Interval Fuzzy Segments

In this paper, we bring together two concepts related to uncertainty and vagueness: fuzzy numbers and intervals. With them, we build a new structure whose elements we call interval fuzzy segments. We have undertaken this based on the conviction that the fuzzy numbers are a correct representation of the real numbers under situations of indeterminacy. We also believe that if it makes sense to consider the set of real numbers between two real bounds, then it also makes sense to consider the set of all the fuzzy numbers between two fuzzy number bounds. In this way, we extend the concept of real interval to the concept of interval fuzzy segment defined by two fuzzy bounds and a transition mapping that leads from the lower fuzzy bound to the upper fuzzy bound and this transition mapping generates the set of all the fuzzy numbers comprised between those fuzzy bounds. At the same time, this transition mapping brings the concept of interval fuzzy segment closer to the concept of line segment.


Introduction
We can apply both probabilistic and non-probabilistic methods in order to deal with uncertainty, imprecision, indiscernibility and vagueness. If we focus on non-probabilistic methods, the theory of fuzzy sets and interval theory have been widely used over the past few decades to quantify both uncertainty and imprecision.
The theory of fuzzy sets, introduced by Zadeh [1], provides a non-probabilistic way to represent uncertainty using membership functions. Interval analysis is mainly based on studies by Moore [2], using real intervals as a tool to treat non-probabilistic uncertainty and also imprecision.
In most traditional research, fuzzy set theory and interval theory are applied separately to carry out non-probabilistic analysis of uncertainty and imprecision. However, in some situations, it may be useful to combine fuzzy uncertainty and interval uncertainty in some way.
There are studies in which fuzzy numbers and intervals are combined to provide new tools to describe and treat uncertainty. For instance, in the field of engineering, to analyze structural response in special situations, the method called fuzzy interval uncertainty [3] has been proposed. In that method, structures simultaneously have fuzzy parameters and interval parameters, but these are independent. That is to say, the method works with variables that have a degree of uncertainty, evaluated with fuzzy numbers, and it also works with some other variables that have a degree of inaccuracy, which are quantified by intervals. In this way, all the operations necessary to be able to work simultaneously with fuzzy numbers and intervals are defined.
In some studies, we can also find the concept of fuzzy-boundary interval [4][5][6][7]. In this case, the uncertain parameters of structures are treated as interval variables, but the intervals, instead of having two real (determined) bounds, the lower and upper bounds are considered to be fuzzy numbers.
interval equation has a solution, this solution cannot be obtained by any interval syntactic computation on I (R).
With the dual operator introduced by Gardeñes [31] and represented by dual ([a, b]), we can consider: The dual operator allows us to find a solution to some interval problems in which no solution was possible without using this operator. Using the dual operator, we can solve some equations. For instance, the solution to the equation [2,5] + [x, y] = [3,7] is [x, y] = [3,7] − dual ( [2,5]) and hence [x, y] = [1,2] . For more information, see Sainz [32].
Two important relationships in the set of classical intervals are the inequality (≤) and the inclusion (⊆) relationships. The former, between two intervals [a, b] and [c, d] is defined as: Although it is unusual, the interval inequality ≤ can be characterized by: The inequality relationship ≤ between intervals is not a total order, since, given two intervals Hence, the set of classical intervals is a lattice with regard to the ≤ relation.
We have also mentioned the inclusion relationship, defined by set inclusion, which can be expressed using the interval coordinates as: The infimum and supremum of two intervals with regard to the inclusion relationship are named Meet and Join, respectively, and they do not always exist. This is why the set of classical intervals with regard to the inclusion relationship is not a lattice.

Fuzzy Numbers
Fuzzy sets were introduced by Zadeh [1]. Although they are without a doubt the most widely accepted tool to represent uncertainty, there are some other non-probabilistic tools used to represent indiscernibility, vagueness, imprecision and also uncertainty: rough sets [33][34][35]; marks [32]; and numerical clouds [36], 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 a real number µ A (x) ∈ [0, 1] to each element x ∈ X. The value µ A (x) quantifies the level of membership that the element x has of the fuzzy set A.
A fuzzy number A is a fuzzy set of the real line. The set of fuzzy numbers is represented by FN. Its membership function, , upper semi-continuous and such that the closure of the set {x ∈ R | µ A (x) > 0} is bounded [37].
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: A fuzzy number A can be represented by its membership function or alternatively by the set of its α-cuts: 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 and is denoted by core (A).
When the core of a fuzzy number is point-wise, we will refer to it as a punctual-core fuzzy number.
The set of the punctual-core fuzzy numbers will be represented by • FN. The arithmetic operations on fuzzy numbers can be approached either by the direct use of the membership function by Zadeh's extension principle [1] or in terms of arithmetic operations on their α-cuts that we will use in this paper. Thus, given A and B two fuzzy numbers expressed by 1] , if ω is an interval binary operator, the extension of ω on A and B, denoted by W, is defined as The partial ordering of fuzzy numbers is defined from the partial ordering of their α-cuts (see [38]), that is, let A and B be two fuzzy numbers: Note that on this basis of the concept of the partial order, we can also introduce the concept of the fuzzy number with definite sign: positive or negative.

