An Outline of New Approach to Computation with Z-Numbers Based on the Concept of Lower Prevision

1. Introduction

Prof. Zadeh introduced the concept of the Z-number to formalize partially reliable information. Namely, Z-number denoted Z(A, B) is an ordered tuple of fuzzy numbers A and B which encode a linguistic (imprecise) description of a value of a random variable and the related reliability degree [1]: For example: “Air temperature will be about 30 °C, I am sure at about 80%”. The reason to use an imprecise evaluation is that the actual probability distribution of a random variable is not known. An extension principle for computation with continuous Z-numbers is introduced based on fuzzy and probabilistic arithmetic. As Zadeh noted in [1], a problem of computation with Z-numbers is quite easier to formulate than to solve.

Since the concept's introduction, a substantial body of work has been devoted to both theory and applications of Z-numbers [2,3,4,5,6]. From the computational point of view, the studies have developed along two main directions. One of the earliest practical schemes was proposed by Kang et al. [2], who converted a Z-number into a classical fuzzy number by using fuzzy expectation. This greatly simplifies subsequent calculations, since one can then work within the standard framework of fuzzy arithmetic. However, this convenience comes at a cost: once the original two-component structure is compressed into an equivalent fuzzy representation, part of the information carried by the Z-number is inevitably lost, especially in the reliability part [2].

A different line of research follows Zadeh’s original semantics more closely and attempts to perform arithmetic through hidden probability information. In this direction, Aliev et al. proposed the arithmetic of discrete Z-numbers [3] and later extended it to the continuous case [7]. These works provided one of the first systematic frameworks for basic operations such as addition, subtraction, multiplication, and division without collapsing the Z-number into an ordinary fuzzy number at the outset. The strength of this approach is that it preserves the internal semantics of Z-information more faithfully. Its weakness is the substantially higher computational burden [3,7]. An alternative approach was proposed by Pirmuhammadi et al. [8], who developed a parametric approach to Z-numbers by representing them as pairs (A, N) where A is triangular and N is a normal reliability function, and defined componentwise arithmetic operations. This formulation makes analytical treatment more convenient and was further used to support developments such as Z-number differentiation and Z-number initial value problems. However, it departs from hidden-distribution-based semantics and therefore follows a different interpretation of Z-number computation [8].

Further attempts to reduce the computational burden have also been proposed. Zhu et al. [9] introduced an approximate calculation method based on kernel density estimation and a utility measure of Z-numbers. However, this approach is primarily intended to reduce the number of Z-numbers to be processed in large datasets, rather than to simplify the internal structure of arithmetic operations. More recently, Li et al. [10] proposed an arithmetic framework for triangular Z-numbers with reduced calculation complexity by extending the triangular distribution used to represent hidden probabilistic information. Although this is an important step toward more efficient computation, the method remains tied to a particular distributional setting. Another important difficulty is that, in hidden-distribution-based approaches, recovering hidden probabilistic information may itself require solving optimization problems [3,7]. This is already implicit in frameworks that reconstruct admissible hidden distributions before applying arithmetic operations. The point becomes even clearer in later work. For example, Liu et al. [11] considered optimization models based on Maximum Shannon Entropy and a Genetic Algorithm for handling hidden probability distributions of discrete Z-numbers.

A review of the literature [12,13,14] shows that the main approaches to arithmetic with Z-numbers can be grouped into two directions: conversion-based methods [2] and direct arithmetic over discrete or continuous Z-numbers(e.g., [10,11]). The second direction also includes works on informativeness-preserving approaches based on specificity measures[5]. Conversion-based methods are attractive for their low computational cost, but they inevitably reduce the original Z-number information. Direct methods preserve the structure of Z-information more faithfully, albeit at the cost of considerably greater computational difficulty [15,16]. Hence, the main computational difficulty of Z-number arithmetic is clear: conversion-based approaches are simpler but lead to information loss, whereas hidden-distribution-based approaches preserve more of the original semantics but require more demanding computational procedures [2,3,11].

In a broader sense, a Z-number represents uncertainty with respect to the actual probability distribution of a random variable, which is often the case in real-world problems. Formally, this uncertainty is described by a set (mixture) of possible probability distributions G, which can be constructed from A and B. In turn, studies on uncertainty about probability were initiated long ago by Boole [17], Keynes [18] (who proposed describing this uncertainty as interval estimates), and other authors. The first fundamental works in this direction were proposed by Walley [19], Kuznetsov [20], and Weichselberger [21]. Walley introduced the term ‘imprecise probability,’ and one of the main ideas was using lower or upper previsions (which are non-additive measures). Particularly, lower (or upper) previsions can be constructed as the lower (upper) envelope of a mixture of distributions. In [15,16,22], the authors consider the relation between Z-numbers and non-additive measures.

A substantial part of literature, especially approaches that remain close to Zadeh’s original semantics, relies on operations over hidden or induced probability distributions in sets G, which leads to high computational complexity. However, according to the principle of the imprecise probability methods, a set of probability distributions can be described by a lower prevision (or upper prevision) w.l.o.g. [19]. In this work, we propose an outline of a new approach to computation with Z-numbers relying on this principle. Assume that given two Z-numbers $Z_1,Z_2$ we need to compute a result of a binary operation $\circ$ (e.g., sum operation): $Z_{12}=Z_1\circ Z_2$. We propose to construct a lower prevision ${\underset{\_}{P}}_i$ for each set $G_i,i=\mathrm{1,2}$ and then to compute resulting lower prevision ${\underset{\_}{P}}_{12}$. Then, set of ${\underset{\_}{G}}_{12}$ can be computed based on ${\underset{\_}{P}}_{12}$. This approach reduces computational complexity as compared to the case when operation $\circ$over all probability distributions in sets $G_i,i=\mathrm{1,2}$ is conducted. The suggested procedure is intended to reduce the internal computational burden of Z-number arithmetic without collapsing the original two-component structure of Z-information. We provide an example comparing the proposed technique with two alternative approaches. The results show that the same accuracy is achieved with reduced computational complexity.

