On a total order on the set of Z-numbers based on discrete fuzzy numbers

Abstract

Z-numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as a measure of certainty or sureness of A. From the initial proposal, several other approaches have been introduced in order to reduce the computational cost of the involved operations. One of such approaches is called discrete Z-numbers where A and B are modelled as discrete fuzzy numbers. In this paper, the construction of total orders on the set of discrete Z-numbers is investigated for the first time. Specifically, the total order is designed for discrete Z-numbers where the second component has membership values belonging to a finite and prefixed set of values. The method relies on solid and coherent linguistic criteria and several linguistic properties are analyzed. The order involves the transformation of the first components of the discrete Z-numbers by using the credibility of the second components in the sense that a lower credibility enlarges in a greater extent the uncertainty of the first component. Then a total order on the set of discrete fuzzy numbers is applied. Finally, a practical example on how to order discrete Z-numbers is presented and a comparison with other ranking methods is performed from which the strengths of our method are stressed.

1 Introduction

Computing With Words (CWW) is an interesting hot topic that has not stopped growing (Herrera et al. 2009; Li et al. 2017; Morente-Molinera et al. 2015; Zuheros et al. 2018; Doukas et al. 2010) since the pioneering works by Lotfi Zadeh (1975a, 1975b, 1975c). Among the main reasons for the interest on this research, besides the need to properly model natural language itself, there exist multiple applications that we can find in the literature and that cover many fields such as control systems, decision making problems and consensus process, artificial intelligence, risk assessment, etc. (see Banerjee et al. (2022); Chiclana et al. (2015); Yuan et al. (2023); Nazari-Shirkouhi et al. (2023)). In this way, with the intention of providing linguistic models that describe common natural language sentences such as “Usually, the travel time between Madrid and Barcelona is approximately three hours” or “Very likely, the anticipated budget deficit for the 2018–2019 school year will be reduced closed to 2 million of euros”, Zadeh (2011) introduced the concept of Z-numbers in 2011, as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy constraint on the values that an uncertainty variable X can take, while B is referred as a measure of certainty or sureness of A, referring to the triple (X, A, B) as a Z-valuation. Therefore, if we take into account the two examples we have mentioned above, we have that the associated Z-valuations are $Z_1=(X=$ travel time between Madrid and Barcelona, approximately three hours, usually) and $Z_2=(X=\text {the anticipated budget deficit for the 2018--2019 school year},$ $\text {closed to 2 million}, \text {very likely})$. Although, as we have already mentioned, the Z-number structure allows modeling many imprecise sentences of natural language, it has the disadvantage of the complexity and the high computational cost of its operations as shown in the literature (Aliev et al. 2015; Pal et al. 2013; Yager 2012). As it was earlier mentioned by Zadeh, “Problems involving computation with Z-numbers are easy to state but far from easy to solve” (Zadeh 2011). Recently, to overcome the limitations of the traditional approach and with the aim to simplify the computational process, a new perspective on Zadeh’s Z-numbers has been proposed by some authors in Massanet et al. (2016, 2020). In this new approach, Z-numbers are represented by pairs of discrete fuzzy numbers (dfn for short) (A, B) where the second component which determines the degree of sureness (certainty, confidence, reliability, strength of belief) of the first component is not interpreted from a strictly probabilistic point of view, but as an evaluation based on a dfn (Massanet et al. 2014; Riera et al. 2015) facilitating the computations between Z-valuations using aggregation operators on the set of dfns (Casasnovas and Riera 2011; Riera and Torrens 2012, 2014). This new perspective, called discrete Z-numbers, largely resolves the issue of the extremely high computational costs associated with the classical definition of Z-numbers, keeping the established linguistic essence of the original idea proposed by Zadeh.

The research on ordering methods among Z-numbers is a common topic in the literature, mainly due to the multiple applications that this linguistic model has. For instance, in the case of a classical multicriteria decision making problem, a hierarchical order is established among the alternatives according to the value of the collective (or aggregated) linguistic interpretation in order to choose the best alternative (Herrera and Herrera-Viedma 2000). For this reason, different methods have been proposed by researchers. In this sense, one of the methods is based on transforming the Z-number into a fuzzy number (Bakar and Gegov 2015; Ezadi et al. 2018; Kang et al. 2012). In this way, Bakar and Gegov (2015) use a multi-layer methodology for Z-number classification where the first layer conveys the conversion of the Z-number into a fuzzy number and the second layer provides the classification of the converted Z-number. A similar idea is proposed by Chai et al. (2023) considering an score and acurracy fuunction to each Z-number or by Ezadi et al. (2018) using sigmoid functions. Other proposals are based on the use of Generalized Fuzzy Numbers (GFNs) instead of couples of fuzzy numbers (Chutia 2021; Jiang et al. 2017). Thus, Chutia (2021) ranking method is based on the use of the value and ambiguity index of the GFNs. In this same framework, Jiang et al. (2017) propose an approach based on the centroid points, degrees of fuzziness and spreads of GFNs. The idea of proposing new methods for ranking fuzzy numbers has also led to the construction of methods for ranking Z-numbers. Ezadi and Allahviranloo (2017) propose a new method for ranking fuzzy numbers based on an hyperbolic tangent function and convex combination. Using this same idea, they apply it to rank Z-numbers. Recently, following Yager’s ideas to rank interval values based on the golden rule (Yager 2016), Cheng et al. (2022) extend this rule and establish a new Z-number ranking method that preserves the initial information of the Z-number. As discussed in the literature (Cheng et al. 2022), many of the ranking methods based on transforming a Z-number into a fuzzy number lose information in the conversion process and this is mainly due to the fact that a Z-number is interpreted as a single combination of two fuzzy numbers, and the meaning of each of the components that form the Z-number is not taken into account.

This paper proposes for the first time a discrete Z-numbers ranking method based on admissible orders on the set of discrete fuzzy numbers. Admissible orders over the set of closed subintervals of [0, 1] were first investigated by Bustince et al. (2013), and are those total orders that refine the natural order of the set of closed subintervals. Since then, several studies on this topic have been presented and this notion has been adapted to different contexts (De Miguel et al. 2016; Lima et al. 2021; Santana et al. 2020). In the framework of the set of discrete fuzzy numbers, the construction of admissible orders is investigated in Riera et al. (2021). Based on these ideas, this paper aims to construct a total order on the set of discrete Z-numbers. More specifically, the order is established on the subset of discrete Z-numbers where the second component has membership values from a finite and prefixed set of values. Moreover, it is proved that this total order is consistent with the linguistic meaning of the two components of the discrete Z-number according to Zadeh’s original proposal, which is one of the main shortcomings that the known orders on Z-numbers in the literature have (Cheng et al. 2022).

The structure of the paper is as follows. In Sect. 2 all the necessary definitions and concepts about orders, discrete fuzzy numbers and Z-numbers to make the paper as self-contained as possible are recalled. In Sect. 3, we compute the total number of discrete fuzzy numbers with a finite set of possible membership values and we introduce the concept of relative rank on these discrete fuzzy numbers. By using these concepts, in Sect. 4 we introduce a new total order defined over the set of discrete Z-numbers and several properties of this order are proved which guarantee that the order has a significant linguistic meaning. Section 5 provides an example to show an application of the order. Finally, in Sect. 6 we perform a comparative study with other ranking methods in the literature.

2 Preliminaries

In order to facilitate a comprehensive understanding of the paper, it is necessary to introduce some preliminary concepts. First of all, let us introduce some basic concepts about orders.

Definition 1

(Bustince et al. (2013)) Given a non-empty set A, a partial order $\preceq$ on the set A is a binary relation on A which is reflexive, antisymmetric and transitive, i.e., a binary relation that satisfies the following properties for each $a,b,c\in A$:

• 1. $a\preceq a$ (reflexivity).

• 2. if $a\preceq b$ and $b\preceq a$, then $a=b$ (antisymmetry).

• 3. if $a\preceq b$ and $b\preceq c$, then $a\preceq c$ (transitivity).

We will write $a \prec b$ if $a\preceq b$ but $a\ne b$. A set A with a partial order $\preceq$ is called a partially ordered set (poset, for short) and denoted by $(A,\preceq )$. If in a poset $(A,\preceq )$ any two elements a, b are comparable, i.e., either $a \preceq b$ or $b \preceq a$ hold, the partial order $\preceq$ is called a linear (or total) order (in this case A is called a chain).

Let us denote by L([0, 1]) the set of all closed subintervals of the unit interval, $L([0, 1]) = \{[a, b] \ | \ 0\le a\le b\le 1\}$.

Definition 2

(Bustince et al. (2013)) Let $(L([0, 1]),\preceq )$ be a poset. The order $\preceq$ is called an admissible order, if

• (i) $\preceq$ is a total (linear) order on L([0, 1]),

• (ii) for all $[a, b], [c, d] \in L([0, 1])$, $[a, b] \preceq [c, d]$ whenever $[a, b]\le _2 [c, d]$ where $\le _2$ denotes the classical partial order of intervals $[a, b]\le _2 [c, d] \iff (a\le c) \hbox { and } (b\le d)$.

Three classical examples of admissible orders on L([0, 1]) are the lexicographic order (lex1), the antilexicographic order (lex2) and the order (XY) proposed by Xu and Yager in (2006).¹

Definition 3

(Bustince et al. (2013); Xu and Yager (2006)) Given $[a,b],[c,d]\in L([0,1])$, we define:

$$\begin{aligned}{} & {} {[}a,b]\le _{lex 1} [c,d]\iff a< c \text { or } (a=c \text { and } b\le d), \\{} & {} {[}a,b]\le _{lex 2} [c,d]\iff b< d \text { or } (b=d \text { and } a\le c), \\{} & {} {[}a,b]\le _{XY}[c,d]\iff (a+b<c+d)\text { or } [(a+b=c+d)\text { and } (b-a\le d-c)]. \end{aligned}$$

Now let us introduce some concepts related to discrete fuzzy numbers. By a fuzzy set of $\mathbb {R}$, we mean a function $A:\mathbb {R}\rightarrow [0,1]$. For each fuzzy set A, let $A^{\alpha }=\left\{ x\in \mathbb {R}\ | \ A(x)\ge \alpha \right\}$ for any $\alpha \in (0,1]$ be its $\alpha$-cut. The support of a fuzzy set A is the set $\hbox {supp}(A)=\left\{ x\in \mathbb {R}\ | \ A(x)>0\right\}$. The core of a fuzzy set A is the $\alpha$-cut $A^1$, that is, $\hbox {core}(A)=\left\{ x\in \mathbb {R} \ | \ A(x)=1\right\}$. In this paper we focus on the discrete version of fuzzy numbers that we call discrete fuzzy numbers.