Interval Fuzzy Segments
In this section, we will introduce a new set whose elements will be called interval fuzzy segments, as a generalization of a real interval. In the interval fuzzy segments, we will consider fuzzy numbers whose core is a point-wise interval instead of considering real numbers. To obtain a consistent definition of interval fuzzy segments, we will first introduce what we define as a fuzzy numbers S-transformation.

Definition 1. (Fuzzy numbers S-transformation)
Let A and B be two punctual-core fuzzy numbers, that is A, B ∈ • FN, such that A ≤ B. The fuzzy numbers S-transformation from A to B is a mapping that satisfies the following properties: Given A, B ∈ • FN, such that A ≤ B, the existence of fuzzy numbers S-transformation from A to B is proved in Proposition 8.

Definition 2. (Interval fuzzy segment)
Given A, B ∈ • FN such that A ≤ B and given T, a fuzzy numbers S-transformation, if we consider the set of all the punctual-core fuzzy numbers between A and B obtained by the S-transformation T, that is: Then, the interval fuzzy segment with bounds A and B and S-transformation T is represented by [A, B] T and is defined as the pair: The set of the interval fuzzy segments is represented by IFS. We call the set: the support of an interval fuzzy segment with bounds A and B and S-transformation T, and we represent it by supp([A, B] T ).
Notice that given A, B ∈ • FN such that A ≤ B, there may be various S-transformations that generate the same support. For instance, the S-transformations T λ (α) = A α + λ (B α − dual (A α )) and T λ (α) = A α + λ 2 (B α − dual (A α )) generate the same support, that is supp([A, B] T ) = supp([A, B] T ); however, the interval fuzzy segments will not be the same: [A, B] T = [A, B] T as the interval fuzzy segments will take into account not only the support generated, but also how that support has been generated. We will study different S-transformations in Section 6.  1], the (λ, α)-cut of A is the interval denoted by A α λ and defined as: Notice that the set of (0, α)-cuts A α 0 α∈[0,1] and the set of (1, α)-cuts A α 1 α∈[0,1] correspond to the fuzzy bounds of the interval fuzzy segment A, that is, A α 0 α∈[0,1] = A and A α 1 α∈[0,1] = B. An interval fuzzy segment A can be represented as the pair that consists of the set of its (λ, α)-cuts and the fuzzy numbers S-transformation T, thus: Using the fuzzy numbers S-transformation from A to B, we can evaluate the membership function of every fuzzy number C ∈supp([A, B] T ) , taking into account that there will exist a value λ ∈ [0, 1] such that C = {T λ (α)} α∈[0,1] (see Figure 1). As the infimum and supremum of a real interval [a, b] are defined by inf ([a, b]) = a and sup ([a, b]) = b, it follows that: (15) and hence:

Arithmetic Operations on IFS
Arithmetic operators on the set of the interval fuzzy segments are based on the following two properties:

1.
Each interval fuzzy segment can fully and uniquely be represented by its (λ, α)-cuts.
These properties enable us to define arithmetic operations on IFS in terms of the arithmetic operations of their (λ, α)-cuts.

Definition 4. (Arithmetic operations)
Let A = A, A T and B = B, B T be two IFS. Then, the four arithmetic operations on IFS are defined as follows:  an operator, and ∈ {+, −, * , /} , then given A and B, two IFS, the calculus C = A B, where C = C, C T is an interval fuzzy segment, that is, the application T is a fuzzy numbers S-transformation as it satisfies: Proof. We will prove the cases (b) , (c.3) and (c.4) (Definition 4). The demonstration in the other cases follows in a similar way. 1] is a fuzzy number, it will be T λ (α 2 ) ⊆ T λ (α 1 ). In the same way, as 1] is also a fuzzy number and Applying the inclusivity property of the interval difference (see [32]), it follows that 1] are punctual-core fuzzy numbers. 4.
If 1] is also a fuzzy number and Applying the inclusivity property of the product, it follows that 1] is also a fuzzy number, and, consequently, T 1 (α 2 ) ⊆ T 1 (α 1 ). Applying the inclusivity property of the product, it follows that 1] are punctual-core fuzzy numbers. 4. If In all the cases, C and C are fuzzy numbers as they are obtained operating fuzzy numbers. Moreover, C ≤ C since T verifies the properties 1, 2 and 4.

Inequality and Inclusion Relationship
In this section, we define the inequality and the inclusion relationships between interval fuzzy segments and we analyze some of their properties.