The paper is structured as follows. Necessary definitions are given in Section 2. In Section 3 we describe the proposed approach to operations of Z-numbers with reduced computational complexity. Numerical experiments used to illustrate the proposed approach are given in Section 4. Section 5 concludes.

2. Materials and Methods

2.1. Preliminaries

Definition 1 ([24]). A discrete fuzzy number. A fuzzy subset A of the real line R with membership function ${\mu }_A:R\to \left[\mathrm{0,1}\right]$ is a discrete fuzzy number if its support is finite, i.e., there exist $x_1,...,x_n\in R$ with$x_1<x_2<...<x_n$, such that $\mathrm{supp}\left(A\right)=\{x_1,...,x_n\}$ and there exist natural numbers $s,t$ with $1\le s\le t\le n$ satisfying the following conditions:

${\mu }_A\left(x_i\right)=1$for any natural number i with $s\le i\le t$${\mu }_A\left(x_i\right)\le {\mu }_A\left(x_j\right)$for each natural numbers i,j with $1\le i\le j\le s$${\mu }_A\left(x_i\right)\ge {\mu }_A\left(x_j\right)$for each natural numbers$i,j$with

Definition 2 ([25]). Random variables and probability distributions. A random variable, X, is a variable whose possible values x are numerical outcomes of a random phenomenon. Random variables are of two types: continuous and discrete. Consider a discrete random variable X with outcomes space $\mathit{X}=\left\{\hspace{0.17em}{x}_1,\dots ,x_n\right\}$. A probability of an outcome X = x_i, denoted $P\left(X=x_i\right)$ is defined in terms of a probability distribution. A function p is called a discrete probability distribution or a probability mass function if $P(X=x_i)=p\left(x_i\right)$, where $p\left(x_i\right)\in \left[\mathrm{0,1}\right]$ and ${\sum }_{i=1}^{n}p\left(x_i\right)=1$

Definition 3 ([26,27]). Probability measure of a discrete fuzzy number. Let A be a discrete fuzzy number and p be a probability distribution defined over $\mathit{X}=\{\hspace{0.17em}{x}_1,...,x_n\}$. A probability measure of A denoted P(A) is defined as

$$P\left(A\right)={\sum }_{i=1}^n{\mu }_A\left(x_i\right)p\left(x_i\right).$$

Often, in real-world problems, it is not possible to determine actual probability distribution p due to lack of information. Prof. Zadeh introduced the concept of Z-number to describe imprecise information on possible probability distributions p.

Definition 4. A discrete Z-number [1,3]. Given two discrete fuzzy numbers A and B such that supp A⊆R and supp B⊆[0,1], a discrete Z-number is considered as Z(A, B). Z-number Z(A, B) represents information on a value of X in the form:

$$“\mathrm{probability} \mathrm{measure} P\left(A\right)={\sum }_{i=1}^n{\mu }_A\left(x_i\right)p\left(x_i\right)\mathrm{is}B″$$

As information on P(A) is imprecise, one considers a set of probability distributions of X, where an actual probability distribution for X is unknown. Denote this set of probability distributions as $G$:

$$G=\left\{p:{\sum }_{i=1}^n{\mu }_A\left(x_i\right)p\left(x_i\right)=b∊suppB\right\} ,$$

where $supp B$ is a support of B. Thus, values $b∊$ $supp B$ induce set G.

In existing literature, various approaches are used to describe a set $G$ of possible probability distributions p. If for all $x_i∊\mathit{X}$ lower (or upper) bounds $\underset{\_}{p}\left(x_i\right)$ for $p\left(x_i\right)$ are known, then one can define $G=\{p:p\left(x_i\right) \le \underset{\_}{p}\left(x_i\right),{\forall x}_i∊\mathit{X}\}$. Denote P a probability measure defined over $\sigma$-algebra $\mathcal{X}$ of crisp subsets of X. Denote $\mathcal{C}$ a set of probability measures P induced by G (in existing literature such set is referred to as credal set[23]). The following definition applies(see, for example, [19]):

Definition 5.  

A lower envelope [19]. A lower envelope is a function $\underset{\_}{P}:\mathcal{X}\to \left[\mathrm{0,1}\right]$

which is defined as follows:

$$\underset{\_}{P}\left(H\right)=\underset{P\in \mathcal{C}}{min}P\left(H\right),H\subset \mathit{X}$$

Thus, a lower envelope is determined as a minimal probability of $H\subset \mathit{X}$. The dual concept, upper envelope is defined as $\overline{P}\left(H\right)=1- \underset{\_}{P}\left(H^c\right)$, where $H^c$ is the complement of $H$. Thus, $\underset{\_}{P}\left(H\right)\le P\left(H\right)\le \overline{P}\left(H\right),$ $\forall H\subset \mathit{X},P\in \mathcal{C}$. A lower envelope is a kind of lower probability. The latter is a special case of lower prevision. These concepts are defined for the concept of a gamble[19]. Formally, a gamble is a bounded mapping from a set of values of a random variable to set of real numbers: $f:\mathit{X}\to R$

Definition 6.  