Definition 4

(Voxman (2001)) A fuzzy set A on $\mathbb {R}$ is called a discrete fuzzy number (dfn) if its support is finite, i.e., there exist $x_1,\ldots , x_{n} \in \mathbb {R}$ with $x_1< x_2<\ldots < x_{n}$ such that $\hbox {supp}(A) = \{x_1,\ldots , x_{n}\}$, and there are natural numbers s, t with $1 \le s \le t \le n$ such that:

• 1. $A(x_i)=1$ for any natural number i with $s \le i \le t$ (core).

• 2. $A(x_i)\le A(x_j)$ for each natural number i, j with $1 \le i \le j \le s$.

• 3. $A(x_i) \ge A(x_j)$ for each natural number i, j with $t \le i \le j \le n$.

A dfn A with $\hbox {supp}(A) = \{x_1,\ldots , x_{n}\}$ is denoted for short as $A=\{A(x_1)/x_1,\ldots ,A(x_{n})/x_{n}\}$.

Starting from this point forward, we use the notation $L_n$ to refer to the finite chain $L_n=\{0,1,\ldots ,n\}$. The set of all dfns whose support lies within $L_n$ is referred to as $\mathcal {D}_{L_n}$. Additionally, the set of dfns having a closed subinterval of the finite chain $L_n$ as their support is denoted by $\mathcal {A}_1^{L_n}$. The significance of exploring $\mathcal {A}_1^{L_n}$ lies in the ability of this category of discrete fuzzy numbers to serve as linguistic expressions that accurately represent the viewpoints of an expert in a decision-making scenario (Massanet et al. 2014; Riera et al. 2015).

Now, we recall the total order on dfns which will constitute the basis to construct a total order on discrete Z-numbers. Let us start with a preliminary definition:

Definition 5

(Riera et al. (2021)) Let $A\in \mathcal {A}_1^{L_n}$ such that $\hbox {supp}(A) =[i,j]$, $i\le j,\ i,j\in L_n$. Then, $\alpha \in (0,1]$ is called a relevant $\alpha$-level for A if there exists $x_i\in supp(A)$ such that $A(x_i) = \alpha$.

Definition 6

(Riera et al. (2021)) Let $A,B\in \mathcal {A}_1^{L_n}$ be two discrete fuzzy numbers whose sets of relevant $\alpha$-levels are $S_A=\{\alpha _1<\cdots <\alpha _k=1\}$ with $k\le n+1$, $S_B=\{\beta _1<\cdots <\beta _m=1\}$ with $m\le n+1$ respectively, and $S_{AB}=S_A\cup S_B=\{\gamma _1<\gamma _2<\cdots <\gamma _t=1\}$ with $1\le t\le k+m-1$. Let us consider an admissible order $\preceq _\delta$ on $\Pi [L_n]=\{[a,b]\;:\; 0\le a\le b\le n,\;a,b\in L_n\}$. We will say $A\prec _{\Delta ^\downarrow _\delta } B$ if and only if $A\not =B$ and there exists a natural number $j\in I$ such that $A^{\gamma _{j}}\prec _{\delta } B^{\gamma _{j}}$ and $A^{\gamma _{i}}=B^{\gamma _{i}}$ for all $i>j$. We will say $A\preceq _{\Delta ^\downarrow _\delta } B$ if and only if $A=B$ or $A\prec _{\Delta ^\downarrow _\delta } B$.

Remark 1

It is easy to check that $A=B$ if and only if all their level sets in $S_{AB}$ are equal, that is, $A^{\gamma _i}=B^{\gamma _i}$ for all $i\in I=\{1,\ldots , t\}$.

Theorem 1

(Riera et al. (2021)) The binary relation $\preceq _{\Delta ^\downarrow _\delta }$ is a total order on $\mathcal {A}_1^{L_n}$.

Remark 2

Although the total order recalled in Definition 6 relies on the use of an admissible order on $\Pi [L_n]$, the binary relation $\preceq _{\Delta ^\downarrow _\delta }$ remains a total order when considering any total (non necessarily admissible) on $\Pi [L_n]$.

Finally, let us recall the concept of Z-number introduced by Zadeh (2011) as the basis of a new linguistic computational model capable of modeling natural language in a precise way.

Definition 7

(Zadeh (2011)) An ordered pair of fuzzy numbers (A, B) is a Z-number. A Z-number is associated with a real-valued uncertain variable, X, with the first component A, playing the role of fuzzy restriction on X, while the fuzzy number B is an imprecise estimation of reliability of A. The ordered triple, (X, A, B), is referred as a Z-valuation and it is equivalent to an assignment statement, X is (A, B).

However, as it is stated in the introduction, the original idea proposed by Zadeh has some drawbacks related to the high computational cost of its associated operations. For this reason, in the literature, several new approaches have been presented which drastically reduce the computational cost. In this way in Massanet et al. (2016) a new vision of Z-numbers where the Z-number is expressed as a pair of discrete fuzzy numbers is proposed.

Definition 8

(Massanet et al. (2016)) Let us consider $L_n$ and $L_m$ two finite chains. An ordered pair of discrete fuzzy numbers (A, B) with $A \in A_1^{L_n}$, $B \in A_1^{L_m}$ is a $(L_n,L_m)$-discrete Z-number. An $(L_n, L_m)$-discrete Z-number is associated with an uncertain variable, X, with the first component, A, playing the role of fuzzy restriction on X, while the dfn, B is an imprecise estimation of reliability of A. The ordered triple (X, A, B) is referred as a Z-valuation and it is equivalent to an assignment statement, X is (A, B).

The set of these discrete Z-numbers is denoted by $\mathcal Z(A,B)$ and an arbitrary element will be denoted by Z(A, B).

Unlike Zadeh’s Z-numbers, which require integration and calculus to operate, these discrete Z-numbers can be manipulated using simple arithmetic and logical operations (Massanet et al. 2016).

3 On discrete fuzzy numbers on $\mathcal {A}_1^{L_{n}}$ with membership values from a finite and prefixed set of values

The main goal of the paper is to define a total order on the set of discrete Z-numbers. In fact, we will define a total order on the subset of discrete Z-numbers defined as pairs of discrete fuzzy numbers on $\mathcal {A}_1^{L_{n}}$ where the second component has membership values from a finite and prefixed set of values. Let us introduce this subset formally.

We will denote by $\mathcal {A}_1^{L_n\times Y_m}$ the subset of discrete fuzzy numbers A with support a closed subinterval of the finite chain $L_n$ and whose membership values A(x) with $x\in L_n$ are in the set $Y_m =\{y_1=0,y_2,\ldots ,y_m=1\}$, with $0=y_1<y_2<\cdots<y_{m-1}<y_m=1$.

Definition 9

Let $L_{n_A}=\{0,1,\ldots ,n_A\}$ and $L_{n_B}=\{0,1,\ldots ,n_B\}$ be two finite chains and $Y_m =\{y_1=0,y_2,\ldots ,y_m=1\}$, with $0=y_1<y_2<\ldots<y_{m-1}<y_m=1$. A second component finite-valued $(L_{n_A},L_{n_B})$-discrete Z-number is a discrete Z-number Z(A, B) with $A\in A_1^{L_{n_A}}$ and $B \in A_1^{L_{n_B}\times Y_m}$.

The set of these discrete Z-numbers is denoted by $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ and an arbitrary element will be denoted by Z(A, B).

The main reason to restrict this study to this subset of discrete Z-numbers is that the proposed total order will rely on the existence of a finite number of potential second components of the discrete Z-numbers. This fact, which seems at a first glance a potential shortcoming for applying the order in applications, is not a significant restriction due to two reasons. On the one hand, human experts are not able to differentiate among an infinite number of possible membership values and each expert always considers a finite subset of possible membership values. On the other hand, since the support of the discrete fuzzy numbers is a closed subinterval of the finite chain $L_n$, each single discrete fuzzy number has a finite set of membership degrees naturally. This implies that in any decision-making problem, the set of membership values considered by the experts for providing their evaluations is finite and this set is going to be the $Y_m$ that will be used for ranking the opinions. Therefore, in real applications, second component finite-valued discrete Z-numbers will not be a restriction at all. In fact, discrete Z-numbers defined as pairs of discrete fuzzy numbers on $\mathcal {A}_1^{L_{n}}$ where each component has membership values from a finite and prefixed set of values would be even more realistic, but for the sake of generality, we will consider only that the second component is finite-valued since the order that we will define does not need a finite number of potential first components of the discrete Z-numbers.

Our total order which will be introduced in the next section will rely on the computation of the relative rank of a discrete fuzzy number in the finite set $\mathcal {A}_1^{L_{n}\times Y_m}$. For this reason, in this section we will compute the cardinal of the set $\mathcal {A}_1^{L_{n}\times Y_m}$ and we will define the relative rank function on this set.

Note that the set $\mathcal {A}_1^{L_{n} \times Y_m}$ forms a totally ordered lattice with the order defined in Definition 6. Let us compute the cardinality of $\mathcal {A}_1^{L_n \times Y_m}$, that we will denote by $|\mathcal {A}_1^{L_n \times Y_m}|$.

We need first a result on the number of monotone mappings between two finite sets.

Lemma 1

Let $m_1,m_2\in \mathbb {N}$. Let $A=\{x_1, \ldots , x_{m_1}\}$ with $x_1<\cdots < x_{m_1}$ and $B=\{y_1, \ldots , y_{m_2}\}$ with $y_1<\cdots < y_{m_2}$ be finite sets with $m_1$ and $m_2$ ordered real numbers, respectively. Then there are exactly $\left( {\begin{array}{c}m_1+m_2-1\\ m_2-1\end{array}}\right)$ increasing (or decreasing) mappings $f: A\longrightarrow B$.

