He has published more than 400 articles in journals, books and conference proceedings. He is on the editorial board of several journals. He has also been a guest editor for several international journals, member of the scientific committee of several conferences and reviewer in a wide range of international journals. Recently (2015–2018), Clarivate Analytics (previously Thomson & Reuters) has distinguished him as a highly cited researcher in computer science. He is currently interested in decision making, aggregation operators, computational intelligence, bibliometrics and applications in business and economics.

In this paper, we develop a new flexible method for interval-valued intuitionistic fuzzy decision-making problems with cosine similarity measure. We first introduce the interval-valued intuitionistic fuzzy cosine similarity measure based on the notion of the weighted reduced intuitionistic fuzzy sets. With this cosine similarity measure, we are able to accommodate the attitudinal character of decision-makers in the similarity measuring process. We study some of its essential properties and propose the weighted interval-valued intuitionistic fuzzy cosine similarity measure.

Further, the work uses the idea of GOWA operator to develop the ordered weighted interval-valued intuitionistic fuzzy cosine similarity (OWIVIFCS) measure based on the weighted reduced intuitionistic fuzzy sets. The main advantage of the OWIVIFCS measure is that it provides a parameterized family of cosine similarity measures for interval-valued intuitionistic fuzzy sets and considers different scenarios depending on the attitude of the decision-makers. The measure is demonstrated to satisfy some essential properties, which prepare the ground for applications in different areas. In addition, we define the quasi-ordered weighted interval-valued intuitionistic fuzzy cosine similarity (quasi-OWIVIFCS) measure. It includes a wide range of particular cases such as OWIVIFCS measure, trigonometric-OWIVIFCS measure, exponential-OWIVIFCS measure, radical-OWIVIFCS measure. Finally, the study uses the OWIVIFCS measure to develop a new decision-making method to solve real-world decision problems with interval-valued intuitionistic fuzzy information. A real-life numerical example of contractor selection is also given to demonstrate the effectiveness of the developed approach in solving real-life problems.

Atanassov (

A similarity measure is an essential tool for determining the degree of similarity between two objects. In 2002, Denfeng and Chuntian (

In many complex decision-making problems, the preference information provided by the decision-makers is often imprecise or uncertain due to the increasing complexity of the social-economic environment or a lack of data about the problem domain or the expert’s lack of expertise to precisely express their preferences over the considered objects. In such cases, it is suitable and convenient to express the decision-maker’s preference information in terms of IVIFSs. Therefore, it is necessary to pay attention to the study of the similarity measure for IVIFSs. There is some progress in this direction. Xu (

Note that the cosine similarity measures introduced by Singh (

To do so, we first propose a new cosine similarity measure for IVIFSs based on the weighted reduced intuitionistic fuzzy sets (Ye,

The paper is organized as follows. Section

In this section, we present some basic concepts related to fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and OWA operators, which will be needed in the following analysis.

A fuzzy set

A cosine similarity measure is defined as the inner product of two vectors divided by the product of their lengths. This is nothing but the cosine of the angle between the vectors representation of two fuzzy sets.

Let

Atanassov (

An intuitionistic fuzzy set

For convenience, we abbreviate the set of all IFSs defined in

In 2011, Ye (

Let

A cosine similarity measure between two intuitionistic fuzzy sets

Atanassov and Gargov (

Let

For any

Clearly, if

In the study of IVIFSs, the set-theoretic operations are defined as follows:

Let

Singh (

Let

If

If

If

The OWA operator was introduced by Yager (

An OWA operator of dimension

The OWA operator is commutative, monotonic, bounded, and idempotent. Especially, if

Furthermore, in 2004, Yager (

A GOWA operator of dimension

The quasi-arithmetic means are an important class of parameterized aggregation operators that have been used extensively in different application areas. It includes a wide range of aggregation operators such as arithmetic, quadratic, geometric, harmonic, root-power, and exponential. Fodor

A quasi-OWA operator of dimension

The quasi-OWA operator is monotonic, commutative, bounded, and idempotent. If we consider different functions

In the next section, using the idea of weighted reduced IFS of an IVIFS, we propose a new similarity measure on interval-valued intuitionistic fuzzy sets, called ‘

We proceed with the following formal definition:

Let

Analogous to the cosine similarity measure for IFSs given in Eq. (

The new measure

(a) It is evident that the property is true according to the cosine value of Eq. (

(b) This follows from the symmetry of

(c) First, let

This proves the theorem. □

For proof of the further properties, we will consider separation of

Using Definition

We prove (i) only, (ii) can be proved analogously.