Inequality Relationship: The Lattice (IFS, ≤)
From the expression for the interval inequality relationship ≤ described in Equation (3), we define the inequality relationship between interval fuzzy segments as follows:
The inequality relationship ≤ on the interval fuzzy segments is not a total order, as neither the interval nor the fuzzy inequality are. Proof.

1.
As A ≤ A and M ≤ M, then inf {A, M} ≤ inf A, M and as all the cores are point-wise intervals, their infimum will also be a point-wise interval. Moreover, the conditions required in Definition 1 are fulfilled, as we prove below: FN as both T λ (α) and T λ (α) are punctual-core fuzzy numbers.
Thus, X, X T ≤ A, A T and X, X T ≤ M, M T .
Moreover, if P = P, P T ∈ IFS is such that P, P T ≤ A, A T and P, P T ≤ M, M T , then it will follow that: thus P, P T ≤ X, X T . 2.
In a similar way:

Inclusion Relationship
The definition of the inclusion of two interval fuzzy segments is based on the inclusion relationship between their supports.

Definition 6. (Inclusion in IFS)
Given A = A, A T and M = M, M T two interval fuzzy segments, we define: Proposition 5. Given A = A, A T and M = M, M T two interval fuzzy segments, from Definition 6 above, it follows that: , which is As 0 ≤ µ, then, by applying Definition 1, we have: and thus A = T µ (α) In the same way, A ≤ M.

Proposition 7. The inclusion relationship studied in Definition 6 is a partial order in IFS.
Proof. Given A = A, A T , B = B, B T , C = C, C T ∈ IFS, the following properties are fulfilled:  1] , and consequently A ⊆ C.
The inclusion relationship cannot be a total order in the set IFS, as the interval inclusion is not. Thus, the structure (IFS, ⊆) is not a lattice, as the structure (I (R) , ⊆) is not either.

Some Fuzzy Number S-Transformations
Next, we will study interval fuzzy segments by considering different fuzzy number S-transformations. We will illustrate each of these S-transformations by an application. The first S-transformation we present is called linear S-transformation and it represents the transition from the lower bound of an interval fuzzy segment to the upper bound that corresponds to a linear mapping. The second S-transformation we consider has been named exponential S-transformation, as it represents the transition from the lower bound of an interval fuzzy segment to the upper bound that corresponds to an exponential mapping.
) is a fuzzy number S-transformation (Definition 1) from A to B. Proof.

1.
If Moreover, In the same way, As A and B are punctual-core fuzzy numbers, then A 1 and B 1 are real numbers, hence T λ (1) is also a real number. 4.

Linear Interval Fuzzy Segment
Let A and B be two fuzzy numbers such that A ≤ B. Let us consider the following linear mapping: Applying Proposition 8 to this mapping, T, is a fuzzy number S-transformation and it holds that: In a particular way, we can illustrate these linear interval fuzzy segments considering that the bounds A and B are triangular fuzzy numbers, A = (a 1 , a 2 , a 3 ) and B = (b 1 , b 2 , b 3 ), then the characteristic functions are: and the respective α−cuts are: thus, the linear S-transformation is expressed by: The membership function of a fuzzy number C ∈supp([A, B] T ) can be evaluated as: which results in: which results in: It is obvious that if α = 1, we obtain Example 2. Let us suppose that we have a digital thermometer and the temperature range to be measured is from −15 • C to +45 • C. Every temperature reading with this thermometer is subject to inaccuracy, such as the sensitivity of the device and/or the accuracy of the display. If x is the real temperature we want to measure and p is the temperature read from the thermometer, we require that they satisfy |x − p| ≤ . Thus, for every temperature p read from the display of the thermometer, we can consider we have obtained a triangular fuzzy number P whose core is the temperature reading p and whose support is the real interval [p − , p + ]. If the thermometer display has a single decimal place, and the accuracy of the device is complete, we can then consider a value for such as 0.1. However, if we want to take into account the accuracy of the device, we should increase the value > 0. As this is a descriptive example, we can take, for example, = 0.11. In this way, the thermometer would measure a temperature that can be expressed as an IFS, thus: The linear fuzzy number S-transformation is: Example 3. Considering the same situation as in the previous example, let us now suppose that the temperature range we want to measure is from +10 • C to +30 • C. Let us also suppose a margin of related error (e r ) of the device, and e r = 5%. Thus, the fuzzy bounds A and B of the interval fuzzy segment are: where T λ (a) = A α + λ (B α − dual (A α )) and thus: As all the temperatures in the established range are positives, it is easy to prove that this transition function maintains the related error e r of 5%, as for every λ ∈ [0, 1] it is A 0 λ = [9.5 + 19λ , 10.5 + 21λ] and also A 1 λ = [10 + 20λ , 10 + 20λ]. Thus, e r = |10.5+21λ−(10+20λ)| 10+20λ = 0.5+λ 10+20λ = 0.05 = 5%.