Proof

The proof can be found in Lemma 3.2 in Munar et al. (2023) for the case $m_2\ge m_1$. Let us prove the case $m_1> m_2$. Since f is monotone, without loss of generality we can assume that it is increasing. Therefore, the mapping f is fully determined by counting the number of times that a value $x_i$ with $1\le i \le m_1$ is assigned to the same value $y_j$ with $1\le j\le m_2$. This combinatorial problem is equivalent to determine in how many ways $m_1$ equal objects can be distributed into $m_2$ different sets. The formula to compute this combinatorial problem can be found in Theorem 5.3.2 in Biggs (2002):

$$\begin{aligned} \left( {\begin{array}{c}m_1+m_2-1\\ m_2-1\end{array}}\right) . \end{aligned}$$

$\square$

Now, we are in position of providing the cardinal of the set $\mathcal {A}_1^{L_n \times Y_m}$.

Proposition 1

The number of discrete fuzzy numbers in the set $\mathcal {A}_1^{L_n \times Y_m}$ is

$$\begin{aligned} \left| \mathcal {A}_1^{L_n \times Y_m}\right| =\left( {\begin{array}{c}n+2m-2\\ 2m-2\end{array}}\right) =\frac{(n+2m-2)!}{(2m-2)!n!}. \end{aligned}$$

Proof

To compute the number of discrete fuzzy numbers in the set $\mathcal {A}_1^{L_n \times Y_m}$, we consider four cases according to the values of the core:

• 1. The core of the dfn is $\{0,\ldots ,i\}$, with $i\in \{0,\ldots ,n-1\}$. Let $\mathcal {A}_{1,1}^{L_n \times Y_m}$ be the subset of these dfns.

• 2. The core of the dfn is $\{i,\ldots ,j\}$, with $i\in \{1,\ldots ,n-1\}$ and $j\in \{i,\ldots ,n-1\}$. Let $\mathcal {A}_{1,2}^{L_n \times Y_m}$ be the subset of these dfns.

• 3. The core of the dfn is $\{i,\ldots ,n\}$, with $i\in \{1,\ldots ,n\}$. Let $\mathcal {A}_{1,3}^{L_n \times Y_m}$ be the subset of these dfns.

• 4. The dfn with core $L_n =\{0,\ldots ,n\}$. Let $A_1$ be this dfn, that is, $A_1(i)=1$, for all $i\in L_n$.

It can be easily checked that

• $\mathcal {A}_{1}^{L_n \times Y_m}=\mathcal {A}_{1,1}^{L_n \times Y_m}\cup \mathcal {A}_{1,2}^{L_n \times Y_m}\cup \mathcal {A}_{1,3}^{L_n \times Y_m}\cup \{A_1\}$.

• $\mathcal {A}_{1,i}^{L_n \times Y_m}\cap \mathcal {A}_{1,j}^{L_n \times Y_m}=\emptyset$, for $i,j=1,2,3$, with $i\ne j$.

• $A_1\not \in \mathcal {A}_{1,i}^{L_n \times Y_m}$, $i=1,2,3$.

Therefore, the cardinal of $\mathcal {A}_{1}^{L_n \times Y_m}$ is:

$$\begin{aligned} \bigg |\mathcal {A}_{1}^{L_n \times Y_m}\bigg |=\bigg |\mathcal {A}_{1,1}^{L_n \times Y_m}\bigg |+\bigg |\mathcal {A}_{1,2}^{L_n \times Y_m}\bigg |+\bigg |\mathcal {A}_{1,3}^{L_n \times Y_m}\bigg |+1. \end{aligned}$$

Let us compute $\big |\mathcal {A}_{1,i}^{L_n \times Y_m}\big |$, for $i=1,2,3$.

a) Computation of $\big |\mathcal {A}_{1,1}^{L_n \times Y_m}\big |$:

$$\begin{aligned} \bigg |\mathcal {A}_{1,1}^{L_n \times Y_m}\bigg |=\sum _{i=0}^{n-1} \text{ Number } \text{ of } \text{ decreasing } \text{ mappings } f:\{i+1,\ldots ,n\}\longrightarrow \{y_1,\ldots ,y_{m-1}\}. \end{aligned}$$

Using Lemma 1, we have that the number of decreasing mappings $f:\{i+1,\ldots ,n\}\longrightarrow \{y_1,\ldots ,y_{m-1}\}$ is $\left( {\begin{array}{c}n-i+m-2\\ m-2\end{array}}\right)$. So, we have:

$$\begin{aligned} \bigg |\mathcal {A}_{1,1}^{L_n \times Y_m}\bigg |&=\sum _{i=0}^{n-1} \left( {\begin{array}{c}n-i+m-2\\ m-2\end{array}}\right) =\left( {\begin{array}{c}m+n-2\\ m-2\end{array}}\right) \sum _{i=0}^{n-1} \frac{\left( {\begin{array}{c}n-i+m-2\\ m-2\end{array}}\right) }{\left( {\begin{array}{c}m+n-2\\ m-2\end{array}}\right) }. \end{aligned}$$

Now, using the lookup algorithm (page 36 in Petkovsek et al. (1996)), we have:

where ${}_2F_1(a,b;c;z)$ is the hypergeometric function defined for $a=-n$, $b=1$, $c=2-m-n$ and $z=1$. A full description of hypergeometric series and hypergeometric functions can be found in Bailey (1964). The value of is²:

So,

$$\begin{aligned} \bigg |\mathcal {A}_{1,1}^{L_n \times Y_m}\bigg | = \left( {\begin{array}{c}m+n-2\\ m-2\end{array}}\right) \frac{m+n-1}{m-1}-1=\left( {\begin{array}{c}m+n-1\\ m-1\end{array}}\right) -1. \end{aligned}$$

b) Computation of $|\mathcal {A}_{1,3}^{L_n \times Y_m}|$:

$$\begin{aligned} \bigg |\mathcal {A}_{1,3}^{L_n \times Y_m}\bigg |=\sum _{i=1}^n \text{ Number } \text{ of } \text{ increasing } \text{ mappings } f:\{0,\ldots ,i-1\}\longrightarrow \{y_1,\ldots ,y_{m-1}\}. \end{aligned}$$

Similarly, using again Lemma 1, we have that the number of increasing mappings $f:\{0,\ldots ,i-1\}\longrightarrow \{y_1,\ldots ,y_{m-1}\}$ is $\left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right)$. So, we have:

$$\begin{aligned} \bigg |\mathcal {A}_{1,3}^{L_n \times Y_m}\bigg |&=\sum _{i=1}^n \left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right) =\sum _{i=0}^{n-1} \left( {\begin{array}{c}i+m-1\\ m-2\end{array}}\right) =\sum _{j=0}^{n-1} \left( {\begin{array}{c}n-j+m-2\\ m-2\end{array}}\right) =\bigg |\mathcal {A}_{1,1}^{L_n \times Y_m}\bigg |\\ {}&=\left( {\begin{array}{c}m+n-1\\ m-1\end{array}}\right) -1. \end{aligned}$$

c) Computation of $|\mathcal {A}_{1,2}^{L_n \times Y_m}|$:

$$\begin{aligned} \bigg |\mathcal {A}_{1,2}^{L_n \times Y_m}\bigg |&=\sum _{i=1}^{n-1} \text{ Number } \text{ of } \text{ increasing } \text{ mappings } f:\{0,\ldots ,i-1\}\longrightarrow \{y_1,\ldots ,y_{m-1}\}\cdot \\ {}&\quad \sum _{j=i}^{n-1}\text{ Number } \text{ of } \text{ decreasing } \text{ mappings } f:\{j+1,\ldots ,n\}\longrightarrow \{y_1,\ldots ,y_{m-1}\}\\ {}&= \sum _{i=1}^{n-1}\left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right) \sum _{j=i}^{n-1} \left( {\begin{array}{c}n-j+m-2\\ m-2\end{array}}\right) . \end{aligned}$$

First of all, using the lookup algorithm we obtain:

$$\begin{aligned} \begin{array}{rl} \displaystyle \sum _{j=i}^{n-1} \left( {\begin{array}{c}n-j+m-2\\ m-2\end{array}}\right) &{}= \displaystyle \sum _{j=0}^{n-1-i} \left( {\begin{array}{c}n-j-i+m-2\\ m-2\end{array}}\right) =\displaystyle \frac{(m+n-i-1)!}{(m-1)!(n-i)!}-1 \\ {} &{}=\displaystyle \left( {\begin{array}{c}m+n-i-1\\ m-1\end{array}}\right) -1. \end{array} \end{aligned}$$

Therefore,

$$\begin{aligned} \bigg |\mathcal {A}_{1,2}^{L_n \times Y_m}\bigg |&=\sum _{i=1}^{n-1}\left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right) \left( \left( {\begin{array}{c}m+n-i-1\\ m-1\end{array}}\right) -1\right) \\&= \sum _{i=1}^{n-1}\left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right) \left( {\begin{array}{c}m+n-i-1\\ m-1\end{array}}\right) -\sum _{i=1}^{n-1}\left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right) . \end{aligned}$$

Using again the lookup algorithm, we obtain:

$$\begin{aligned} \sum _{i=1}^{n-1}\left( {\begin{array}{c}i+m-2\\ m-2\end{array}}\right) =\left( {\begin{array}{c}m+n-2\\ m-1\end{array}}\right) -1. \end{aligned}$$

The other summand can be computed as follows:

The value of the previous hypergeometric function is³:

Collecting the previous results, we get that

$$\begin{aligned} \bigg |\mathcal {A}_{1,2}^{L_n \times Y_m}\bigg |&=\left( {\begin{array}{c}m+n-2\\ m-1\end{array}}\right) \frac{m+n-1}{n}\left( \frac{(2m+n-2)! (m-1)!}{((2m-2)!(m+n-1)!}-1\right) -\left( {\begin{array}{c}m+n-2\\ m-2\end{array}}\right) \\&\quad -\left( {\begin{array}{c}m+n-2\\ m-1\end{array}}\right) +1 =\left( {\begin{array}{c}2m+n-2\\ 2m-2\end{array}}\right) -2\left( {\begin{array}{c}m+n-1\\ m-1\end{array}}\right) +1. \end{aligned}$$

To sum up, the number of dfns is:

$$\begin{aligned} \bigg |\mathcal {A}_{1}^{L_n \times Y_m}\bigg |&= \bigg |\mathcal {A}_{1,1}^{L_n \times Y_m}\bigg |+\bigg |\mathcal {A}_{1,2}^{L_n \times Y_m}\bigg |+\bigg |\mathcal {A}_{1,3}^{L_n \times Y_m}\bigg |+1\\ {}&= \left( {\begin{array}{c}2m+n-2\\ 2m-2\end{array}}\right) -2\left( {\begin{array}{c}m+n-1\\ m-1\end{array}}\right) +1+2\left( {\begin{array}{c}m+n-1\\ m-1\end{array}}\right) -2+1\\&=\left( {\begin{array}{c}n+2m-2\\ 2m-2\end{array}}\right) . \end{aligned}$$

$\square$

In Table 1, we show some examples to see how $\left| \mathcal {A}_1^{L_n \times Y_m}\right|$ grows in terms of the values given to m and n.