(i) Let us consider the expression

From Definition

This proves the theorem. □

(a) It follows from the relation of membership and non-membership of an element in a set and its complement.

(b) It directly follows from Definition

(c) It simply follows (a) and (b).

This proves the theorem. □

By adjusting the values of

If

If

If

If

Assume that the elements in the universe of discourse

(i) If

Obviously, the

These properties can be proved easily on lines similar to the proof of Theorem

Note that the cosine similarity measures defined in Eq. (

Let

A OWIVIFCS measure based on the weighted reduced IFSs of IVIFSs is a mapping

The main advantages of the OWIVIFCS measure are that it is not only a straightforward generalization of measure defined in Eq. (

Now consider the following numerical example to understand the computation procedure more clearly.

Let

Taking different values of

Values of

0.2 | 0.7 | 1 | 2 | 5 | 7 | 9 | 15 | 25 | |

0.9303 | 0.9304 | 0.9305 | 0.9309 | 0.9317 | 0.9325 | 0.9331 | 0.9349 | 0.9374 |

The OWIVIFCS measure is commutative, monotonic, bounded, idempotent, non-negative, and reflexive. These properties can be proved with the following theorems:

Note that the proofs of these theorems are straightforward and thus omitted here.

By using the different manifestation of the weighting vector

When we consider different values of the parameter

If

If

If

If

By considering the different selections of the weighting vector, we are able to analyse the cosine similarity measure between two interval-valued intuitionistic fuzzy sets from min. similarity to max. similarity.

If

If

More generally, if

If

The WIVIFCS measure is obtained when the ordered position of the

If

If

If

If

A quasi-OWIVIFCS measure based on the weighted reduced IFSs of IVIFSs is a mapping

As we can see, when

Further, by assigning different functions to

when

When

If

The OWIVIFCS measure can be applied to solve different problems, including decision-making, medical diagnosis, pattern recognition, engineering, and economics. In the next section, we present an application of the proposed OWIVIFCS measure to solve the multiple criteria decision-making problem with the interval-valued intuitionistic fuzzy information.

IVIFS is a suitable tool for better modelling the imperfectly defined facts and data, as well as imprecise knowledge. In this section, we present a 5-step method to solve a multiple criteria decision-making problem under an interval-valued intuitionistic fuzzy environment.

Let

Using the OWIVIFCS measure defined in Eq. (

In the following, we are going to consider a real-life numerical example to demonstrate the applicability of the proposed method to multiple criteria decision making. To do so, we consider below a contractor selection decision-making problem for road development.

Chile is a South American country occupying a long, narrow strip of land between the Andes to the east and the Pacific Ocean to the west. In Chile, tourism has become one of the main sources of income for the people, especially living in its most extreme areas. In 2018, a record of a total of 7 million international tourists visited Chile. The online guestbook Lonely Planet listed ‘

The step-wise decision-making process as follows:

