Modified level operator 2 1 N γ γ applied over interval valued intuitionistic fuzzy sets

The recently proposed intuitionistic fuzzy level operator Nγ generates a subset of an intuitionistic fuzzy set A, where the elements of the subset are those elements of A, for which the ratio of their degrees of membership to their degrees of non-membership is greater than or equal to a given constant γ > 0. Here we propose a continuation of this idea from the case of intuitionistic fuzzy sets to the case of interval-valued intuitionistic fuzzy sets. This modification requires us to introduce a second constant, i.e. γ1, γ2 > 0. We show that there are twenty possible scenarios for the mutual position of the intervalized level operator 2 1 N γ γ and the element of the interval-valued intuitionistic fuzzy set, and give the respective formulas which calculate in each case the membership and non-membership degrees with which the IVIFS element belongs to the set defined by the operator 2 1 N γ γ . These twenty scenarios are graphically interpreted in the intuitionistic fuzzy interpretational triangle, and the respective formulas have been derived. In conclusion, further ideas of research have been suggested.


Introduction
Extending the concept of fuzzy sets of level α, in [3] K. Atanassov introduced the concept of (α,β)-set, generated by the intuitionistic fuzzy set A in a universe E, where α, β are fixed numbers in the [0, 1]-interval, for which α + β ≤ 1. The formal notation of the operator that produces this (α,β)-set is the following: Hence, this operator reduces the number of elements of the set A, retrieving only those elements whose degrees of membership are above a given level (threshold) α and their degrees of non-membership are below a given level β. A series of properties of the operator Nα,β are checked, involving the set-theoretic operations "negation", "union", "intersection" and the relation "inclusion". Nα,β is also called to be a 'level operator'.
In continuation of this idea, in [5] the author proposed a new level operator Nγ, which employs the ratio γ of the membership to non-membership values of the elements of the set, instead of thresholds for these functions. Thus the new operator returns in the resultant subset only those elements of the set that maintain a ratio greater than or equal to a predefined number γ.
As it was noted in [5], the reader may find it interesting that the idea about this new operator was inspired by the theory of the American psychologist John M. Gottman, stating that the marital relationships are likely to be stable if they exhibit the "magic ratio" of 5:1 of positive to negative interactions between the partners (see [7]). In comparison with the resultant set of the level operator Nα,β that of the new level operator Nγ, the elements of the γ-set may exhibit higher uncertainty, as long as the ratio between their membership and non-membership values are also maintained high enough (see Figure 1.1). In this paper, the idea of modifying the level operator Nγ from the case of intuitionistic fuzzy sets to the case of interval-valued intuitionistic fuzzy sets. Section 2 gives some preliminaries and Section 3 gives the main results of the paper.

Preliminaries
Here we will remind the reader of some definitions from the area of intuitionistic fuzzy sets (see [1], Chapter 1 in [3]) and interval-valued intuitionistic fuzzy sets (see [4], Chapter 2 in [3]). Analogously to the IFS case, in case of sup MA*(x) + sup NA*(x) < 1, this gives rise to the interval PA*(x), whose length equals 1 -sup MA*(x) -sup NA*(x), which attributes to the uncertainty of the element x ∈ E to the IVIFS A*.
Obviously, the definition of IVIFS is constructed analogously to the definition of an IFS. The geometrical interpretation is slightly more complex than that of an IFS (see [2]), and both geometrical interpretations are given below in  The other definition which we will be working with throughout this paper is that of the level operator Nγ., which we will modify as a next step of research.

Definition 3 (see [7]): Let us call an IFS
> 0). Let us define for eachν-positive IFS A the following operator where γ is an arbitrary non-negative real number.

Main results
Here we asked ourselves, what if we consider not just one segment that represents the threshold γ forming the Nγ operator, but a whole interval [γ1; γ2], whose interpretation in the intuitionistic fuzzy triangle will be that of an angle. We will only remark, that so far, we have used the traditional notations in literature, where in the context of operators over IFS N was used for the level operator Nα,β and respectively Nγ, while in the context of IVIFS N denotes the interval within the non-membership function takes its values. In order to avoid duplicate notation and confusion, we will denote here the modified level operator over IVIFS as 2 1 N γ γ where γ1 stays for the upper line closer to the 〈0,1〉 point, or the logical Falsity, and γ2 stays for the lower line closer to the 〈1,0〉 point, or the logical Truth. In the Figures 3.1-3.20 below, the lines will be denoted respectively by u and l (for "upper" and "lower").
The careful investigation of the intersection between an element of an IVIFS (the rectangle) and the angle formed by "intervalized" operator 2 1 N γ γ shows that there are exactly 20 possible cases, which are different from each other and unrepresentable by each other. What is interesting here are the surfaces of the figures (triangles, trapezoids or pentagons), which occur as a results of the intersection of the rectangle, representing and IVIFS element, with the two lines, representing the level operator. The proportion of the surfaces of the figures thus obtained to the surface of the rectangle can be used as an intuitionistic fuzzy measure of the membership and non-membership (as well as uncertainty) of the IVIFS element to the set defined by the level operator 2 1 N γ γ . Why 20 cases? If we take one line and one rectangle with the desired properties, inscribed in the intuitionistic fuzzy interpretational triangle, we see that there are 6 possible positions of the line against the rectangle. Given that the two lines needed for representing the "intervalized" level operator are strictly arranged (upper and lower), this gives 21 combinations. One of these however is impossible, when the two lines intersect, due to their intersection in the 〈0,0〉 point, i.e. cannot again intersect within the rectangle. For all these 20 cases we provide below the graphic representation and the formulas for the surfaces of the figures, obtained as a result of section between the rectangle and the two components of the Case 9.
Case 13.  their boundary values ranging from 0 to 1, we can arrange the twenty above described cases in a 6 × 6 table as shown on

Conclusion and next steps of research
The present paper is an attempt to modify and extend the recently proposed level operator Nγ over intuitionistic fuzzy sets for the case of interval-valued intuitionistic fuzzy sets. While the ordinary element of an intuitionistic fuzzy set is graphically interpreted as a point plotted on the intuitionistic fuzzy interpretational triangle, in the IVIFS case, the element of the set is graphically interpreted as a rectangle, defined by the intervals of its membership MA(x) and non-membership NA(x) for MA(x) ⊂ [0, 1], NA(x) ⊂ [0, 1] and for all x ∈ E, sup MA(x) + sup NA(x) ≤ 1. The "intervalization" of the level operator Nγ requires us to introduce an additional second constants and work with the interval of the membership-to-non-membership ratios γ1, γ2, interpreted graphically as an angle in the intuitionistic fuzzy interpretational triangle. Thus we define the extended level operator 2 1 N γ γ . Depending on the mutual position of the IVIFS element and the two lines forming this angle, staying for γ1, γ2, we can possibly have 20 different cases, and for all of them the graphical interpretation and the respective formulas are given for the surfaces of the rectangle (IVIFS element) cut off by the lines. In the next step of research, we will investigate the properties of 2 1 N γ γ similarly to what has been done for the level operator Nγ in [5]. We will also research the usability of the new operator in particular decision making problems, for instance, problems handled with intercriteria analysis, as shown for the case of Nγ , [6].