Table 1 Cardinality of $\mathcal {A}_1^{L_n \times Y_m}$ for different values of n and m

Once we know the total number of dfns that belong to the set $\mathcal {A}_1^{L_n\times Y_m}$, let us define the so-called relative rank function, which uses the rank of a particular dfn in the set $\mathcal {A}_1^{L_n\times Y_m}$. This function will be essential for the total order proposed in the next section.

Definition 10

Let $\preceq$ be a total order on $\mathcal {A}_1^{L_n\times Y_m}$. Let $C_{A}=\{X\in \mathcal {A}_1^{L_n\times Y_m}\ | \ X\preceq A\}$. We define the function $\omega$ that gives the relative rank of a dfn A as follows:

$$\begin{aligned} \begin{array}{lccc} \omega :&{} \ \mathcal {A}_1^{L_n \times Y_m}&{} \longrightarrow &{} (0,1]\\ &{} A&{}\longrightarrow &{}\omega (A)=\frac{|C_{A}|}{\left| \mathcal {A}_1^{L_n \times Y_m}\right| }=\frac{|C_{A}|}{\left( {\begin{array}{c}n+2m-2\\ 2m-2\end{array}}\right) }, \end{array} \end{aligned}$$

where $|C_A|$ denotes the cardinality of $C_A$.

Note that given a total order on $\mathcal {A}_1^{L_{n} \times Y_m}$ we know that $C_{A}$ is not the empty set because $A\in C_{A}$. Furthermore, $C_A$ is finite because $\mathcal {A}_1^{L_{n} \times Y_m}$ is a finite set and $C_{A}\subseteq \mathcal {A}_1^{L_{n} \times Y_m}$. Therefore, $\omega (A)$ is well defined. Moreover, it satisfies the following property.

Proposition 2

Let $A,B\in \mathcal {A}_1^{L_n \times Y_m}$ and $\preceq$ be a total order on $\mathcal {A}_1^{L_n \times Y_m}$. The function $\omega$ defined in Definition 10 satisfies that $A \preceq B$ if, and only if, $\omega (A)\le \omega (B).$

Proof

Let $A,B\in \mathcal {A}_1^{L_n \times Y_m}$ such that $A \preceq B$. Since $\preceq$ is an order relation, it satisfies the transitivity property, therefore it holds that for all $C\in \mathcal {A}_1^{L_n \times Y_m}$ such that $C \preceq A$ then $C \preceq B$, and this implies $C_A\subseteq C_B$ and then, $\omega (A)\le \omega (B)$.

Now suppose $\omega (A)\le \omega (B)$. By definition it can be deduced that $|C_{A}|\le |C_{B}|$, therefore for being $\preceq$ a total order we can deduce that $C_{A} \subseteq C_{B}$, this implies $A\in C_{B}$ and consequently, $A \preceq B$. $\square$

It is clear that if one wants to compute the relative rank of a dfn, all the dfns in $\mathcal {A}_1^{L_n\times Y_m}$ need to be computed. In Algorithm 3 we can see the pseudocode of the construction of every element of $\mathcal {A}_1^{L_n \times Y_m}$. This algorithm uses Algorithms 1 and 2 which generate all the possible increasing and decreasing functions used to generate a dfn.

Algorithm 1

Algorithm for generating increasing functions

Algorithm 2

An algorithm for generating decreasing functions

Algorithm 3

Algorithm for generating all dfns in $\mathcal {A}_{1}^{L_n \times Y_m}$

At this point, we can introduce in Algorithm 4 the algorithm to compute the relative rank of a dfn. Note that this algorithm needs as input the list of all dfns in $\mathcal {A}_{1}^{L_n \times Y_m}$ increasingly sorted according to a total order $\preceq$ on $\mathcal {A}_{1}^{L_n \times Y_m}$. This ordered list can be obtained applying any efficient sorting algorithm such as merge sort (based on $\preceq$) to the output of Algorithm 3.

Algorithm 4

Algorithm for computing the relative rank $\omega$ according to a total order $\preceq$ on $\mathcal {A}_{1}^{L_n \times Y_m}$

All the necessary functions to generate and order all the dfns in the set $\mathcal {A}_1^{L_{n} \times Y_m}$ are implemented in the Github repository https://github.com/aarnaumir/znumbers. The readme.md file contains all the details of such an implementation, as well as examples of how to use these functions.

4 Total order on discrete Z-numbers

Now we have all the necessary background to define our total order on the subset of second component finite-valued discrete Z-numbers. The key idea is to order these discrete Z-numbers by obtaining first discrete fuzzy numbers from them and then applying a total order on discrete fuzzy numbers (for instance the one given in Definition 6). As we will prove later, this order has linguistic sense when an adequate underlying interval order is chosen.

As a first step, let us see how we obtain the discrete fuzzy number that will be used later in the definition of the total order.

Definition 11

Let $Z(A,B)\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$. We define ${A}_{B}$, the transformation of A by B as the function given by:

$$\begin{aligned} {A}_{B}(x)= \left\{ \begin{array}{ll} A(x)+f(\omega (B))(1-A(x)), &{} \text{ if } x\in \hbox {supp}(A), \\ \\ 0 &{} \text{ otherwise, } \end{array} \right. \end{aligned}$$

where $f:(0,1]\longrightarrow [0,1)$ is a strictly decreasing function such that $f(1)=0$ and $\omega$ is the relative rank in $\mathcal {A}_1^{L_{n_B}\times Y_m}$.

Let us prove that ${A}_{B}$ is a discrete fuzzy number satisfying some additional properties.

Theorem 2

Let $Z(A,B)\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ and let ${A}_{B}$ be the transformation of A by B. Then ${A}_{B}\in \mathcal {A}_1^{L_{n_A}}$ having the same support and core of A. Moreover, it satisfies that for any $\alpha \in (0,1]$ $A_B^{(1-f(\omega (B)))\alpha +f(\omega (B))} =A^{\alpha }.$

Proof

It is clear that

$$\begin{aligned} \begin{array}{rl} \{x\in L_{n_A} \ | \ A_B(x)>0\} &{} =\{x\in \hbox {supp}(A) \ | \ A(x)+f(\omega (B))(1-A(x))>0\}\\ {} &{} =\hbox {supp}(A), \end{array} \end{aligned}$$

(1)

since $f(\omega (B))(1-A(x))\ge 0$ for all $x\in L_{n_A}$.

Now let us prove that

$$\begin{aligned} \{x\in L_{n_A} \ | \ A_B(x)=1\}=\hbox {core}(A). \end{aligned}$$

(2)

Let $A(x)=1$, then ${A}_{B}(x)=1+f(\omega (B))(1-1)=1.$ Now suppose that ${A}_{B}(x)=1$ then $1=A(x)+f(\omega (B))(1-A(x))$, or, equivalently $A(x)(1-f(\omega (B)))=1-f(\omega (B))$. Since $f(\omega (B))<1$, this only holds if $A(x)=1.$

Now, let us prove that ${A}_{B}\in \mathcal {A}_1^{L_{n_A}}$ with $\hbox {supp}({A}_{B})=\hbox {supp}(A)$ and $\hbox {core}({A}_{B})=\hbox {core}(A)$. Consider $x_i, x_j\in L_{n_A}$ such that $\min (\hbox {supp}(A))\le x_i \le x_j<\min (\hbox {core}(A))$. Since $A\in \mathcal {A}_1^{L_{n_A}}$, it holds that $A(x_i) \le A(x_j)$. So,

$$\begin{aligned} {A}_{B}(x_i)=A(x_i)(1-f(\omega (B)))+f(\omega (B))\le A(x_j)(1-f(\omega (B)))+f(\omega (B))={A}_{B}(x_j). \end{aligned}$$

On the other hand, consider $x_i, x_j\in L_{n_A}$ such that $\max (\hbox {core}(A))< x_i \le x_j\le \max (\hbox {supp}(A))$. Since $A\in \mathcal {A}_1^{L_{n_A}}$, it holds that $A(x_i) \ge A(x_j)$. So,

$$\begin{aligned} {A}_{B}(x_i)=A(x_i)(1-f(\omega (B)))+f(\omega (B))\ge A(x_j)(1-f(\omega (B)))+f(\omega (B))={A}_{B}(x_j). \end{aligned}$$

Taking into account Eqs. (1) and (2), the result holds.

Finally, let us see that for any $\alpha \in (0,1]$, it holds $A_B^{(1-f(\omega (B)))\alpha +f(\omega (B))} =A^{\alpha }.$ Since $1-f(\omega (B))>0$, we have that:

$$\begin{aligned} A_B(x)&\ge (1-f(\omega (B)))\alpha +f(\omega (B)) \quad \Longleftrightarrow \\ A(x)(1-f(\omega (B)))+f(\omega (B))&\ge (1-f(\omega (B)))\alpha +f(\omega (B)) \quad \Longleftrightarrow \\ A(x)(1-f(\omega (B)))&\ge (1-f(\omega (B)))\alpha \quad \Longleftrightarrow \\ A(x)&\ge \alpha \end{aligned}$$

and therefore $A_B^{(1-f(\omega (B)))\alpha +f(\omega (B))} =A^{\alpha }$. $\square$

The transformation of A by B does not only maintain the same core and support of the initial discrete fuzzy number A. As it can be seen in its definition, the positive value $f(\omega (B))(1-A(x))$ is added to the membership of every element of the support of A. This value is higher as smaller is the relative rank of B in $L_{n_B}\times Y_m$, i.e., as smaller is the reliability/confidence of A. In this way, this transformation changes the uncertainty of A (increasing the membership values outside of the core) according to the reliability/confidence of A provided by B.