A lower prevision [19]. A lower prevision is a function $\underset{\_}{P}$

satisfying for all$f,g∊\mathcal{F}$, where$\mathcal{F}$a linear space of all gambles

$$\underset{\_}{P}\left(f\right)\ge inf\{f\left(x\right):x∊\mathit{X}\},\mathrm{br}\mathrm{to}\mathrm{break}\underset{\_}{P}\left(cf\right)=\underset{\_}{cP}\left(f\right), c∊R^{+},\mathrm{br}\mathrm{to}\mathrm{break}\underset{\_}{P}\left(f+g\right)\ge \underset{\_}{P}\left(f\right)+\underset{\_}{P}\left(g\right).$$

A lower probability is a special case of lower prevision obtained when $\mathcal{F}$ is a set of all indicator functions defined over X. Thus, if $\mathcal{F}$ is a set of membership functions (when a membership function of a fuzzy set is considered as a gamble), one obtains a lower prevision as a more general measure. As lower probability can be constructed as lower envelope of probability measures pf crisp sets (Def. 5), a lower prevision can be obtained in form of a lower envelope of probability measures of fuzzy events $A$:

$$\underset{\_}{P}\left(A\right)=\underset{P\in \mathcal{C}}{min}P\left(A\right),A\subset \mathit{X}$$

(1)

A one-to-one correspondence between coherent lower previsions and nonempty closed convex sets of probability measures exists [19, Theorem 3.6.1]. We rely on this result to propose a new approach to computations with Z-numbers by using Defs. 4-6 and Formula (1).

2.2. General Framework of the Approach

Let two Z-numbers $Z_1(A_1,B_1),Z_2(A_2,B_2)$ be given that describe partially reliable information on values of independent random variables random variables $X_1$ and $X_2$. Assume we need to arrive at imprecise and partially reliable information about of a value of random variable $X_{12}$ as a result of a basic arithmetic operation $\circ \in \{+,-,\cdot ,/\}$ over values of $X_1$ and $X_2$:

$$X_{12}=X_1\circ X_2.$$

Thus, we consider a problem of computation of a result $Z_{12}(A_{12},B_{12})$ of a basic arithmetic operation $\circ \in \{+,-,\cdot ,/\}$ over two Z-numbers $Z_1(A_1,B_1),Z_2(A_2,B_2)$:

$$Z_{12}=Z_1\circ Z_2.$$

Solving of the considered problem consists of the following pillars:

• Computation of A part of the result. $A_{12}$

is a fuzzy number computed on the basis of fuzzy numbers $A_1,A_2$ by using fuzzy arithmetic $A_{12}=A_1\circ A_2$[24].

• Construction of sets of $G_l,l=\mathrm{1,2}$

by extracting of probability distributions for Z-numbers $Z_l=\left(A_l,B_l\right),l=\mathrm{1,2}$. This implies that for each $b\in supp{ B}_l$ we need to find probability distribution $p_i$ satisfying:

$$b={\sum }_{i=1}^n{\mu }_{A_l}\left(x_i\right)p\left(x_i\right).$$

Thus, a set of values $b_i$ induces a set of probability distributions $p$

• Construction of set $G_{12}$

on the basis of sets $G_l,l=\mathrm{1,2}$. A set $G_{12}$ is a set of convolutions $p_{12}$ induced by sets $G_l,l=\mathrm{1,2}$: $G_{12}=\left\{r=p\circ q:p\in G_1,q\in G_2\right\}$. It is well-known that convolution $r=p\circ q,$is determined as follows [25]:

$$r\left(x_{12}\right)={\sum }_{x=x_1*x_2}p\left(x_1\right)q\left(x_2\right),$$

In this study, independence is assumed as a baseline case. Dependence-aware extensions may be considered in future research.

• Computation of B part of the result. $B_{12}$

is a fuzzy number computed on the basis of fuzzy number $A_{12}$ and probability distributions $r\in G_{12}$. This implies that each $r\in G_{12}$ induces $b_{12}$:

$$b_{12}={\sum }_{i=1}^n{\mu }_{A_{12}}\left(x_{12i}\right) r\left(x_{12i}\right).$$

In order to reduce computational complexity of operations with Z-numbers, we propose not to deal directly with sets $G_l,l=\mathrm{1,2}$. Instead, we propose to use the lower previsions as lower envelopes of the corresponding sets ${\mathcal{C}}_l,l=\mathrm{1,2}$ (recall that $\mathcal{C}$ is a set of probability measures $P$ induced by set $G$ of probability distributions).

Below we describe the main stages of the proposed approach.

Stage 1. Computation of A part${\mathit{A}}_{12}$. At this stage, a fuzzy number $A_{12}$ is computed based on fuzzy numbers $A_l,l=\mathrm{1,2}$.

To make arithmetic operations computable, we replace each first component by a finite discrete approximation:

$$A_1^d=\left\{\right(x_i^{\left(1\right)},{\mu }_i^{\left(1\right)}){\}}_{i=1}^{n_1},{ A}_2^d=\{(x_j^{\left(2\right)},{\mu }_j^{\left(2\right)}){\}}_{j=1}^{n_2}.$$

Here $x_i^{\left(1\right)}=x_{1i}$ and $x_j^{\left(2\right)}=x_{2j}$ are discretization nodes, while ${\mu }_i^{\left(1\right)}$and ${\mu }_j^{\left(2\right)}$are the corresponding membership values inherited from the original fuzzy numbers. Then, $A_{12}=A_1\circ A_2$ can be computed.

