A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory

We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name “modal interval trapezoidal fuzzy numbers”. In this generalization, we accept that the alpha cuts associated with a trapezoidal fuzzy number can be modal intervals, also allowing that two interval modalities can be associated with a trapezoidal fuzzy number. In this context, it is difficult to maintain the traditional graphic representation of trapezoidal fuzzy numbers and we must use the interval plane in order to represent our modal interval trapezoidal fuzzy numbers graphically. Using this representation, we can correctly reflect the modality of the alpha cuts. We define some concepts from modal interval analysis and we study some of the related properties and structures, proving, among other things, that the inclusion relation provides a lattice structure on this set. We will also provide a semantic interpretation deduced from the modal interval extensions of real continuous functions and the semantic modal interval theorem. The application of modal intervals in the field of fuzzy numbers also provides a new perspective on and new applications of fuzzy numbers.


Introduction
The classical numerical system of real numbers is not efficient to express the vagueness, uncertainty and imprecision of real life.Fuzzy numbers and modal intervals are useful tools to get over those deficiencies.
The introduction of fuzzy sets by Zadeh [1] was a novelty as they provide a graduation of the membership relation.When considering fuzzy numbers, the expression "to be an element of a set" makes no sense, whereas many expressions in which this membership relation is relativized do indeed make sense.
Fuzzy numbers can be considered from two different points of view: with their membership function or with their α-cuts.The two ways of considering fuzzy numbers are equivalent, and, depending on the details we want to study, one can be better than the other.Among all types of fuzzy numbers, triangular and trapezoidal ones, whose names are derived from the shape obtained when their membership function is represented in the Cartesian plane, are the most commonly used.However, when we observe a fuzzy number from the point of view of its α-cuts, what we are indeed getting is an intervalar point of view of the fuzzy number.
Fuzzy sets theory has evolved since its appearance in 1965.Nowadays, we can find applications of fuzzy sets in the most part of scientific disciplines, such as decision-making [2][3][4], probability [5], control theory [6], medical sciences [7], characterization of complex systems [8], among others.
Modal intervals were introduced by Gardeñes [9] and they implied a new treatment of interval analysis, providing new resources to solve problems and systematize their resolution, as well as to interpret correctly an intervalar calculus.
Fuzzy numbers and intervals are not far removed from each other.This is because a fuzzy number can be identified by its α-cuts, which are intervals, and intervals can also be considered fuzzy numbers.
The set of intervals, in their classical point of view, has some deficiencies in the operative sense and also in the interpretation of the calculus.These deficiences remain in the set of fuzzy numbers.Modal intervals, which are an extension of classical intervals, solve some of the operative deficiencies (although the distributive property is neither satisfied), and they give, without ambiguity, the right semantical interpretations of the calculus.Moreover, modal intervals are a lattice with regard to the incluson relationship, while classical intervals are not.
The advantages that present modal intervals in front of classical intervals have motivated us to deepen the relationship between fuzzy sets theory and interval analysis theory using modal intervals.We are certain that, with this tool, we open new lines of research.
If the connection between intervals and fuzzy numbers is so obvious, why not use modal intervals when working with fuzzy numbers, especially if we take into account the fact that modal intervals are an efficient extension of classical intervals?This is what we want to do in this paper: we will focus on the modal interval point of view of the α-cuts of trapezoidal fuzzy numbers.
Modal intervals have been used combined with fuzzy sets [10,11].The study that we present is based on the set of trapezoidal fuzzy numbers, expanding this set using modal intervals.The new fuzzy numbers obtained by this extension are named modal interval trapezoidal fuzzy numbers.
In this new set of modal interval trapezoidal fuzzy numbers (MITFNs), we study the inclusion relationship and we prove that this relationship provides a lattice structure.At the same time, in the set of MITFNs, we define the dual operator, inherited from the dual operator of modal intervals.The dual operator is an internal operator in the set of MITFNs, and it gives us the possibility to solve problems that, until now, had no solution in the set of traditional fuzzy numbers.
The rest of this paper is organized as follows.In Section 2, some basic concepts related to modal intervals and to fuzzy numbers are given.In Section 3, we provide the main definitions of this work and we also study the lattice structure of MITFNs with regard to the inclusion relation.In Section 4, we define the modal extension of a real function, we study some properties of this extension and we present the semantic interpretability theorem.Section 4 also includes an example to show some advantages when working on MITFNs instead of working with trapezoidal fuzzy numbers in their traditional sense.The conclusions and future research are described in Section 5.