Let us illustrate this idea with some examples.

Example 1

In both cases, let $Z(A,B)\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ be a discrete second component finite-valued Z-number with $n_A=n_B=5$, $m=4$ and $Y_m=\{0,0.33,0.66,1\}$. The number of discrete fuzzy numbers in $\mathcal {A}_1^{L_{n_B} \times Y_m}$ is $\left( {\begin{array}{c}5+2\cdot 4-2\\ 2\cdot 4-2\end{array}}\right) =\left( {\begin{array}{c}11\\ 6\end{array}}\right) =462$ and we will use function $f(x)=1-x$ which satisfies the conditions of Definition 11.

• a) Consider $A=\{0.2/1,0.6/2,1/3,1/4,0.6/5\}$ and $B=\{0.33/3,1/4,1/5\}$. By considering the total order $\preceq _{\Delta ^\downarrow _\delta }$ given in Definition 6 generated from the interval order $lex_1$, B is in position 440 and $\omega (B)=\frac{440}{462}\approx 0.9523$. In this case, ${A}_{B}$ is given by:

  $$\begin{aligned} {A}_{B}=\{0.2380952/1, 0.6190476/2, 1/3, 1/4, 0.6190476/5\}, \end{aligned}$$

  which is quite similar to A since the confidence/reliability of A given by B was high.

• b) Consider now $A=\{0.6/1,0.8/2,1/3,1/4,0.3/5\}$ and $B=\{1/5\}$. By considering the same total order used in the previous case, B is in position 462 and therefore $\omega (B)=\frac{462}{462}=1$. The value of ${A}_{B}$ is:

  $$\begin{aligned} {A}_{B}=\{0.6/1,0.8/2,1/3,1/4,0.3/5\}, \end{aligned}$$

  which is equal to A since the confidence is the highest possible.

Now, that we have already introduced the background necessary to define the total order on the set $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$, we define the binary relation as follows:

Definition 12

Consider two total orders $\preceq _{\mathcal {A}}$ on $\mathcal {A}_1^{L_{n_A}}$ and $\preceq _{\mathcal {B}}$ on $\mathcal {A}_1^{L_{n_B} \times Y_m}$. We define the binary relation on $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ associated to $\preceq _{\mathcal {A}}$ and $\preceq _{\mathcal {B}}$ as follows: given $Z(A,B),Z(A',B')\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$,

$$\begin{aligned} Z(A,B)\preceq Z(A',B')\iff ({A}_{B}\prec _{\mathcal {A}} {A'}_{B'}) \hbox { or } ({A}_{B}={A'}_{B'} \hbox { and } B\preceq _{\mathcal {B}}B'). \end{aligned}$$

This binary relation works as follows. Given two second component finite-valued discrete Z-numbers Z(A, B) and $Z(A',B')$, we first compare ${A}_{B}$ and ${A'}_{B'}$ because these two discrete fuzzy numbers represent, in each case, the information given by the fuzzy restriction over the variable that we want to model transformed by means of the reliability of this fuzzy restriction. In the case where these two dfns are equal, we consider more relevant the one with a higher reliability.

To prove that this binary relation on $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ is a total order, we need the following lemma.

Lemma 2

Let $Z(A,B), Z(A',B')\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ be such that ${A}_{B}={A'}_{B'}$ and $B=B'$. Then, it holds that $Z(A,B)=Z(A',B')$.

Proof

First, by Theorem 2, it can be deduced that $\hbox {supp}(A)=\hbox {supp}(A')$ because

$$\begin{aligned} \hbox {supp}(A)=\hbox {supp}({A}_{B})=\hbox {supp}({A'}_{B'})=\hbox {supp}(A'). \end{aligned}$$

Therefore, let us consider $x\in \hbox {supp}(A)$ and let us see that $A(x)=A'(x)$ to prove the lemma.

Since $B=B'$, we have $\omega (B)=\omega (B')$. Also, since ${A}_{B}(x)={A'}_{B'}(x)$ for all $x\in \hbox {supp}(A)$ we have:

$$\begin{aligned} A(x)+f(\omega (B))(1-A(x))={A}_{B}(x)={A'}_{B'}(x)=A'(x)+f(\omega (B))(1-A'(x)). \end{aligned}$$

Therefore,

$$\begin{aligned} (1-f(\omega (B)))A(x)=(1-f(\omega (B)))A'(x). \end{aligned}$$

Taking into account that $f(\omega (B))< 1$, this equation holds if $A(x)=A'(x)$. Consequently, $A=A'$ and $Z(A,B)=Z(A',B')$. $\square$

Theorem 3

The binary relation given in Definition 12 is a total order on the set $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$.

Proof

First of all, let us see that the relation defined in Definition 12 is an order relation, that is, it satisfies the reflexivity, antisymmetry and transitivity properties:

• Reflexivity: Let $Z(A,B)\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$. Since ${A}_{B}={A}_{B}$ and $B\preceq _{\mathcal {B}}B$ (in fact, $B=B$), by definition $Z(A,B)\preceq Z(A,B)$.

• Antisymmetry: Suppose $Z(A,B)\preceq Z(A',B')$ and $Z(A',B')\preceq Z(A,B)$ and consider the following cases:

  • ${A}_{B} = {A'}_{B'}$: Since $Z(A,B)\preceq Z(A',B')$ we have that $B\preceq _{\mathcal {B}}B'$ and since $Z(A',B')\preceq Z(A,B)$ we have that $B'\preceq _{\mathcal {B}}B$ . Using the antisymmetry of the order relation $\preceq _{\mathcal {B}}$, we have that $B=B'$ and by Lemma 2, $Z(A,B)=Z(A',B')$.

  • ${A}_{B}\ne {A'}_{B'}$: Since $Z(A,B)\preceq Z(A',B')$ we have that ${A}_{B}\prec _{\mathcal {A}}{A'}_{B'}$. Since $Z(A',B')\preceq Z(A,B)$ we have ${A'}_{B'}\prec _{\mathcal {A}}{A}_{B}$. But this is a contradiction, so it cannot hold that ${A}_{B}\ne {A'}_{B'}$.

• Transitivity: Suppose $Z(A,B)\preceq Z(A',B')$ and $Z(A',B')\preceq Z(A'',B'')$ and consider the following cases:

  • $A_B={A'}_{B'}={A''}_{B''}$: It holds that $B\preceq _{\mathcal {B}}B'\preceq _{\mathcal {B}}B''$, which implies $A_B={A''}_{B''}$ and $B\preceq _{\mathcal {B}}B''$ using the transitivity of order $\preceq _{\mathcal {B}}$.

  • ${A}_{B}={A'}_{B'}\ne {A''}_{B''}$: In this case, ${A'}_{B'}\prec _\mathcal {A} {A''}_{B''}$ which implies $A_B = {A'}_{B'}\prec _\mathcal {A} {A''}_{B''}$.

  • ${A}_{B}\ne {A'}_{B'}={A''}_{B''}$: This implies ${A}_{B} \prec _{\mathcal {A}} {A'}_{B'}= {A''}_{B''}$, and consequently, $A_B \prec _{\mathcal {A}} {A''}_{B''}$.

  • ${A}_{B}\ne {A'}_{B'}\ne {A''}_{B''}$: Now we have that ${A}_{B} \prec _{\mathcal {A}} {A'}_{B'}\prec _{\mathcal {A}} {A''}_{B''}$, and consequently $A_B \prec _{\mathcal {A}} {A''}_{B''}$ using the transitivity of order $\preceq _{\mathcal {A}}$.

  In any case, $Z(A,B)\preceq Z(A'',B'')$.

It remains to be proved that it is a total order, that is, given $Z(A,B), Z(A',B')\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$, it holds that $Z(A,B)\preceq Z(A' ,B')$ or $Z(A',B')\preceq Z(A,B)$. So, suppose $Z(A',B')\npreceq Z(A,B)$, and we will see that $Z(A,B)\preceq Z(A',B')$. Consider two cases:

• ${A}_{B}\ne {A'}_{B'}$: In this case, since $Z(A',B')\npreceq Z(A,B)$ we have ${A'}_{B'}\nprec _{\mathcal {A}}{A}_{B}$. Using that $\preceq _{\mathcal {A}}$ is a total order, it can be deduced that ${A}_{B}\prec _{\mathcal {A}} {A'}_{B'}$ and therefore $Z(A,B)\preceq Z(A',B')$.

• ${A}_{B}={A'}_{B'}$: In this case, we consider two more subcases:

  • $B=B'$: Using Lemma 2, we have $Z(A,B)=Z(A',B')$, a contradiction with the hypothesis $Z(A',B')\npreceq Z(A,B)$.

  • $B\ne B'$: Since $Z(A',B')\npreceq Z(A,B)$ and $B\ne B'$, we have $B'\npreceq _{\mathcal {B}}B$. Using that $\preceq _{\mathcal {B}}$ is a total order, it can be deduced that $B \prec _{\mathcal {B}} B'$ and therefore $Z(A,B)\preceq Z(A',B')$.

$\square$

Let us see an example illustrating the operating of the order. In some of the cases, the transformations of the first component by the second component coincide and we only take on account the second component of the discrete Z-numbers.

Example 2

Let us consider the set of discrete second component finite-valued Z-numbers $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ with $n_A=n_B=5$, $m=4$ and $Y_m=\{0,0.33,0.66,1\}$. Consider now the following discrete fuzzy numbers

$$\begin{aligned} \begin{array}{l} A=\{0.2/1,0.6/2,1/3,1/4,0.6/5\},\\ A'=\{0.6/1,0.8/2,1/3,1/4,0.3/5\},\\ A''=\{1/3,1/4\},\\ A'''=\{0.2380952/1, 0.6190476/2, 1/3, 1/4, 0.6190476/5\},\\ B=\{0.33/3,1/4,1/5\},\\ B'=\{1/5\}. \end{array} \end{aligned}$$

The number of discrete fuzzy numbers in $\mathcal {A}_1^{L_{n_B} \times Y_m}$ is $\left( {\begin{array}{c}5+2\cdot 4-2\\ 2\cdot 4-2\end{array}}\right) =\left( {\begin{array}{c}11\\ 6\end{array}}\right) =462$. By considering the total order $\preceq _{\Delta ^\downarrow _\delta }$ given in Definition 6 generated from the interval order $lex_1$ as $\preceq _{\mathcal {B}}$, we have that B is in position 440, $\omega (B)=\frac{440}{462}\approx 0.952381$ and $B'$ is in position 462, $\omega (B')=\frac{462}{462}=1$.

