for every ,  and  and

(ii)  ,  and  implies

,

,

And

 

The property (i) ensures that the interval equation

If a solution exists, it has of the form

 

Example 4.4. Let  and  in our equation be the following triangular shape picture fuzzy numbers

, ,

.

And

, ,

.

Then the corresponding  cuts of  and  are 

,

.

It is easy to verify that,

,

,

.

Or,

,

,

.

Consequently,

.

It is easy to check that,  ,  and  implies

 for each pair  ,   and .

Therefore, the solution of the picture fuzzy equation is

     implies

, ,

.

 

Fig. 5: Solution  of the equation 3.

5.       Conclusion

Picture fuzzy number plays an important role in many branches of science and engineering where the situation is not controlled by fuzzy number and intuitionistic fuzzy number. The cut is the bridge between classical sets and picture fuzzy sets. The cut method is a standard method for performing different arithmetic operations like addition, subtraction, multiplication and division. Here the arithmetic operations on generalized triangular picture fuzzy numbers by cut method are developed and some related properties of them are explored. Finally, applying these arithmetic operations picture fuzzy linear equations are solved.

Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

[1]	Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87-96. http://dx.doi.org/10.1016/S0165-0114(86)80034-3.
[2]	Atanassov, K. T., &  Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31 (3), 343-349. http://dx.doi.org/10.1016/0165-0114(89)90205-4.
[3]	Buckley, J., &  Qu, Y. (1991). Solving systems of linear fuzzy equations. Fuzzy Sets and Systems, 43, 33-43.
[4]	Burillo, P., Bustince, H. &  Mohedano, V. (1994). Some definition of intuitionistic fuzzy number. Fuzzy based expert systems, fuzzy Bulgarian enthusiasts, Sofia, Bulgaria, 28-30.
[5]	Chakraborty, D.,  Jana D. K., &  Roy T. K. (2014). Arithmetic operations on generalized intuitionistic fuzzy number and its applications to transportation problem, Springer.DOI: 10.1007/s12597-014-0194-1.
[6]	Chen, S.J., &  Hwang, C.L. (1992). Fuzzy Multiple Attribute Decision Making. Springer, Berlin.
[7]	Chang, S. S. L., &  Zadeh, L. A. (1972). On fuzzy mapping and control. IEEE Transaction on Systems, Man and Cybernetics, 2 (1), 30-34. http://dx.doi.org/10.1109/TSMC.1972.5408553.
[8]	Cuong, B.C., Kreinovich, V. (2013). Picture Fuzzy Sets- a new concept for computational intelligence problems. Proceedings of the Third World Congress on Information and Communication Technologies WIICT, 1–6.
[9]	Cuong, B.C. (2014). Picture Fuzzy Sets. Journal of Computer Science and Cybernetics, 30(4), 409-420.
[10]	Dutta P., & Ganju, S. (2018). Some aspects of picture fuzzy set. Transactions of A. Razmadze Mathematical Institute, 172, 164–175.
[11]	Dutta. P. et al. (2020). Operations on Picture Fuzzy Numbers and Their Application in Multi-criteria Group Decision Making Problems. Springer Nature Switzerland AG 2020,169–188, 2020. https://doi.org/10.1007/978-3-030-39033-4_17.
[12]	Dubois D. &  Prade, H. (1980). Fuzzy Sets and Systems Theory and Applications. Academic Press, 36-41.
[13]	Dubois D. &  Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9, 613-626. http://dx.doi.org/10.1080/00207727808941724.
[14]	Hasan M.K., Ali M. Y., Sultana A., & Mitra N. K. (2023). Extension principles for picture fuzzy sets. Journal of Intelligent & Fuzzy Systems, 44, 6265-6275. DOI: 10.3233/JIFS-220616 .