Stage 2. Extraction of probability distributions for Z-numbers$Z_l=\left(A_l,B_l\right),l=\mathrm{1,2}$ and construction of lower previsions.

For Z-numbers $Z_l=\left(A_l,B_l\right),l=\mathrm{1,2}$, the corresponding sets $G_1=\left\{p\right\}, G_2=\left\{q\right\}$ of probability distributions are constructed on the basis of $\left(A_l,B_l\right),l=\mathrm{1,2}$. Thus, it is induced latent probability distributions:

$$p=(p_1,\dots ,p_{n_1}),q=(q_1,\dots ,q_{n_2}),$$

subject to

$$p_i\ge 0,{\displaystyle \sum _{i=1}^{n_1}}{p}_i=1,$$

$$q_j\ge 0,{\displaystyle \sum _{j=1}^{n_2}}{q}_j=1.$$

Here $p_i=p\left(x_i^{\left(1\right)}\right)$ and $q_j=q\left(x_j^{\left(2\right)}\right)$.

These distributions represent possible latent probability patterns on the discretized supports of $A_1$and $A_2$. To keep the probabilistic layer aligned with the first components, the compatibility conditions are imposed[1]:

$${\displaystyle \sum _{i=1}^{n_1}}{x}_i^{\left(1\right)}{p}_i=C_1,{\displaystyle \sum _{j=1}^{n_2}}{x}_j^{\left(2\right)}{q}_j=C_2.$$

The constants $C_1$and $C_2$may be chosen either as the centroids of the original triangular fuzzy numbers

$$C_1={\displaystyle \frac{l_1+m_1+r_1}{3}},C_2={\displaystyle \frac{l_2+m_2+r_2}{3}},$$

or as the centroids of their discrete approximations:

$$C_1={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n_1}}{x}_i^{\left(1\right)}{\mu }_i^{\left(1\right)}}{{\displaystyle \sum _{i=1}^{n_1}}{\mu }_i^{\left(1\right)}}},C_2={\displaystyle \frac{{\displaystyle \sum _{j=1}^{n_2}}{x}_j^{\left(2\right)}{\mu }_j^{\left(2\right)}}{{\displaystyle \sum _{j=1}^{n_2}}{\mu }_j^{\left(2\right)}}}.$$

For each latent distribution, we define the induced probability measures of fuzzy events:

$$P_p\left(A_1\right)={\displaystyle \sum _{i=1}^{n_1}}{\mu }_i^{\left(1\right)}{p}_i,P_q\left(A_2\right)={\displaystyle \sum _{j=1}^{n_2}}{\mu }_j^{\left(2\right)}{q}_j.$$

These quantities measure how strongly the latent distributions are concentrated in the high-valued membership regions of $A_1$and $A_2$.

Now recall that values of $P_p\left(A_1\right)$ and${ P}_q\left(A_2\right)$ are restricted by $\alpha$-cuts of the second components of $Z_i=\left(A_i,B_i\right),i=\mathrm{1,2}$:

$$[B_1{]}^{\alpha }=[L_1\left(\alpha \right),U_1\left(\alpha \right)],[B_2{]}^{\alpha }=\left[L_2\right(\alpha ),U_2(\alpha \left)\right].$$

This implies that at level $\alpha$, these intervals define the following sets:

$$K_1^{\alpha }=\left\{p:\hspace{0.25em} p_i\ge 0,\hspace{0.25em} {\displaystyle \sum _{i=1}^{n_1}}{p}_i=1,\hspace{0.25em} {\displaystyle \sum _{i=1}^{n_1}}{x}_i^{\left(1\right)}{p}_i=C_1,\hspace{0.25em} L_1\left(\alpha \right)\le P_p\left(A_1\right)={\displaystyle \sum _{i=1}^{n_1}}{\mu }_i^{\left(1\right)}{p}_i\le U_1\left(\alpha \right)\right\},$$

$$K_2^{\alpha }=\left\{q:\hspace{0.25em} q_j\ge 0,\hspace{0.25em} {\displaystyle \sum _{j=1}^{n_2}}{q}_j=1,\hspace{0.25em} {\displaystyle \sum _{j=1}^{n_2}}{x}_j^{\left(2\right)}{q}_j=C_2,\hspace{0.25em} L_2\left(\alpha \right)\le P_q\left(A_2\right)={\displaystyle \sum _{j=1}^{n_2}}{\mu }_j^{\left(2\right)}{q}_j\le U_2\left(\alpha \right)\right\}.$$

Note that $K_i^{\alpha }$ is $\alpha$-cut of $G_i$, $i=\mathrm{1,2}$. For each set K, a lower prevision $\underset{\_}{P}\left(A\right)$ is computed based on formula (1). For example, one has for $K_1^{\alpha }$:

$$\underset{\_}{P}\left(A_1\right)={min}_{pϵK_1^{\alpha }}\left(P_p\left(A_1\right)\right).$$

Stage 3. Construction of lower prevision of resulting Z-number based on lower previsions for Z-numbers$Z_i=\left(A_i,B_i\right),i=\mathrm{1,2}$

Given lower previsions $\underset{\_}{P}\left(A_i\right)$ obtained for $Z_i=\left(A_i,B_i\right),i=\mathrm{1,2}$, the lower prevision of $A_{12}$ (A part of Z-number $Z_{12}$), $\underset{\_}{P}\left(A_{12}\right)$, is computed. For each pair of nodes $\left(x_i^{\left(1\right)}\right|x_j^{\left(2\right)})$, we compute the pairwise outcome

$$z_{ij}=x_i^{\left(1\right)}\circ x_j^{\left(2\right)}.$$

All distinct values $z_{ij}$ can be considered as set $\{z_s{\}}_{s=1}^N$ – set of the discrete support of the first component of the result. These points $z_s$ are all possible outcomes obtained by applying the arithmetic operation to pairs of discretization nodes.