Modal Intervals
The set of classical intervals is represented by I (R) and it has been extensively studied.We can highlight the preliminary studies by Warmus [12] and Sunaga [13], and further consolidation of classical interval theory by Moore [14], Nickel [15] and Alefeld [16].The operations between classical intervals have been studied by Kaucher [17].
Modal intervals were introduced by Gardeñes [9,18].Some authors, such as Wang [19], refer to modal intervals as generalized intervals.A modal interval is defined as a pair consisting of a classical interval and a quantifier.The set of modal intervals is represented by , where [a, b] ∈ I (R) and Q ∈ {∃, ∀} .We can refer to [a, b] as the subtractum of A, and it would be represented by set (A); and we will refer to Q as the modality of A, which would be represented by mod (A).
In the set of modal intervals, we distinguish proper intervals as those modal intervals whose modality is ∃; and improper intervals as those whose modality is ∀.
We identify proper intervals as the classical intervals.We will denote the proper interval A = ([a, b] , ∃) by A = [a, b], and the improper interval A = ([a, b] , ∀) by A = [b, a].Using this notation, the interval [2, 4] is the proper interval ([2, 4] , ∃) and the interval [3, 1]  is the improper interval ([1, 3] , ∀).A pointwise interval can be considered as either proper or improper.Moore's semiplane is useful to represent classical intervals, but it is not detailed enough to represent modal intervals, whose graphic representation must be in the interval plane (see Figure 1).A modal interval A must be identified with the set of predicates accepted by A. Given a predicate P, this predicate is accepted by the modal interval such that P (x) is true.In the same way, if A = ([a, b] , ∀) , the predicate P is accepted by A if ∀x ∈ [a, b] , P (x) is true.We denote by pred (A) the set of the predicates accepted by A.
The dual operator of a modal interval A = ([a, b] , Q), which we will represent by du (A), is defined by The inclusion relation between two modal intervals A and B is defined by: and, using the canonical coordinates of A and B, the inclusion A ⊆ B holds in the same way as in the set of classical intervals, that is, In the set of modal intervals, we define the meet (∧) and join (∨) operators between two modal intervals.If A = [a, b] and B = [c, d], then A ∧ B = [max {a, c} , min {b, d}] and A ∨ B = [min {a, c} , max {b, d}].The meet and join operators correspond to the intervalar infimum and supremum of two modal intervals with regard to the inclusion relation.Thus, the set of modal intervals is a lattice with regard to the inclusion relation, while the set of classical intervals is not.
Using the meet and join operators, we define the modal extension of a real continuous function f : R n −→ R which is represented by f * as: where X P are the proper components of X and X I are the improper ones.The calculus of the modal extension of a real continuous function can be semantically interpreted using the semantic modal interval theorem [20].

Fuzzy Numbers
Fuzzy sets were introduced by Zadeh [1].Although they are surely the most accepted tool to represent uncertainty, there are some other tools used to represent indiscernibility, vagueness, imprecision and also uncertainty: rough sets [21][22][23]; marks [24] and numerical clouds [25], among others.
If X is a universal set, a fuzzy set A, in X can be defined by its membership function.The membership function of a fuzzy set A is a mapping µ A : X → [0, 1] which assigns to each element x ∈ X, a real number µ A (x) ∈ [0, 1].The value µ A (x) quantifies the level of membership of the fuzzy set A, of the element x.
A fuzzy number A is a fuzzy set of the real line.Its membership function, , upper semi-continuous and such that the closure of the set {x ∈ R | µ A (x) > 0} is bounded [26].
The membership function of a fuzzy number A can be described as: where a 1 , a 2 , a 3 and a 4 are real numbers such that a 1 < a 2 ≤ a 3 < a 4 ; f L is a real-valued strictly increasing and right-continuous function; and f U is a real-valued strictly decreasing and left-continuous function.
Given a fuzzy set A of X with membership function µ A , and given a real number α ∈ [0, 1], the α-cut of A is the crisp set denoted by A α and is defined by: The α-cut A 0 is called the support of A and it is denoted by supp (A) .The α-cut A 1 is called the core of A.
A fuzzy number A can be represented by its membership function or alternatively by the set of its The expected interval of a fuzzy number is given by Dubois and Prade [27], Grzegorzewski [28], and Heilpern [29].For a trapezoidal fuzzy number A = (a 1 , a 2 , a 3 , a 4 ), the expected interval is EI (A) = a 1 +a 2 2 , a 3 +a 4 2 .