Values of

0.7468 | 0.9883 | 0.8986 | 1.0000 | 0.6623 | 0.9883 |

1.0000 | 0.9437 | 0.9883 | 0.9910 | 0.7349 | 0.9883 |

0.9492 | 0.9437 | 0.9437 | 0.9883 | 0.9272 | 0.8547 |

1.0000 | 0.9883 | 1.0000 | 0.7218 | 0.9832 | 1.0000 |

0.9832 | 0.8022 | 0.9883 | 1.0000 | 0.9815 | 0.9650 |

Values of

0.7175 | 0.9832 | 0.8619 | 1.0000 | 0.5870 | 0.9492 |

1.0000 | 0.9492 | 0.9832 | 0.9487 | 0.8043 | 0.8165 |

0.9338 | 1.0000 | 0.9239 | 0.9832 | 0.9338 | 0.7235 |

1.0000 | 0.9492 | 1.0000 | 0.6623 | 0.9806 | 1.0000 |

0.9783 | 0.7285 | 0.9832 | 0.9847 | 1.0000 | 0.9239 |

Values of

0.7276 | 0.9858 | 0.8867 | 1.0000 | 0.6261 | 0.9766 |

1.0000 | 0.9492 | 0.9858 | 0.9866 | 0.7674 | 0.9358 |

0.9410 | 1.0000 | 0.9337 | 0.9913 | 0.9329 | 0.7995 |

1.0000 | 0.9772 | 1.0000 | 0.6910 | 0.9815 | 1.0000 |

0.9805 | 0.7670 | 0.9913 | 0.9970 | 0.9949 | 0.9509 |

Values of

IVIFMAXCS | IVIFMINCS | IVIFNORCS | IVIFMEDCS | IVIFOLMCS | IVIFWINCS | ||

1.0000 | 0.6623 | 0.8723 | 0.9883 | 0.9010 | 0.9010 | ||

1.0000 | 0.7349 | 0.9364 | 0.9883 | 0.9770 | 0.9770 | ||

1.0000 | 0.8547 | 0.9430 | 0.9492 | 0.9521 | 0.9521 | ||

1.0000 | 0.7218 | 0.9439 | 1.0000 | 0.9929 | 0.9929 | ||

1.0000 | 0.8022 | 0.9512 | 0.9832 | 0.9795 | 0.9795 | ||

1.0000 | 0.6623 | 0.8755 | 0.9883 | 0.9027 | 0.9027 | ||

1.0000 | 0.7349 | 0.9380 | 0.9883 | 0.9771 | 0.9771 | ||

1.0000 | 0.8547 | 0.9434 | 0.9492 | 0.9522 | 0.9522 | ||

1.0000 | 0.7218 | 0.9458 | 1.0000 | 0.9929 | 0.9929 | ||

1.0000 | 0.8022 | 0.9521 | 0.9832 | 0.9795 | 0.9795 | ||

1.0000 | 0.6623 | 0.8807 | 0.9883 | 0.9055 | 0.9055 | ||

1.0000 | 0.7349 | 0.9406 | 0.9883 | 0.9771 | 0.9771 | ||

1.0000 | 0.8547 | 0.9440 | 0.9492 | 0.9524 | 0.9524 | ||

1.0000 | 0.7218 | 0.9489 | 1.0000 | 0.9929 | 0.9929 | ||

1.0000 | 0.8022 | 0.9534 | 0.9832 | 0.9795 | 0.9795 | ||

1.0000 | 0.6623 | 0.8904 | 0.9883 | 0.9109 | 0.9109 | ||

1.0000 | 0.7349 | 0.9452 | 0.9883 | 0.9773 | 0.9773 | ||

1.0000 | 0.8547 | 0.9452 | 0.9492 | 0.9526 | 0.9526 | ||

1.0000 | 0.7218 | 0.9543 | 1.0000 | 0.9929 | 0.9929 | ||

1.0000 | 0.8022 | 0.9558 | 0.9832 | 0.9795 | 0.9795 | ||

1.0000 | 0.6623 | 0.9142 | 0.9883 | 0.9247 | 0.9247 | ||

1.0000 | 0.7349 | 0.9560 | 0.9883 | 0.9779 | 0.9779 | ||

1.0000 | 0.8547 | 0.9486 | 0.9492 | 0.9535 | 0.9535 | ||

1.0000 | 0.7218 | 0.9664 | 1.0000 | 0.9930 | 0.9930 | ||

1.0000 | 0.8022 | 0.9619 | 0.9832 | 0.9797 | 0.9797 | ||

1.0000 | 0.6623 | 0.9388 | 0.9883 | 0.9409 | 0.9409 | ||

1.0000 | 0.7349 | 0.9666 | 0.9883 | 0.9788 | 0.9788 | ||

1.0000 | 0.8547 | 0.9536 | 0.9492 | 0.9549 | 0.9549 | ||

1.0000 | 0.7218 | 0.9774 | 1.0000 | 0.9931 | 0.9931 | ||

1.0000 | 0.8022 | 0.9689 | 0.9832 | 0.9798 | 0.9798 |

Values of

IVIFMAXCS | IVIFMINCS | IVIFNORCS | IVIFMEDCS | IVIFOLMCS | IVIFWINCS | ||

1.0000 | 0.5870 | 0.8380 | 0.9492 | 0.8729 | 0.8729 | ||