For any admissible pair

$$p\in K_1^{\alpha },q\in K_2^{\alpha },$$

the latent distribution of the result is defined by

$$r\left(z_s\right)=\sum _{z_{ij}=z_s}{p}_i{q}_j.$$

So $r\left(z_s\right)$ is the probability assigned to the result point $z_s$ by the latent distributions $p$ and $q$. $p_i{q}_j$corresponds to the standard independent combination of the two latent distributions.

Let $A_{12}=A_1\circ A_2$ be the first component of the result and $A_{12}^d=\left\{\right(z_s,{\mu }_s^{\left(12\right)}){\}}_{s=1}^N$ be its discretized representation. Then the inherited reliability of the result is defined by

$$P_{p,q}^{(°)}\left(A_{12}\right)={\displaystyle \sum _{s=1}^N}{\mu }_s^{\left(12\right)}\hspace{0.17em}r\left(z_s\right).$$

This formula has a simple meaning: each point $z_s$of the resulting support has two attributes — its latent probability $r\left(z_s\right)$and its membership value ${\mu }_s^{\left(12\right)}$. The inherited reliability is just the weighted average of these membership values with respect to the latent distribution of the result.

The same quantity can be written directly in terms of the original node pairs. Indeed,

$$P_{p,q}^{(°)}\left(A_{12}\right)={\displaystyle \sum _{i=1}^{n_1}}{\displaystyle \sum _{j=1}^{n_2}}{\mu }_{A_{12}}(x_i^{\left(1\right)}\circ x_j^{\left(2\right)})\hspace{0.17em}{p}_i{q}_j.$$

This leads naturally to the weight matrix

$$M^{(°)}=\left({\mu }_{ij}^{(°)}\right)\in {\mathbb{R}}^{n_1\times n_2},{\mu }_{ij}^{(°)}={\mu }_{A_{12}}(x_i^{\left(1\right)}\circ x_j^{\left(2\right)}).$$

The entries of $M^{(°)}$ can be called weights because they tell us how strongly each pairwise outcome agrees with the first component of the result. Each of them is simply the membership value of the pairwise outcome $x_i^{\left(1\right)}\circ x_j^{\left(2\right)}$in the fuzzy number $A_{12}$. With this notation,

$$P_{p,q}^{(°)}\left(A_{12}\right)=p^T{M}^{(°)}q.$$

So the whole mechanism is:

$$(p,q)\hspace{0.25em} ⟶\hspace{0.25em} r\left(z_s\right)\hspace{0.25em} ⟶\hspace{0.25em} P_{p,q}^{(°)}\left(A_{12}\right),$$

where $r\left(z_s\right)$ is the latent distribution of the result, and $M^{(°)}$ is the matrix that contains the membership grades of all pairwise outcomes, and $p=(p_1,\dots ,q_{n_1}{)}^T$ and $q=(q_1,\dots ,q_{n_2}{)}^T$ denote hidden probability distributions on the discretized supports of $A_1$and $A_2$, respectively.

Finally, the second component of the resulting Z-number is reconstructed level by level:

$$\left[B_{12}^{(°)}{]}^{\alpha }\right|\left[{\underset{‾}{b}}_{12}^{(°)}\left(\alpha \right),\hspace{0.25em} {\bar{b}}_{12}^{(°)}\left(\alpha \right)\right],$$

where

$${\underset{‾}{b}}_{12}^{\left(°\right)}\left(\alpha \right)=\underset{p\in K_1^{\alpha },\hspace{0.25em} q\in K_2^{\alpha }}{\mathit{min}}{p}^T{M}^{\left(°\right)}q$$

(2)

$${\bar{b}}_{12}^{(°)}\left(\alpha \right)=\underset{p\in K_1^{\alpha },\hspace{0.25em} q\in K_2^{\alpha }}{max}{p}^T{M}^{(°)}q$$

(3)

It can be easily shown that distributions p and q obtained as solution of problem (2) underly lower previsions:

$$\underset{\_}{P}\left(A_1\right)={min}_{pϵK_1^{\alpha }}\left(P_p\left(A_1\right)\right)=L_1\left(\alpha \right),$$

$$\underset{\_}{P}\left(A_2\right)={min}_{qϵK_2^{\alpha }}\left(P_q\left(A_2\right)\right)=L_2\left(\alpha \right).$$

In turn, ${\underset{‾}{b}}_{12}^{(°)}\left(\alpha \right)$ is a lower prevision $\underset{\_}{P_{p,q}^{(°)}}\left(A_{12}\right)$. Thus, lower previsions $\underset{\_}{P}\left(A_i\right)$ obtained for $Z_i=\left(A_i,B_i\right),i=\mathrm{1,2}$, induce the lower prevision of $A_{12}$ (A part of Z-number $Z_{12}$), $\underset{\_}{P}\left(A_{12}\right)$. Analogously, upper probabilities $\overline{P}\left(A_i\right)$ induce upper probability $\overline{P}\left(A_{12}\right)$. Based on $\underset{\_}{P}\left(A_{12}\right)$ and $\overline{P}\left(A_{12}\right)$, the corresponding credal set $K_{12}^{\alpha }$, and, as a result, $G_{12}$ set are constructed.