Modal Interval Trapezoidal Fuzzy Numbers Definition 1. (Modal interval trapezoidal fuzzy number
The modal interval [a 1 , a 4 ] that corresponds to A 0 is the support of A, and the modal interval [a 2 , a 3 ] that corresponds to A 1 is the core of A. Thus, A = A 0 , A 1 . We denote the set of MITFNs by TI * (R) , extending the expression I * (R) , which denotes the set of modal intervals.
) is an MITFN, we can define set (A) as the trapezoidal fuzzy number (in its standard sense):

Definition 2. (Modality of an MITFN)
Given A = ([a 1 , a 4 ] , [a 2 , a 3 ]) which is an MITFN, we define: 1.A is a proper-improper MITFN (MITFN I P ) if supp (A) is a proper interval and core (A) is an improper interval.We denote it by A = (set (A) , ∃, ∀); 2. A is an improper-proper MITFN (MITFN P I ) if supp (A) is an improper interval and core (A) is a proper interval.We denote it by A = (set (A) , ∀, ∃); 3. A is a proper-proper MITFN (MITFN P P ) if both the support and core of A are proper intervals.We denote it by A = (set (A) , ∃, ∃); 4. A is an improper-improper MITFN (MITFN I I ) if both the support and core of A are improper intervals.We denote it by A = (set (A) , ∀, ∀).
In general, an MITFN A will be: where A is a trapezoidal fuzzy number in its traditional sense; Q 1 ∈ {∃, ∀} is the interval modality of the support of A, that is, the modality of [a 1 , a 4 ]; and Q 2 ∈ {∃, ∀} is the interval modality of the core of A, that is, the modality of [a 2 , a 3 ].
If A is an MITFN P P , we will refer to A as a proper trapezoidal fuzzy number; while if A is an MITFN I I , then we will refer to A as an improper trapezoidal fuzzy number [10].Notice that if A is an MITFN P P , then A is a trapezoidal fuzzy number in its classical sense.
) , which is an MITFN I P or an MITFN P I , there exists a unique value We must impose: which has a unique solution as If γ ∈ α 0 , 1 , then γ = α 0 + ξ, ξ ≥ 0 and the demonstration would be equivalent.Similar reasoning holds for the case in which A is an MITFN P I .
We will refer to α 0 as the transition modality value of the MITFN, A. The pointwise interval A α 0 = [p, p] is the transition α−cut of A, and its graphical representation is shown in Figure 4.
) is a modal interval triangular non-normalized fuzzy number and its modality is the same as the interval modality of the support of A, that is, if A is an MITFN I P , then ) will be a proper triangular fuzzy number; while, if A is an MITFN P I , then ) will be an improper triangular fuzzy number [10].

If A is an MITFN
The proof of both case 3 and case 4 is trivial.

Proposition 3. (Interval modality of the expected interval
) is an MITFN, then the interval modality of the expected interval is the same as the interval modality of the support of A, that is: mod (EI (A)) = mod (supp (A)) .

Proof. The expected interval for a trapezoidal fuzzy number
• If both supp (A) and core (A) are proper intervals, then a 1 ≤ a 4 and a 2 ≤ a 3 so a 1 + a 2 ≤ a 3 + a 4 and a 1 +a 2 2 , a 3 +a 4 2 is a proper interval.
• If both supp (A) and core (A) are improper intervals, then a 1 ≥ a 4 and a 2 ≥ a 3 so a 1 + a 2 ≥ a 3 + a 4 and a 1 +a 2 2 , a 3 +a 4 2 is an improper interval.
The graphical representation of the expected interval is shown in Figure 5.
It is possible to consider the membership function of an MITFN, but it is difficult to represent improper intervals in the Cartesian plane.However, a graphical visualization of the functions f L and f U is provided in Figure 2. Next, we will define some operators on the set of MITFNs.Dual operator.If A = (A , Q 1 , Q 2 ) ∈ TI * (R) the dual operator on A, dual (A) is defined as: We will distinguish between the dual operator of an MITFN, which we will represent by dual ( ) and the dual operator of an interval, which we will represent by du ( ) Moreover, (dual (A)) α = du (A α ) and, consequently, supp (dual (A)) = du (supp (A)) and core (dual It is obvious that if A is an MITFN I P , then dual (A) is an MITFN P I ; if A is an MITFN P I , then dual (A) is an MITFN I P ; if A is an MITFN P P , then dual (A) is an MITFN I I ; and if A is an MITFN I I , then dual (A) is an MITFN P P .
Proper and improper operators.

