Fermatean fuzzy sets (FFSs), proposed by Senapati and Yager (

Orthopair fuzzy sets are fuzzy sets in which the membership grades of an element

PFSs have attracted the attention of many researchers within a short period of time. For example, Yager (

Senapati and Yager (

Senapati and Yager (

The subtraction and division operations of IFS have been introduced by some authors (Atanassov,

In this section, we have given some definitions, and that, hopefully will come in handy in our work.

IFS, which was invented or formerly explained by Atanassov (

(See Atanassov,

Currently, Yager (

(PFSs) The Pythagorean fuzzy sets defined on a non-empty set

In this section, Fermatean fuzzy sets are defined in detail and their corresponding properties are discussed. Score and accuracy functions of these sets have been defined and compared.

(See Senapati and Yager,

For any FFS

For convenience, Senapati and Yager called

We shall point out the membership grades related to Fermatean fuzzy sets as Fermatean membership grades (FMGs).

(See Senapati and Yager,

Comparison of space of FMGs, PMGs and IMGs.

This development can be evidently recognized from Fig.

(See Senapati and Yager,

(See Senapati and Yager,

(See Senapati and Yager,

(See Senapati and Yager,

(See Senapati and Yager,

(See Senapati and Yager,

In order to rank FFNs, we define the score function of the FFN:

For any FFN

By Definition

Let

Clearly,

From Definitions

Depending upon these score and accuracy functions of FFNs, the ranking technique for any two FFNs can be described as:

Let

If

If

If

If

If

If

The subtraction and division operations in IFSs had been firstly suggested by Atanassov and Riecan (

Let

Let

We will present the proofs of (i) and (iii). Let

(i) Since

(iii)

The other assertions are proved analogously. □

Let

We will present the proof of (i). Let

Let

We will present the proof of (i). Let

Let

We will present the proof of

Let

We will present the proofs of (i) and (iii). Let

Since

Let

We will present the proofs of (i) and (iii). Let

(i) Since

(iii) Since

The other assertions are proved analogously. □

Let

We will present the proofs of (i) and (iii). Let

(i)

(iii) Since

The other assertions are proved analogously. □

Let

We will present the proof of (i), and (ii) can be proved analogously. Let

Since

In this section, Fermatean arithmetic mean operations over FFNs are defined in detail and two corresponding theorems are proved.

Let

Let

(i)

(iii)

Assertion (ii) is proved analogously. □

Let

We know that for any two real numbers

The weighted product model (WPM) is one of the best known and often applied MCDM methods for evaluating a number of alternatives in terms of a number of decision criteria. Each decision alternative is compared with the others by multiplying a number of ratios, one for each decision criterion. Each ratio is raised to the power equivalent to the relative weight of the corresponding criterion. Some of the first references to this method are due to Bridgman (

Its application first requires development of a decision/evaluation matrix,

According to WPM the total relative importance of alternative

In this section, I am going to introduce the MCDM problem under Fermatean fuzzy environment. Then, an effective decision-making method is hereby indicated to handle such MCDM problems.

The main work done in most of the MCDM problem is to rank one or more alternatives from a collection of possible alternatives regarding multiple criteria. For a stated MCDM problem under Fermatean fuzzy domain, presume that there are

For an effective solution of the MCDM problem, mentioned above, we suggest a Fermatean fuzzy WPM (FF-WPM) method.

The first step in FF-WPM is to normalize the

And the linear normalization for any

By applying score functions, we can easily get the result that

According to WPM the total relative importance of alternative

In Eq. (

In the last step, the score and accuracy functions are calculated for

In this section, we expand the implementation of the proposed WPM approach with a numerical example, which is discussed in Chen (

Fermatean fuzzy decision matrix.

Criteria | Optimization direction | Weight |
max/min |
|||

max 0.142875 | ||||||

min 0.059524 | ||||||

min 0.214251 | ||||||

min 0.095238 | ||||||

max 0.142875 | ||||||

min 0.119048 | ||||||

min 0.059524 | ||||||

min 0.166665 |

The normalized decision matrix will be obtained by applying Eq. (

Normalized decision matrix.

Criteria | ||||

Finally, the FF-WPM is calculated for all of the alternatives. The results are shown in Table

Numerical results obtained by FF-WPM.

Alternatives | FF-WPM | Score | Rank |

0.182312972 | 1 | ||

3 | |||

2 | |||

4 |

In this article, we have introduced the subtraction, division and Fermatean arithmetic mean operations over FFNs. A few operative rules of these three operations have been provided. In addition, the relationships between these operations have also been established. These operations can be instantly prolonged into interval-valued Fermatean fuzzy sets. Finally, we propose a FF-WPM method and apply it to bridge construction selection. In the future, we will combine others methods like VIKOR Method (Liu and Qin,