Consider now the function $f(x)=1-x$ which satisfies the conditions of the Definition 11 and the total order $\preceq _{\Delta ^\downarrow _\delta }$ given in Definition 6 generated from the interval order $lex_1$ as $\preceq _{\mathcal {A}}$.

• a) It holds that $Z(A',B')\prec Z(A,B)$ because ${A}_{B}$ is given by:

  $$\begin{aligned} {A}_{B}=\{0.2380952/1, 0.6190476/2, 1/3, 1/4, 0.6190476/5\}, \end{aligned}$$

  and ${A'}_{B'}$ is:

  $$\begin{aligned} {A'}_{B'}=\{0.6/1,0.8/2,1/3,1/4,0.3/5\}, \end{aligned}$$

  and ${A'}_{B'}\prec _{\Delta ^\downarrow _\delta } {A}_{B}.$

• b) It holds that $Z(A'',B)\prec Z(A'',B')$. Let us check it. In this case, we have that ${A''}_{B}=A''=A''={A''}_{B'}$ since $A''$ only has elements on the core. Therefore we must consider the total order on the second components to order these discrete Z-numbers. Indeed, it is immediate to see that $B\prec _{\Delta ^\downarrow _\delta } B'$.

• c) It holds that $Z(A,B)\prec Z(A''',B')$. Again we have that

  $$\begin{aligned} {A}_{B}=A'''={A'''}_{B'} \end{aligned}$$

  but this case is different from the previous one because one of the transformations ${A}_{B}$ is different from A. Anyway, we must resort to the total order on the second components to order these discrete Z-numbers. Again, we have that $B\prec _{\Delta ^\downarrow _\delta } B'$.

Although we have already proved that the order defined in Definition 12 is a total order, it is of great interest to investigate if this order has a linguistic meaning, that is, if it presents a behavior which is compatible with the intuition of a user when sorting a set of discrete Z-numbers.

Let us start with a first lemma which will be useful for the subsequent results.

Lemma 3

Let Z(A, B), $Z(A',B)\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ be two discrete Z-numbers with the same second component. If $\alpha =\max \{\gamma \in (0,1] \ | \ A^{\gamma }\ne A'^{\gamma }\}$, then $\alpha _{B}:=\alpha +f(\omega (B))(1-\alpha )=\max \{\beta \in (0,1] \ | \ {A}_{B}^{\beta }\ne {A}_{B}^{'\beta }\}.$

Proof

Since according to Theorem 2, for all $\alpha \in (0,1]$, $A_B^{(1-f(\omega (B)))\alpha +f(\omega (B))} =A^{\alpha }$, the proof is straightforward. $\square$

Let us introduce two linguistic properties that, in our opinion, it is advisable that any total order on discrete Z-numbers fulfill in order to have linguistic meaning. First of all, given two discrete Z-numbers $Z(A,B),Z(A',B')\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$, a desirable linguistic property would be that when the values of the first component are the same, that is, $A=A'$, i.e., the fuzzy restrictions over the real-valued uncertain variable X coincide, these discrete Z-numbers should be ranked according to the degree of certainty/realiability of such restriction, in the sense that the one with the higher certainty is greater. Formally, this is translated to:

$$\begin{aligned} \hbox {If } A=A', \hbox { then } B\preceq _{\mathcal {B}}B'\iff Z(A,B)\preceq Z(A',B'). \end{aligned}$$

On the other hand, if the values of the second component are the same, that is, $B=B'$, we have that the degrees of certainty/reliability are the same, so we only have to take into account the order on the values of the first component to compare the two discrete Z-numbers. Formally, this is translated to:

$$\begin{aligned} \hbox {If } B=B', \hbox { then } A\preceq _{\mathcal {A}}A'\iff Z(A,B)\preceq Z(A',B'). \end{aligned}$$

We have to take into account that we have introduced a whole family of total orders on the set $\mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ since this total order is based on two total orders on the sets $\mathcal {A}_1^{L_{n_A}}$ and $\mathcal {A}_1^{L_{n_B}}$, which in turn, depend on the choice of admissible orders within the set of intervals of $L_{n_A}$ and $L_{n_B}$. Unfortunately, the two properties do not hold for any configuration, but we will see that certain configurations fulfill both properties.

Given two discrete Z-numbers $Z(A,B), Z(A',B')\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$, notice that, by construction, if $A=A'$ and $B\preceq _{\mathcal {B}}B'$, we have that for all $\alpha \in (0,1]$, ${A}_{B'}^{\alpha }\subseteq {A}_{B}^{\alpha }$. This fact tells us that if we want $B\preceq _{\mathcal {B}}B'\iff Z(A,B)\preceq Z(A,B')$, we need to choose a total order within the set of intervals of $L_{n_A}$ where being contained is equivalent to being greater. This means that we require an order that satisfies that if ${A}_{B'}^{\alpha }=[x_1,x_2]$ and ${A}_{B}^{\alpha }=[x_3,x_4]$ such that $x_1 \le x_3$ and $x_4\le x_2$, then the interval $[x_3,x_4]$ must be greater. This can be achieved considering a total order that extends the following partial order defined over the intervals.

Definition 13

Given $[a,b],[c,d]\in \Pi [L_n]$ we define the binary relation $\le _{\text {inc}}$ on the set $\Pi [L_n]$ as:

$$\begin{aligned}{}[a,b]\le _{\text {inc}} [c,d]\iff (a\le c \hbox { and } d\le b) \end{aligned}$$

It is clear that the binary relation defined in Definition 13 is a partial order on the set $\Pi [L_n]$. This order ensures that the interval-inclusion property is satisfied, in the sense that if an interval is contained within another interval, it is considered to be greater with respect to the interval ordering. Now we will search for a total order that fulfills the interval-inclusion property, that is an order that refines the partial order defined in Definition 13. We understand that a total order refines a given partial order as whenever two elements are comparable in the partial order, they are compared in the same way by using the total order. Looking at the definition of an admissible order defined over the intervals in Bustince et al. (2013) one can make a reconstruction of the definition based on the one defined in Definition 13 in the following way.

Definition 14

Let $\le _{\text {inc}}$ be the partial order defined in Definition 13. We say that the binary relation $\le$ on the set $\Pi [L_n]$ is an admissible order with respect to $\le _{\text {inc}}$ if it fulfills the following properties:

• 1. It is a total order.

• 2. It refines $\le _{\text {inc}}$, i.e., for any $[a,b],[c,d]\in \Pi [L_n]$ such that $[a,b]\le _{\text {inc}} [c,d]$, it holds that $[a,b]\le [c,d]$.

Let us define an interval order that fulfills the properties of Definition 14 to see an example of an admissible order with respect to $\le _{\text {inc}}$.

Definition 15

Given $[a,b],[c,d]\in \Pi [L_n]$ we define the binary relation $\le _{\text {t-inc}}$ on the set $\Pi [L_n]$ as:

$$\begin{aligned}{}[a,b]\le _{\text {t-inc}} [c,d]\iff a<c \hbox { or }(a=c \hbox { and } d\le b). \end{aligned}$$

It is a matter of straightforward computation to check that $\le _{\text {t-inc}}$ is a total order on $\Pi [L_n]$ and that refines $\le _{\text {inc}}$.

Now, let us prove the two linguistic properties.

Theorem 4

Let $Z(A,B), Z(A',B')\in \mathcal{Z}(L_{n_A},L_{n_B}\times Y_m)$ and consider $\preceq$ the total order defined in Definition 12 generated from the total orders $\preceq _{\Delta ^\downarrow _{\delta _A}}$ on $\mathcal {A}_1^{L_{n_A}}$ (with underlying total order $\preceq _{\delta _A}$ on $\Pi [L_{n_A}]$) and $\preceq _{\Delta ^\downarrow _{\delta _B}}$ on $\mathcal {A}_1^{L_{n_B}}$ (with underlying total order $\preceq _{\delta _B}$ on $\Pi [L_{n_B}]$). The following statements are true.

• a) If $A=A'$ and $\preceq _{\delta _A}$ is an admissible order with respect to $\le _{\text {inc}}$ in the sense of Definition 14, then $B\preceq _{\Delta ^\downarrow _{\delta _B}}B'\iff Z(A,B)\preceq Z(A',B')$.

• b) If $B=B'$ then $A\preceq _{\Delta ^\downarrow _{\delta _A}}A'\iff Z(A,B)\preceq Z(A',B)$.

Proof

a) Suppose $A=A'$, and consider two cases:

• ${A}_{B}={A'}_{B'}$: In this case, by the definition of $\preceq$, it holds that $B\preceq _{\Delta ^\downarrow _{\delta _B}} B'\iff Z(A,B)\preceq Z(A',B')$.

• ${A}_{B}\ne {A'}_{B'}$: Since ${A}_{B}\ne {A'}_{B'}$, one easily sees that $B \ne B'$. Call $\lambda =f(\omega (B))-f(\omega (B')) \ne 0.$ For any $x\in supp(A)\backslash core(A)$ such that ${A}_{B}(x)\ne {A'}_{B'}(x)$, it holds that

  $$\begin{aligned} {A}_{B}(x)-{A'}_{B'}(x)=(1-A(x))(f(\omega (B))-f(\omega (B')))=(1-A(x))\lambda \ne 0 , \end{aligned}$$

  which has a constant sign. Therefore, we deduce ${A}_{B}(x)<{A'}_{B'}(x)$ or ${A}_{B}(x)>{A'}_{B'}(x)$ for any $x\in supp(A)\backslash core(A)$. This means that ${A}_{B}^{\alpha }\subset {A'}_{B'}^{\alpha }$ or ${A'}_{B'}^{\alpha }\subset {A}_{B}^{\alpha }$ where $\alpha =\max \{\beta \in (0,1]\ |\ {A}_{B}^{\beta }\ne {A'}_{B'}^{\beta }\}$. Now, using that the total order $\preceq _{\Delta ^\downarrow _{\delta _A}}$ on $\mathcal {A}_1^{L_{n_A}}$ is generated from $\preceq _{\delta _A}$, which is an admissible order with respect to $\le _{\text {inc}}$ in the sense of Definition 14, we have that ${A'}_{B'}^{\alpha }\subset {A}_{B}^{\alpha }\iff {A}_{B}\prec _{\Delta ^\downarrow _{\delta _A}}{A'}_{B'}$. Let us prove in further detail this last step.