Graphical Representation of an MITFN in the Interval Plane
Trapezoidal fuzzy numbers, in their traditional sense, can be represented in Moore's semiplane as decreasing segments in which the support is represented by • and the core is represented by • [30].
Moore's semiplane is not detailed enough to represent an MITFN, and we must use the intervalar plane to represent them graphically.Using the same notation, that is, representing the support of an MITFN A by • and representing its core by •, the segment with ends • and • represents the MITFN A. The graphic representation of the four types of MITFNs, described in Definition 2, is shown in Figure 3.

The Lattice of MITFNs
We have defined the MITFN (Definition 1) using the α-cuts, and we will now define the inclusion relation between two MITFNs using the modal interval inclusion of the α-cuts, that is: In the following proposition, we study the inclusion relation between two MITFNs A and B using the modalities of the support and the core of both A and B. There are 16 cases to be considered, but, using the properties of the modal interval duality, we can reduce those 16 cases to 10.

Proof. ⇒) As
if X is proper and Y is improper.
By applying Lemma 1 to the α-cuts A α and B α , which are modal intervals, we obtain the desired result.

Definition 3. (Infimum and supremum)
Given A, B, X, Y ∈ TI * (R), Proof.We should consider the following four cases, depending on the modalities of L 0 and L 1 .
1. L 0 and L 1 are proper intervals.
As L 0 and L 1 are proper intervals, then A and B are MITFNs P P .Thus, A 1 ⊆ A 0 and B 1 ⊆ B 0 so L 1 ⊆ L 0 and as L 1 and L 0 are proper intervals, set 2. L 0 is a proper interval and L 1 an improper interval.
and it follows that L 1 ⊆ L 0 .We distinguish the following two cases according to the inclusion set of L 1 and L 0 : (a) If set L 1 ⊆ set L 0 , we can proceed as in the first case.
Notice that, as L 1 is an improper interval, D 1 will also be an improper interval and then set L 1 ⊆ set D 1 .We must prove that D 0 ⊆ L 1 and, consequently, D ⊆ X.
3. L 0 is an improper interval and L 1 a proper one. As and it follows that set L 1 ⊆ set L 0 ; thus, the demonstration follows as in the first case.4. L 0 and L 1 are improper intervals.
This implies that D 0 and D 1 are improper intervals.
Moreover, as set D 1 ⊆ set D 0 , then due to the modality of D 0 and D 1 it will be the case that D 0 ⊆ D 1 and as As L 1 and L 0 are both improper, D 1 and D 0 will also be improper intervals.
Moreover, as set D 1 ⊆ set D 0 , then due to the modality of D 0 and D 1 , it will be the case that D 0 ⊆ D 1 and as Proof.Applying modal interval properties relating duality and meet-join operators [24], that is: and sup {A, B} = dual (inf {dual (A) , dual (B)}) Let L 0 be L 0 = du A 0 ∧ du B 0 .As set L 0 = set du L 0 , then In the same way, if When we calculate Inf{dual (A) , dual (B)} , we should consider:

Interpretability of the Calculations Definition (Modal extension of a real function)
Let f : R n −→ R be a real continuous function.We represent the modal interval trapezoidal fuzzy extension associated with f by TI f * , and we define it over (X 1 , . . . ,X n ) ∈ (TI * (R)) n using the interval extension f * (see Sainz [24,31]) over the α−cuts of the MITFNs X α 1 , . . ., X α n : As the α-cuts are modal intervals, we can express this modal interval trapezoidal fuzzy extension using the meet and join operators, and splitting the components of X α 1 , . . ., X α n into the proper ones: X α j P and the improper ones: x jp ∈set X α j P max x j i ∈set X α j I f x j p , x j i , max x jp ∈set X α j P min