[15]	Hasan M.K., Sultana A., & Mitra N. K. (2023). Arithmetic Operations of Generalized Trapezoidal Picture Fuzzy Numbers by Vertex Method. American Journal of Computational Mathematics, 13(1), 99-121. DOI: 10.4236/ajcm.2023.131004.
[16]	Islam, S., Saiduzzaman, M., Shafiqul Islam, M. and Sultana, A. (2019). Comparison of Classical Method, Extension Principle and α-Cuts and Interval Arithmetic Method in Solving System of Fuzzy Linear Equations. American Journal of Computational Mathematics, 9, 1-24. doi: 10.4236/ajcm.2019.91001.”
[17]	Klir G. &Yaun B. (1977). Fuzzy Set and Fuzzy logic: Theory and Application.Upper Saddle River. NJ: Prentice Hall.
[18]	Li, D. F. (2010). A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to madm problems. Computer and Mathematics with Applications, 60,1557-1570.
[19]	Li, D.F. &  Nan, J. (2011). Extension of the TOPSIS for multi-attribute group decision making under Atanassov IFS environment. International Journal of Fuzzy System Applications,1(4), 47-61.
[20]	Mahapatra, G. S., &  Roy, T. K. (2009). Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations. Proceedings of World Academy of Science, Engineering and Technology, Malaysia, 38, 587-585.
[21]	Mahapatra, G. S. &  Mahapatra, B. S. (2010). Intuitionistic fuzzy fault tree analysis using intuitionistic fuzzy numbers. International Mathematical Forum, 5(21), 1015-1024.
[22]	Mitchell, H. B. (2004). Ranking intuitionistic fuzzy numbers. International Journal of Uncertainty, Fuzzyness and Knowledge Based Symtems,12, 377-386.
[23]	Parvathi, R., &  Malathi, C. (2012). Arithmetic operations on symmetric trapezoidal intuitionistic fuzzy numbers. International Journal of Soft Computing and Engineering, 2 (2), 268-273.
[24]	Robinson, J. &  Amirtharaj, E. C. (2011). Vague correlation coefficient of interval vague sets. International Journal of Fuzzy System Applications, 2(1), 18-34.
[25]	Shu, M. H.,  Cheng, C. H., &  Chang, J. R. (2006). Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly. Microelectronics Reliability, 46, 2139-2148.
[26]	Seikh, M. R.,  Nayak, P.K. &  Pal., M. (2013). Notes on triangular intuitionistic fuzzy numbers.  International Journal of Mathematics in Operational Research, 5, 446-465.https://doi.org/10.1504/IJMOR.2013.054730.
[27]	Venketeshwari, V. L. G., &  Sivaraman, G. (2006). Ranking of intuitionistic fuzzy numbers. IEEE International Conference on Fuzzy Systems, 1971-1974.
[28]	Wang, X. and Kerre,  E.E. (2001). Reasonable properties for the ordering of fuzzy quantities. Fuzzy Sets and Systems, 118, 375-385.
[29]	Wang, J. Q., & Zhang, Z. (2009). Multi-criteria decision-making method with incomplete certain information based on intuitionistic fuzzy number. Control and Decision, 24(2), 226-230.
[30]	Xu, Z. S. (2007). Intuitionist fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 15(6), 1179-1187. http://dx.doi.org/10.1109/TFUZZ.2006.890678.
[31]	Yager, R.R. (1980). On choosing between fuzzy subsets. Kybemetes, 9, 151-154.
[32]	Yager, R.R. (1981). A procedure for ordering fuzzy subests of the unit interval. Inf. Sci., 24, 143-161.
[33]	Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8 (3), 338-356. http://dx.doi.org/10.1016/S0019-9958(65)90241-X.
[34]	Zadeh, L.A. (1975). The concept of a linguistic variable and its application to approximate reasoning. Parts 1, 2, and 3. Information Science, 8: 199–249; 8: 301–357;, 9: 43–80.

 

---

 

* Correspondig author

E-mail address: [e-mail]  (Mohammad Kamrul Hasan)

 

DOI:

---