  $\Rightarrow$):

  If ${A'}_{B'}^{\alpha }\subset {A}_{B}^{\alpha }$, we know that ${A}_{B}^{\alpha }<_{\text {inc}} {A'}_{B'}^{\alpha }$ and since $\preceq _{\delta _A}$ is an admissible order with respect to $\le _{\text {inc}}$, we get ${A}_{B}^{\alpha }\prec _{\delta _A} {A'}_{B'}^{\alpha }$. Finally, taking into account that $\alpha =\max \{\beta \in (0,1]\ |\ {A}_{B}^{\beta }\ne {A'}_{B'}^{\beta }\}$ and the total order $\preceq _{\Delta ^\downarrow _{\delta _A}}$ on $\mathcal {A}_1^{L_{n_A}}$ is generated from $\preceq _{\delta _A}$, then ${A}_{B}\prec _{\Delta ^\downarrow _{\delta _A}}{A'}_{B'}$.

  $\Leftarrow$):

  Consider now ${A}_{B}\prec _{\Delta ^\downarrow _{\delta _A}}{A'}_{B'}$. Recall that we had that either ${A}_{B}^{\alpha }\subset {A'}_{B'}^{\alpha }$ or ${A'}_{B'}^{\alpha }\subset {A}_{B}^{\alpha }$ where $\alpha =\max \{\beta \in (0,1]\ |\ {A}_{B}^{\beta }\ne {A'}_{B'}^{\beta }\}$. If ${A}_{B}^{\alpha }\subset {A'}_{B'}^{\alpha }$, this implies that ${A'}_{B'}^{\alpha }\le _{\text {inc}} {A}_{B}^{\alpha }$, which implies ${A'}_{B'}^\alpha \prec _{\delta _A} {A}_{B}^\alpha$. Therefore, ${A'}_{B'}\prec _{\Delta ^\downarrow _{\delta _A}}{A}_{B}$ and a contradiction arises. So, it must hold that ${A'}_{B'}^{\alpha }\subset {A}_{B}^{\alpha }$.

  Finally, the result follows since

  $$\begin{aligned} \begin{array}{rl} B\prec _{\Delta ^\downarrow _{\delta _B}} B'&{}\iff \omega (B)<\omega (B')\iff {A'}_{B'}^{\alpha }\subset {A}_{B}^{\alpha }\iff {A}_{B}\prec _{\Delta ^\downarrow _{\delta _A}} {A'}_{B'}\\ {} &{}\iff Z(A,B)\prec Z(A',B').\end{array} \end{aligned}$$

b) Suppose $B'=B$, therefore $\omega (B)=\omega (B')$. Let $A^{\alpha }$, $(A')^{\alpha }$ be the $\alpha$-cuts where $\alpha =\max \{\beta \in (0,1] \ | \ A^{\beta }\ne (A')^{\beta }\}$ and ${A}_{B}^{\alpha _{B}}$, $({A'}_{B})^{\alpha _{B}}$ be the $\alpha$-cuts where $\alpha _{B}=\alpha +f(\omega (B))(1-\alpha )$. By Lemma 3 we have that $\alpha _{B}=\max \{\beta \in (0,1] \ | \ {A}_{B}^{\beta }\ne ({A'}_{B})^{\beta }\}$. Moreover, by Theorem 2 it holds that $A^{\alpha }={A}_{B}^{\alpha _{B}}$ and $(A')^{\alpha }=({A'}_{B})^{\alpha _{B}}$. Therefore, we have

$$\begin{aligned} A\preceq _{\Delta ^\downarrow _{\delta _A}} A'\iff A^{\alpha }\preceq _{\delta _A} (A')^{\alpha }\iff {A}_{B}^{\alpha _{B}}\preceq _{\delta _A} ({A'}_{B})^{\alpha _{B}}\iff Z(A,B)\preceq Z(A',B). \end{aligned}$$

$\square$

5 Practical example

Let us suppose that an NBA Team wants to analyze how good are the six best draft prospects of a given year. Hence, they decide to hire a professional scout that analyzes the six prospects $E=\{P_1,P_2,P_3,P_4,P_5,P_6\}$ to determine the capability of each player on the court. This expert evaluates the variable $X=\{\text{ Performance } \text{ on } \text{ the } \text{ court }\}$ with some degree of confidence and for each of the six prospects, the scout makes an evaluation (using second component finite-valued Z-numbers) in order to determine which player would be the most appropriate to select to join the team.

We will denote by $Z(A_i,B_i)$ the evaluation of the scout on player $P_i$ for all $1\le i \le 6$. In our example, we assume

$$\begin{aligned} n_A=10, n_B= 6, m=6, Y_m =\{0,0.2,0.4,0.6,0.8,1\}, \end{aligned}$$

and $\mathcal {L}_{n_A}=\{\text {N,AN,VL,L,BA,A,AA,H,VH,PL,RS}\}$ and $\mathcal {L}_{n_B}=\{\text {I,NS,M,P,SS,S,AS}\}$ are linguistic chains with the labels collected in Table 2. Using Proposition 1, there are $\left( {\begin{array}{c}6+2\cdot 6-2\\ 2\cdot 6-2\end{array}}\right) =\left( {\begin{array}{c}16\\ 10\end{array}}\right) =8008$ dfns in the set $\mathcal {A}_1^{L_6 \times Y_6}$.

Table 2 Labels of linguistic chains $\mathcal{L}_{n_A}$ and $\mathcal{L}_{n_B}$

Accordingly to this notation, the evaluations provided by the scout for each of the players are the following:

• Player 1: The scout states the following evaluation with regard to Player 1: “I am not sure that his performance on the court is high level.” This sentence can be interpreted as the Z-valuation Z(Performance on the court, High Level, Not Sure), or in a simplified way Z(High Level, Not Sure), and, following the established notation and convention, the evaluation expressed as a couple of dfn’s is:

  $$\begin{aligned} Z(A_1,B_1)=(A_1=\{0.6/\text{AA }, 1/\text{H }, 0.6/\text{VH }\}, B_1=\{0.6/\text{I }, 1/\text{NS }, 0.8/\text{M }\}). \end{aligned}$$

• Player 2: The scout states the following evaluation with regard to Player 2: “I am absolutely sure that his performance on the court is below average.” This sentence can be interpreted as the Z-valuation Z(Performance on the court, Below Average, Absolutely Sure), or in a simplified way Z(Below Average, Absolutely Sure), and, following the established notation and convention,

  $$\begin{aligned} Z(A_2,B_2)=(A_2=\{0.4/\text{L }, 1/\text{BA }, 0.4/\text{A }\}, B_2=\{0.8/\text{S }, 1/\text{AS }\}). \end{aligned}$$

• The scout states the following evaluation with regard to Player 3: “I am not sure that his performance on the court is between below average and above average.” This sentence can be interpreted as the Z-valuation Z(Performance on the court, between Average and Above Average, Not Sure), or in a simplified way Z(between Average and Above Average, Not Sure), and, following the established notation and convention,

  $$\begin{aligned} Z(A_3,B_3)=(A_3=\{0.4/\text{BA },1/\text{A }, 1/\text{AA }, 0.2/\text{H }\}, B_3=\{0.6/\text{I }, 1/\text{NS }, 0.8/\text{M }\}). \end{aligned}$$

• The scout states the following evaluation with regard to Player 4: “My evaluation is that his performance on the court is high level but my certainty is between not sure and maybe.” This sentence can be interpreted as the Z-valuation Z(Performance on the court, High Level, between Not Sure and Maybe), or in a simplified way Z(High Level, between Not Sure and Maybe), and, following the established notation and convention,

  $$\begin{aligned} Z(A_4,B_4)=(A_4=\{0.4/\text{AA }, 1/\text{H }, 0.4/\text{VH }\}, B_4=\{0.4/\text{I }, 1/\text{NS }, 1/\text{M }, 0.8/\text{P }\}). \end{aligned}$$

• The scout states the following evaluation with regard to Player 5: “Possibly his performance on the court is between average and above average” This sentence can be interpreted as the Z-valuation Z(Performance on the court, between Average and Above Average, Possibly), or in a simplified way Z(between Average and Above Average, Possibly), and, following the established notation and convention,

  $$\begin{aligned} Z(A_5,B_5)=(A_5=\{1/\text{A }, 1/\text{AA }, 0.2/\text{H }\}, B_5= \{0.6/\text{M }, 1/\text{P }, 0.8/\text{SS }\}). \end{aligned}$$

• The scout states the following evaluation with regard to Player 6: “I am sure that his performance on the court is between below average and above average.” This sentence can be interpreted as the Z-valuation Z(Performance on the court, between Below Average and Above Average, Sure), or in a simplified way Z(between Below Average and Above Average, Sure), and, following the established notation and convention,

  $$\begin{aligned} Z(A_6,B_6)= (A_6=\{1/\text{BA }, 1/\text{A }, 1/\text{AA }\}, B_6=\{0.8/\text{P }, 0.8/\text{SS }, 1/\text{S }, 0.6/\text{AS }\}). \end{aligned}$$

The first step is to compute the relative rank $\omega$ in $\mathcal {A}_1^{L_6 \times Y_6}$ for all the second components $B_i$ for all $1\le i\le 6$ of the discrete Z-numbers.

With respect to Player 1, the position of dfn $B_1$ is 813. Then, the value of $\omega (B_1)$ is $\omega (B_1)=\frac{813}{8008}\approx 0.1015235.$

Table 3 collects these relative ranks along with the positions of dfns $B_i$ where we have considered the total order $\preceq _{\Delta ^\downarrow _\delta }$ on $\mathcal {A}_1^{L_{6}}$ generated from the total order given in Definition 15.