Stage 4. Computation of B part of Z-number${\mathit{Z}}_{12}$ .

$B_{12}$ (B part of Z-number $Z_{12}$) is computed by using $A_{12}$ and $G_{12}$ according to Definition 4.Thus, Z-number $Z_{12}^{(°)}=(A_{12},B_{12}^{(°)})$is computed, where the first component comes from the arithmetic operation applied to $A_1$and $A_2$, while the second one is reconstructed from all inherited reliability values generated by admissible latent distributions.

This framework allows one to reduce computational complexity operations on Z-numbers. Indeed, in traditional approaches $G_{12}$ is computed on directly based on $G_i$, $,i=\mathrm{1,2}$: $G_{12}$ is constructed as the set of convolutions $p_{12}=p_1\circ p_2$ obtained from all the possible combinations of $p_i∊G_i$, $i=\mathrm{1,2}.$ In general, if one considers at least 3 distributions in each set $G_i$, $i=\mathrm{1,2}$, then he has to deal with 3x3=9 combinations to construct convolutions. In contrast, it is proposed to compute one lower prevision $\underset{\_}{P}\left(A_{12}\right)$ based on $\underset{\_}{P}\left(A_i\right)$ – one combination of two lower previsions only vs 9 or more combinations. Deriving of $\underset{\_}{P}\left(A_i\right)$ from $G_i$, $i=\mathrm{1,2}$ and construction of $G_{12}$ based on $\underset{\_}{P}\left(A_{12}\right)$ are not computationally complex problems.

3. Results

Let's consider numerical experiments to evaluate the proposed approach. For example, we can explore cases involving Z-numbers with triangular fuzzy number-based A and B parts. The choice of triangular fuzzy numbers to express parts A and B is due to their properties that combine a balance between computational efficiency and the ability to represent uncertainty, being intuitive and easier to interpret, aiding decision-makers in understanding the results, and easiness to represent diverse types of uncertainty criteria across different spheres [28,29,30,31,32].

3.1. Addition of Two Z-Numbers

We consider two Z-numbers $Z_1=(A_1,B_1),Z_2=(A_2,B_2),$ where the first components are triangular fuzzy numbers and can be presented in the form below:

$$A_1=(m_1-h_1,\hspace{0.25em} m_1,\hspace{0.25em} m_1+h_1),A_2=(m_2-h_2,\hspace{0.25em} m_2,\hspace{0.25em} m_2+h_2),$$

and the second components are triangular fuzzy numbers

$$B_1=(u_1,v_1,w_1),B_2=(u_2,v_2,w_2).$$

We apply 3-point discretization of A with only the three vertices, 3-α levels for B - $\alpha \in \{0,\hspace{0.25em} 0.5,\hspace{0.25em} 1\}.$ The 3-point scheme is used as the minimal analytically tractable discretization that preserves the left endpoint, modal value, and right endpoint of a triangular fuzzy number. Higher-order discretization can provide more accurate approximations but lead to more complex expressions. 1. Discretization of the first components

For each $A_i$, we take the three points

$$x_1^{\left(i\right)}=m_i-h_i,x_2^{\left(i\right)}=m_i,x_3^{\left(i\right)}=m_i+h_i,$$

with membership values – (0,1,0)

So, the discrete approximation is

$$A_i^d=\left\{\right(m_i-h_i,0),\hspace{0.25em} (m_i,1),\hspace{0.25em} (m_i+h_i,0\left)\right\}.$$

5. Weight matrix for the 3-point scheme

For the standard simple case $h_1=h_2$, the weight matrix is

$$M=\left(\begin{array}{ccc}0& {\displaystyle \frac{1}{2}}& 1\\ {\displaystyle \frac{1}{2}}& 1& {\displaystyle \frac{1}{2}}\\ 1& {\displaystyle \frac{1}{2}}& 0\end{array}\right).$$

Then the reliability of the sum A₁₂ (inherited) is

$$m_{12}(p,q)=p^{T}Mq.$$

Substituting $p=(a,1-2a,a),q=(d,1-2d,d),$ we obtain the expression –

$$m_{12}(p,q)=1-a-d+2ad$$

This is the key analytical formula for the 3-point scheme.

6. Three levels of $B$

For a triangular fuzzy number

$$B_i=(u_i,v_i,w_i),$$

the three chosen $\alpha$-cuts are:

• at $\alpha =0$

: - $[B_i{]}^0=[u_i,w_i],$at $\alpha =0.5$: $\left[B_i{]}^{0.5}\right|\left[{\displaystyle \frac{u_i+v_i}{2}}\right|\hspace{0.25em} {\displaystyle \frac{v_i+w_i}{2}}],$at $\alpha =1$: $[B_i{]}^1=[v_i,v_i].$Let $[B_1{]}^{\alpha }=[L_1\left(\alpha \right),U_1\left(\alpha \right)],[B_2{]}^{\alpha }=\left[L_2\right(\alpha ),U_2(\alpha \left)\right].$Because

$$m_{A_1}\left(p\right)=1-2a,m_{A_2}\left(q\right)=1-2d,$$

the constraints at α-level become

$$L_1\left(\alpha \right)\le 1-2a\le U_1\left(\alpha \right),L_2\left(\alpha \right)\le 1-2d\le U_2\left(\alpha \right).$$

Hence

$${\displaystyle \frac{1-U_1\left(\alpha \right)}{2}}\le a\le {\displaystyle \frac{1-L_1\left(\alpha \right)}{2}},{\displaystyle \frac{1-U_2\left(\alpha \right)}{2}}\le d\le {\displaystyle \frac{1-L_2\left(\alpha \right)}{2}}.$$

7. Analytical formula for each level of $B_{12}$

Since - $m_{12}\left(p,q\right)=1-a-d+2ad$is decreasing in both a and d on the admissible range, we get:

• the minimum at the upper bounds of $a$

and $d$,

• the maximum at the lower bounds of $a$

and $d$.