Exponential Interval Fuzzy Segment
Although the linear S-transformation is one of the easiest to use, there are some situations in which it can be useful to work with other nonlinear S-transformations. In some models, it can be useful to work with an exponential S-transformation, as this can better represent the problem to be solved.
Considering the following exponential mapping: Applying Proposition 8 to this mapping, T, is a fuzzy number S-transformation and it holds that: From the punctual-core fuzzy numbers A, B ∈ • FN and the associated exponential S-transformation T, we can define the interval fuzzy segment [A, B] T .
In particular, we can consider the bounds A and B of these interval fuzzy segments as the triangular fuzzy numbers A = (a 1 , a 2 , a 3 ) and B = (b 1 , b 2 , b 3 ), with their respective membership functions µ A (x) and µ B (x) as laid out previously.
The exponential S-transformation is expressed as: and the membership function of a fuzzy number C ∈supp([A, B] T ) can be evaluated as: If and The exponential S-transformation T is: that is: To complete this section, we will present an example that shows how this model works in a real-life problem, and will compare the results obtained using interval fuzzy segments in front of the results obtained working with intervals and working with fuzzy numbers, highlighting the advantages obtained using interval fuzzy segments.
Example 5. Consider a simplified circuit of voltage regulation using the LM317 circuit and detailed in the following diagram ( Figure 5) We will assume that, for a given input voltage V I N , we obtain a constant value V PQ = 1.25 V. The output voltage V OUT is We will not take into account the value I ADJ because it is small enough not to affect significantly the study we are doing.
Let us consider that R 1 is a resistor whose value is 1000 Ω with a tolerance of 5% and R 2 is a variable resistor whose value is comprised between 0 Ω and 10,000 Ω with a tolerance of 5%. We will study the different values obtained for V OUT using interval calculus, fuzzy numbers and interval fuzzy segments with both linear S-transformation and exponential S-transformation. .
From these data, we obtain V OUT whose value is .
Using the values λ = 0, λ = 1, we obtain the fuzzy bounds for V OUT : .
As can be seen, the information expressed using interval analysis only gives us the margin of variation of the resulting output voltage. When we use fuzzy numbers, we obtain a little more information, as the fuzzy value for V OUT incorporates, from the α−cuts of V OUT , a vagueness component of the output voltage. Although fuzzy numbers provide a better understanding of the problem than intervals, they can not naturally include the modeling of the variation of the resistor R 2 , and therefore they can not express the modeling of the output voltage V OUT . Interval fuzzy segments can describe the modeling of the output voltage as they describe the problem taking into account their uncertainties and inaccuracies, describing the dynamic procedure and not just the end result. Using interval fuzzy segments, we know which is the variation of the output voltage depending on the variation of the resistors. In addition to this variation, the result also reflects the uncertainty and inaccuracy that are propagated throughout the calculation, depending on the uncertainty and inaccuracy in the data.
Note that the value V OUT obtained using a linear interval fuzzy segment and exponential interval fuzzy segment is the same in both cases and the same goes for V OUT . This is explained from the concept of support of an interval fuzzy segment. The support of both IFSs is the same, but for a given λ such that λ = 0 and λ = 1, the fuzzy number obtained using linear fuzzy interval segments and exponential fuzzy interval segments are different.

Conclusions
Both fuzzy numbers and intervals are powerful tools to represent uncertainty, inaccuracy, indiscernibility and vagueness. In this work, we combine these tools to create a new concept: the interval fuzzy segment.
We have generalized intervals, as a set of real values between two real bounds, in terms of the set of fuzzy numbers comprised between two fuzzy numbers and generated by a function that we have treated as a transition function.
We have gone a little beyond the generic concept of interval, since by introducing the S-transformation to generate all the elements of the interval fuzzy segment, this new concept would be closer to the concept of segment or path between two points than to the concept of interval.
Taking advantage of the structures and operations of both fuzzy numbers and intervals, we have built on this new set of interval fuzzy segments the main arithmetic operators. We have also studied the inequality relationship, and proved that this relationship gives a lattice structure to the set of interval fuzzy segments. Furthermore, we introduced the inclusion relationship in the set of interval fuzzy segments.
The interval fuzzy segments that we have defined and studied here reflect, in a very realistic way, situations in which there is uncertainty, inaccuracy and even inaccuracy in the uncertainty.
Many measurements are uncertain and a good way to represent these is to adopt fuzzy numbers. However, when these measurements are lower and upper bounded, a good tool to represent them is obtained using interval fuzzy segments.
If we consider future research lines and applications of the new structure that we have created here, we think that it would be interesting to move beyond the idea of interval and go deeper into the idea of a segment of fuzzy numbers, without any inequality restriction on the bounds.