Proposition 7. (Inclusivity of the modal extension)
If TI f * is the modal extension of a real continuous function f : R n −→ R, given X 1 , . . ., X n , Y 1 , . . ., Y n ∈ TI * (R) such that ∀i ∈ {1, . . . ,n} X i ⊆ Y i , then: Proof.We can express the inclusion of MITFNs by using the inclusion of the α-cuts, that is (TI Given a rational real continuous function f , if X α 1 , . . ., X α n are the α-cuts of the classical trapezoidal fuzzy numbers X 1 , . . ., X n , then ∀α ∈ [0, 1], and the interval extension that we represent by F X α 1 , . . ., X α n = Y α associated with f can be interpreted as: or also as: because f is a continuous function and the value Y α corresponds to: min In most of the cases, ∀α ∈ [0, 1] , the exact values Y α = F X α 1 , . . ., X α n that define the fuzzy number Y = {Y α , α ∈ [0, 1]} are difficult to calculate.This is the reason why we often replace every rational operator in the function f by its corresponding intervalar operator.The result obtained with this replacement is not the same interval Y α , but an interval Z α which conforms to: However, the only valid semantic application to this last calculus will be the interpretation given in Equation ( 1) and the semantic expressed in Equation (2) will not be applicable to this last calculus F X α 1 , . . ., X α n ⊆ Z α .

Theorem 1. (Interpretability theorem)
Let TI f * : (TI * (R)) n −→ TI * (R) be the * -extension of a real continuous function f : R n −→ R, and let X = (X 1 , . . . ,X n ) be a vector of MITFNs sorted by their modality, that is: . . ,X n ) ⊆ Z and we consider γ 0 q , δ 0 r and α 0 z the transition modality values of the MITFNs C q , D r and Z, respectively, then, given α ∈ [0, 1] , it holds that: although the elements within these sets are not ordered in the same way.
Proof.The interval semantic theorem (Sainz [24]), states that if f * is the * -semantic extension of a real function f , and X is a vector of modal intervals expressed as X = (X P , X I ) , where X P are the proper components of X and X I are the improper components of X, if (Y, Q) ∈ I * (R) is such that f * (X P , X I ) ⊆ Y, then: ∀x p ∈ set (X P ) (Qy ∈ set (Y)) (∃x i ∈ set (X I )) such that y = f x p , x i .
Next, let us consider the fuzzy equation A + X = B, where A is the MITFN P P , A = ([2, 9] , [6, 7]) and B is the MITFN P I , B = ([15, 10] , [9, 14]) .The solution for X is: which is an MITFN P I (see Figure 6).The transition modality value for X is α 0 X = 3 4 = 0.75 and the transition modality value for B is α 0 B = 1 2 = 0.5; thus, the interpretation of the calculus A + X = B is:

Conclusions
In this paper, we have used the lattice structure of modal intervals to develop the lattice completion of trapezoidal fuzzy numbers, with regard to the inclusion relation.We have named the set obtained with this completion MITFNs.The elements of this new set have been defined allowing that their α-cuts can be modal intervals and also allowing that the support modality and the core modality are not the same.This reticular completion has not simply been left in a theoretical study of the inclusion relationship between modal trapezoidal fuzzy numbers, but the calculation of the extensions of real continuous functions has also been addressed.
Moreover, we have not simply focused on the calculation of the real extensions on MITFNs, but we have also used the semantic theorem of modal interval analysis so as to interpret the calculus to the α-cuts of the extensions.We are certain that knowing the meaning of a calculation is even more important than the calculation itself.
With the study presented in this paper, we have provided a new tool for fuzzy numbers.We have introduced an extension of traditional trapezoidal fuzzy numbers and we have solved a problem that had no solution in the set of traditional trapezoidal fuzzy numbers, while also providing the semantic interpretation of the result obtained.
Our future lines of research are twofold; on the one hand, further theoretical research will be conducted, and, on the other, some practical applications of our theoretical studies will be developed.
Regarding theoretical studies, we believe it is interesting to look for and implement algorithms that allow us to obtain a good inclusive approach to the semantic extension TI f * .Thus, we would

Figure 3 .
Figure 3. Representation of an MITFN in the interval plane.

Figure 6 .
Figure 6.Graphical solution of the equation A + X = B.