After substitution, we obtain

$$[B_{12}{]}^{\alpha }=\left[{\displaystyle \frac{1+L_1\left(\alpha \right)L_2\left(\alpha \right)}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+U_1\left(\alpha \right)U_2\left(\alpha \right)}{2}}],\alpha \in \left\{\mathrm{0,0.5,1}\right\}.$$

8. Formulas for the three levels

In α=0 the L₁(0)=u₁, U₁ (0)=w₁, L₂ (0)=u₂, U₂ (0)=w₂ we get $B_{12}^0= \left[{\displaystyle \frac{1+u_1{u}_2}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+w_1{w}_2}{2}}]$in α=0.5 the $L_1\left(0.5\right)={\displaystyle \frac{u_1+v_1}{2}},U_1\left(0.5\right)={\displaystyle \frac{v_1+w_1}{2}},L_2\left(0.5\right)={\displaystyle \frac{u_2+v_2}{2}},U_2\left(0.5\right)={\displaystyle \frac{v_2+w_2}{2}},$we obtain $B_{12}^{0.5}=\left[{\displaystyle \frac{1+\left(\frac{u_1+v_1}{2}\right)\left(\frac{u_2+v_2}{2}\right)}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+\left(\frac{v_1+w_1}{2}\right)\left(\frac{v_2+w_2}{2}\right)}{2}}]$or $B_{12}^{0.5}=\left[{\displaystyle \frac{1}{2}}+{\displaystyle \frac{\left(u_1+v_1\left)\right(u_2+v_2\right)}{8}},\hspace{0.25em} {\displaystyle \frac{1}{2}}+{\displaystyle \frac{\left(v_1+w_1\left)\right(v_2+w_2\right)}{8}}\right]$In α=1 the L₁(1)=U₁(1)=v₁, L₂ (1)=U₂ (1)=v₂ and

$$B_{12}^1=\left[{\displaystyle \frac{1+v_1{v}_2}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+v_1{v}_2}{2}}]$$

The result is a single point.

9. Final formula for the sum of two Z-numbers

Therefore, for Z₁+Z₂=Z₁₂(A₁₂,B₁₂) where

A₁₂=(m₁+m₂-(h₁+h₂), m₁+m₂, m₁+m₂+(h₁+h₂))

and the second component $B_{12}$is defined by the three cuts

$${\left[B_{12}{]}^0,[B_{12}{]}^{0.5},[B_{12}\right]}^1$$

given above.

3.2. Numerical Example

Let Z₁=((1,2,3),(0.7,0.8,0.9)),Z₂=((7,8,9),(0.4,0.5,0.6))

Here we have for A₁ = (1,2,3) m₁ =2, h₁ = 1, for A₂ = (7,8,9) m₂ =8, h₂ = 1

So A₁₂ = (1+7,2+8,3+9) = (8,10,12)

For B we have

For α=0 - $B_1^0=\left[\mathrm{0.7,0.9}\right],B_2^0=\left[\mathrm{0.4,0.6}\right].$So $B_{12}^0=\left[{\displaystyle \frac{1+0.7\cdot 0.4}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+0.9\cdot 0.6}{2}}]$ and after computation - 0.7⋅0.4=0.28,0.9⋅0.6=0.54.

$$B_{12}^0=\left[{\displaystyle \frac{1.28}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1.54}{2}}]=[0.64,\hspace{0.25em} 0.77].$$

For α=0.5

$$B_1^{0.5}=\left[{\displaystyle \frac{0.7+0.8}{2}}\right|{\displaystyle \frac{0.8+0.9}{2}}]=\left[\mathrm{0.75,0.85}\right],$$

$$B_2^{0.5}=\left[{\displaystyle \frac{0.4+0.5}{2}}\right|{\displaystyle \frac{0.5+0.6}{2}}]=\left[\mathrm{0.45,0.55}\right].$$

So, $B_{12}^{0.5}=\left[{\displaystyle \frac{1+0.75\cdot 0.45}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+0.85\cdot 0.55}{2}}]$ and after computation

$$B_{12}^{0.5}=\left[{\displaystyle \frac{1.3375}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1.4675}{2}}]=[0.66875,\hspace{0.25em} 0.73375]$$

For α=1 $B_1^1=\left[\mathrm{0.8,0.8}\right],B_2^1=\left[\mathrm{0.5,0.5}\right].$So, $B_{12}^1=\left[{\displaystyle \frac{1+0.8\cdot 0.5}{2}}\right|\hspace{0.25em} {\displaystyle \frac{1+0.8\cdot 0.5}{2}}]$ and we obtain $B_{12}^1=\left[\mathrm{0.7,0.7}\right]$So the sum is Z₁+Z₂ = ((8,10,12), B₁₂) and B₁₂ is described by $B_{12}^0=[0.64,\hspace{0.25em} 0.77]$,

$$B_{12}^{0.5}=[0.66875,\hspace{0.25em} 0.73375]$$

And in form of a triangular fuzzy number approximation B₁₂ = (0.64, 0.7, 0.77)