Table 3 Relative ranks in $\mathcal {A}_1^{L_6 \times Y_6}$ of all dfns $B_i$

The next step is to compute ${{(A_i)}_{B_i}}$ for all $1\le i\le 6$, where the function $f(x)=1-x$ is chosen. With respect to Player 1, the value of the dfn $(A_1)_{B_1}$ is:

$$\begin{aligned} (A_1)_{B_1}(N)&= (A_1)_{B_1}(0) = 0,\\ (A_1)_{B_1}(AN)&= (A_1)_{B_1}(1) =0,\\ (A_1)_{B_1}(VL)&=(A_1)_{B_1}(2) = 0,\\ (A_1)_{B_1}(L)&=(A_1)_{B_1}(3) = 0,\\ (A_1)_{B_1}(BA)&=(A_1)_{B_1}(4) = 0,\\ (A_1)_{B_1}(A)&= (A_1)_{B_1}(5) = 0,\\ (A_1)_{B_1}(AA)&= (A_1)_{B_1}(6) \\ {}&= A_1(6)+f(w(B_1))(1-A_1(6))=0.6+f\left( \frac{813}{8008}\right) (1-0.6)\\ {}&=0.6+\left( 1-\frac{813}{8008}\right) \cdot 0.4\approx 0.6+0.8984765\cdot 0.4= 0.9593906,\\ (A_1)_{B_1}(H)&=(A_1)_{B_1}(7) =1+f\left( \frac{813}{8008}\right) (1-1) =1,\\ (A_1)_{B_1}(VH)&=(A_1)_{B_1}(8) =0.6+f\left( \frac{813}{8008}\right) (1-0.6)= 0.9593906,\\ (A_1)_{B_1}(PL)&=(A_1)_{B_1}(9) = 0,\\ (A_1)_{B_1}(RS)&=(A_1)_{B_1}(10) = 0. \end{aligned}$$

Consequently, the dfn $(A_1)_{B_1}$ will be:

$$\begin{aligned} (A_1)_{B_1}=\{0.9594/\text{AA }, 1/\text{H },0.9594/\text{VH }\}. \end{aligned}$$

The expressions of all dfns $(A_i)_{B_i}$, $i=1,\ldots ,6$ are shown in Table 4.

Table 4 Expressions of the dfns ${{(A_i)}_{B_i}}$, $i=1,\ldots ,6$

Finally, applying the total order $\preceq$ defined in Definition 12 where we have considered the total orders $\preceq _{\Delta ^\downarrow _\delta }$ (generated from the interval total order $\le _{\text {t-inc}}$ given in Definition 15) as $\preceq _{\mathcal {A}}$ and $\preceq _{\Delta ^\downarrow _\delta }$ (generated from the interval total order lex1) as $\preceq _{\mathcal {B}}$, the players are sorted according to the following rank of evaluations:

$$\begin{aligned} Z(A_2,B_2)\prec Z(A_6,B_6) \prec Z(A_3,B_3) \prec Z(A_5,B_5) \prec Z(A_1,B_1) \prec Z(A_4,B_4). \end{aligned}$$

6 Comparison with other methods in the literature

The study of ranking methods, as mentioned in the introduction, is a subject in constant evolution. From the analysis of the different ranking methods of Z-numbers that have been published in the literature, a comparison among these ranking methods is mathematically problematic. Indeed,

• the probabilistic approach that is inherent to most of the Z-numbers approaches and which is not present in our approach,

• the typology of the components of the Z-numbers that depending on the approach can be triangular or trapezoidal fuzzy (generalised) numbers and not discrete fuzzy numbers as in our approach,

• the different ordering methodologies used,

jeopardize the conclusions derived from any comparison study. Taking into account the above discussion, after a thorough analysis of the literature, we have only been able to find two proposals that will allow us to make a comparison according to the interpretation of the Z-numbers proposed in this work. First, in Bilgin and Alci (2022) an extensive study of ranking techniques for Z-numbers is performed and specifically, there is a section devoted to the particular case of discrete Z-numbers, where Gong’s method (Gong et al. 2020) is presented. Second, Xian et al. in (2021) proposed a method to sort Z-mixture numbers, and particularly discrete Z-numbers. These two approaches will be analyzed and compared with our ranking method. Anyway, it is important to highlight that the authors of both papers did not study whether these orders are total, a key property which has been proved for the order presented in our paper.

The discrete Z-numbers we will use in our comparison are shown in Table 5 and coincide with those used in the previous section. According to the sorting method proposed in this paper, the following results were obtained

$$\begin{aligned} Z(A_2,B_2)\prec Z(A_6,B_6) \prec Z(A_3,B_3) \prec Z(A_5,B_5) \prec Z(A_1,B_1) \prec Z(A_4,B_4). \end{aligned}$$

Note that our method allow us to rank Z-numbers whose components are not necessary discrete triangular fuzzy numbers (discrete fuzzy numbers whose core contains a unique value).

Table 5 Expressions of the $Z_i(A_i,B_i)$, $i=1,\ldots ,6$

In Xian et al. (2021), proposed an order for Z-mixture-numbers based on the positive and negative ideal degree. In that paper, given two Z-numbers $Z_1$ and $Z_2$, the ideal degree is defined based on a parameter $\beta \in [0,1]$ that balances the influence of each Z-number, see Xian et al. (2021) for details. All the examples considered in that paper are Z-numbers whose components are discrete triangular fuzzy numbers. When we apply the Xian’s method to order the discrete Z-numbers considered in Table 5, we find that the discrete Z-numbers $Z(A_2,B_2)$ and $Z(A_6,B_6)$ cannot be considered because the computation of the centroid points associated with components $B_2$ and $A_6$ was impossible to perform. This is because $B_2$ has only two values on the support and, in the case of $A_6$, it is described only by the core which contains three values.

With the other discrete Z-numbers, we have obtained the following result depending on the parameter $\beta$:

• $\beta \in [0,0.25):$ $Z(A_3,B_3)\prec Z(A_4,B_4)\prec Z(A_5,B_5)\prec Z(A_1,B_1),$

• $\beta \in [0.25,1]:$ $Z(A_3,B_3)\prec Z(A_4,B_4)\prec Z(A_1,B_1)\prec Z(A_5,B_5).$

Therefore, the only discordant Z-number is $Z(A_4,B_4)$, which is placed in third place while according to our sorting order, it is in first place. We also observe that parameter $\beta$ plays an important role as depending on its value, the Z-number $Z(A_5,B_5)$ is placed in first place. As a conclusion we obtain that our procedure does not present any problem when choosing any discrete fuzzy number as a component of the corresponding discrete Z-number, a problem that Xian’s method has.

In Gong et al. (2020), suggest a ranking method based on transforming each discrete Z-number into a discrete fuzzy subset. Once its corresponding discrete fuzzy subset has been obtained for each discrete Z-number, the latter is ordered according to the method proposed in Basirzadeh et al. (2012). For the transformation of a discrete Z-number, the degree of fuzziness of the second component Kang et al. (2019) is used (for more details see Gong et al. (2020)). In the case of discrete Z-number $Z_4=(A_4,B_4)$, the method proposed by Gong cannot be applied. This is because only discrete Z-numbers whose second components have a unique value as their core are allowed. Note that component $B_4$ has as its core the interval [1, 2].

With the other discrete Z-numbers, we have obtained the following ordering when we take the parameter $\alpha =0$ (see Gong et al. (2020) for more details about the parameter $\alpha$).

• $Z(A_5,B_5)\prec Z(A_6,B_6)\prec Z(A_2,B_2)\prec Z(A_1,B_1)\prec Z(A_3,B_3).$

From the comparison of the two orderings, it can be deduced, from the point of view of linguistic interpretation, that it does not make much sense that $Z_1=(A_1,B_1) \prec Z_3=(A_3,B_3)$ is obtained by applying Gong’s method. Note that the second components are the same. Furthermore, if we compare the first components, we see that the linguistic assessment described by $A_1$ is intuitively more favorable than $A_3$. Finally, if we apply any of the total orders described in Riera et al. (2021) we have that $A_3 \prec A_1$, a fact that coincides with our initial intuition.

In summary, from the analysis of the compared methods it is clear that the ranking method proposed in this paper allows us to order discrete Z-numbers whose components do not necessarily have a unique element as their core, a problem that we have seen present in both cases. Furthermore, the obtained results are consistent with linguistic intuition.

7 Conclusions

Zadeh’s Z-numbers have positioned themselves as an interesting computational linguistic model that has been used in multiple applications, and in particular, in decision-making problems. In this field, it is essential to have adequate procedures, consistent with the linguistic model, that allow ordering experts’ preferences/opinions expressed as Z-valuations. Thus, in this paper we carry out for the first time a study on total orders in the set of discrete Z-numbers (and more specifically, for the subset of second component finite-valued discrete Z-numbers $\mathcal {A}_1^{L_n \times Y_m}$). The order is based on a transformation operator in the set of discrete fuzzy numbers that changes the uncertainty of the first component of a discrete Z-number (increasing the membership values outside of the core) according to the reliability/confidence of this first component provided by the second component of the discrete Z-number. Moreover, this order has a clear linguistic interpretation that is intended to align with human intuition when some of the components of the Z-numbers are equal and a suitable interval order is chosen. For this purpose, the cardinal of the set $\mathcal {A}_1^{L_n \times Y_m}$ has been computed and the construction of a relative order function (constructed from the total order defined in $\mathcal {A}_1^{L_n}$) compatible with this order has been proposed. To ease the computation of the order, we include several algorithms and specifically, an algorithm that allows to generate and order all the dfn of $\mathcal {A}_1^{L_n}$ based on this order. Also, we have provided an illustrative example that shows how the computation of the order works in a practical situation. Finally, a comparative section has been included with the intention of analyzing the validity of our method compared to others that exist in the literature.

In future work, we plan to define aggregation functions for discrete Z-numbers based on the total order we have established. Additionally, we intend to expand this order for the so-called mixed Z-numbers Z(A, B), in which A is not necessarily a dfn but it can be any fuzzy number.