Resulting Z₁₂=(8,10,12) (0.64, 0.7, 0.77)

3.3. Comparison of Results and Complexity Analysis

Let us conduct comparison of the proposed approach with the approaches developed in works [3] and [10]. In both works, computation of a sum $Z_{12}=Z_1+Z_2$ of two triangular Z-numbers Z1=(A1, B1) and Z2=(A2, B2) is considered, where the components are TFNs:

In [3], normal pdfs are used in this example. The result is $Z_{12\left[3\right]}=(\left(\mathrm{8,10,12}\right),(0.62, 0.71, 0.79\left)\right)$. In [10] triangular distributions are used to facilitate computations. The obtained result is $Z_{12\left[10\right]}=(\left(\mathrm{8,10,12}\right),(0.674, 0.692, 0.712\left)\right)$. The result obtained by the proposed approach is: $Z_{12}=(\left(\mathrm{8,10,12}\right),(0.64, 0.7, 0.77\left)\right)$. As one can see, A parts coincide, and B parts differ slightly.

Mean absolute percentage error (MAPE) between B part of the result of the proposed approach $Z_{12}$ and that of the approach in [3] is $\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}\left(B_{12\left[3\right]},B_{12}\right)=2.6\%$. The MAPE w.r.t. the results obtained by the approach proposed in [10] is $\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}\left(B_{12\left[8\right]},B_{12}\right)=4.7\%$. Thus, the proposed approach provides the same accuracy as the approaches proposed in [3] and [10]. At the same time, in the proposed approach, it is not required to assume a type of probability distribution.

For Python code realizing approaches from [3] and [10] the execution time is 0.0560225050 and 0.049157 seconds respectively. For suggested approach Execution time is 0.0000384460.

The computational complexity of the approach proposed in [10] is O(Q+r), where Q denotes the cost of the numerical procedure used to construct the resulting reliability component and r is the number of alpha levels. For a fixed triangular input and fixed numerical settings, this becomes O(1).

The computational complexity of the approach proposed in [3] is O(CLP(n) + m2n2), where n is the number of discretization points of A, m is the number of discretization points of B, and CLP(n) is the cost of solving one LP problem.

The computational complexity of the proposed analytical 3-point/3-alpha method is O(r). Since r=3 in the considered implementation, the complexity is O(1)

The method proposed in [10] has a virtually constant computational structure; its actual execution time is determined by the numerical procedures used to construct the B component of the resulting Z-number. In the case of the approach indicated in [3], the number of convolution operations with 3-point discretization is small; however, the reconstruction of latent probability distributions requires solving LP problems, and it is the costs associated with the LP solver that dominate when the input data size is small. Therefore, the computation times for the methods [3,10] in this small example are similar. In contrast, the proposed analytical 3-point/3-alpha scheme uses neither numerical integration nor LP reconstruction of probability distributions but calculates the B12 component directly using formulas. This explains the significantly shorter execution time.

4. Discussion & Conclusions

In a nutshell, a Z-number represents uncertainty w.r.t. actual probability distribution of a random variable, which is often the case in real-world problems. Formally, this uncertainty is described by a set (mixture) of possible probability distributions. According to imprecise probability concept of Walley, the set of probability distributions may be captured by a lower prevision measure. In this work we propose an outline of a new approach to computation with Z-numbers relying on this principle. This allows to reduce complexity of operations over Z-numbers without imposing restrictions on a type of probability distribution. We provide an example for computation of a sum of two Z-numbers. The results of comparison of the proposed approach with two alternative approaches prove validity of the former.

In the proposed approach, the induced reliability measure is directly related to the central probability mass of the latent distribution. A weight matrix plays a key role in this mechanism. Each element of the matrix reflects the degree of membership of the pairwise arithmetic result in the resulting fuzzy component. In other words, the matrix estimates how well each possible combination of discretized input values matches the fuzzy form of the result. The inherited reliability of the result is then obtained as a weighted average of these compatibility values with respect to the latent distributions of the input Z-numbers. A useful feature of the proposed approach is that the α-cuts of the resulting B12 reliability component can be written analytically. For two triangular reliability components, the lower and upper bounds of the result are obtained directly from the corresponding α-level bounds of the original reliability components. This eliminates the need for repeated numerical optimization in the considered problem formulation. Thus, the method has a simple computational structure and can be useful as a base case for more general Z-number arithmetic.

Future research could extend the proposed framework in several directions. First, the method could be generalized to fuzzy numbers with membership functions of different shapes. Second, higher-order discretization schemes could be considered to improve approximation accuracy. Third, other arithmetic operations, such as subtraction, multiplication, and division, could be studied within the same latent distribution framework. Finally, dependencies between random variables could be incorporated to yield a more flexible reliability propagation model.

The triangular approximation of the resulting reliability component B12, considered as an example, is reconstructed from the selected α-levels and then approximately represented as a triangular fuzzy number. This is a practical and convenient representation. The proposed approach is valuable not because it replaces all existing methods of Z-number arithmetic, but because it provides a transparent analytical case. It demonstrates how the reliability component can be propagated through a latent distribution mechanism and how a weight matrix can link fuzzy arithmetic with reliability inheritance. This provides a basis for further extension to more general discretization schemes and more complex operations with Z-numbers. It provides a useful starting point for developing more general and computationally efficient methods for operations with Z-numbers.