The arithmetic of multidimensional Z-number

Abstract

The Z-number has become a new representation of evaluation information because of its superiority in describing reliability measure. Current studies focuses mainly on the unidimensional Z-number like Z = (about 25 min, usually). Studies on multidimensional cases have not been reported. However, because people often describe things from various aspects, using only one aspect of information to describe uncertain events fully is difficult. In this paper, we propose the concept of the multidimensional Z-number, such as ((about 20 miles, about 25 min), usually) to handle complex information. For this purpose, we first define the basic operations of the multidimensional Z-number. We then propose a feasible comparison method. The effectiveness of the proposed method is demonstrated by a series of numerical examples.

Keywords

Z-number multidimensional Z-number operations of probable events multi-criteria decision-making

1 Introduction

Fuzzy sets (FSs), as proposed by Zadeh [1], are considered useful tools for dealing with fuzzy information and solving multi-criteria decision-making (MCDM) problems [2, 3]. However, the single membership degree function cannot reflect real life information effectively. To this end, Atanassov [4] introduced intuitionistic fuzzy sets (IFSs) that consider membership degree, non-membership degree, and hesitation. To date, IFSs have been studied extensively, and several IFS extensions have been developed and applied to MCDM problems [5, 6]. To preserve the original decision information to a greater extent, Torra and Narukawa [7] proposed hesitant fuzzy sets (HFSs) that allow the membership degree of an element to be a set of several possible values in [0, 1]. The main purpose of HFSs is to model the uncertainty produced by human beings who experience doubt during the extraction of information [8, 9]. With the deepening research and increasingly widespread application of FSs, some key issues still need to be refined, such as overlooking the reliability of the information [10].

To improve and perfect classical FSs, the Z-number, a new fuzzy theoretic concept, was proposed by Zadeh [11]. A Z-number is an ordered pair of fuzzy numbers, Z = (A, B). The fuzzy number A is a fuzzy restriction, and fuzzy number B is a restriction on the probability measure of A. Considerable information is available regarding the form of the Z-number in daily decision making. For a more complete expression, the Z-number can also be expressed as “X is Z = (A, B)”, or as a form of Z-valuation, (X, A, B). The Z-number reflects both the fuzziness of variable X and its randomness and hence, is more consistent with people’s expression habits.

Given the considerable advantages of the Z-number in describing information, it has been studied extensively and applied widely in various fields in recent years. According to different research directions, the current investigations on the Z-number can be divided roughly into three aspects.

The first aspect is the research on special cases of the Z-number, including the fuzzy restriction of the Z-number in linguistic form. Pal, Banerjee, Dutta and Sarma [12] performed a comprehensive investigation of the Z-number approach to compute with words (CWW), which included an algorithm and some simulation experiments for CWW using a Z-number. Pirmuhammadi, Allahviranloo and Keshavarz [13] proposed the concept of the normal Z-number which assumed that the probability distribution in Z-number obeys the normal distribution, and discussed the parameter form and initial value problem of Z-number. Wang, Cao and Zhang [14] proposed the concept of a linguistic Z-number, defined its distance measure and Choquet integral, and then put forward a TODIM method on linguistic Z-number. Peng and Wang [15] developed a method for addressing multi-criteria group decision-making problems based on the normal coud model with Z-number. The aforementioned studies enriched the connotation of the Z-number and extended Z-number theory. Nevertheless, most of these studies have disregarded the underlying probability information implied by the Z-number, which implies a certain loss of information. The Z-number can also be regarded as a bridge between probability theory and fuzzy theory. Hence, ignoring the underlying probability information may prevent the maximization of the advantages of the Z-number.

The second aspect is expansion research based on converting the Z-number to other forms. Kang, Wei, Li and Deng [16] presented a process of transforming the Z-number to a classical fuzzy number according to the fuzzy expectation of the fuzzy set. The conversion scheme in [16] can significantly reduce the computational complexity of the Z-number, but will lead to loss of the original information. Based on the conversion approach in [16], Yaakob and Gegov [17] introduced a modified TOPSIS method to handle stock selection MCGDM problems on the Z-number. Ezadi and Allahviranloo [18] proposed a new method for ranking Z-numbers based on hyperbolic tangent function and convex combination. Aboutorab, Saberi, Rajabi, Hussain and Chang [19] presented an integrated approach combining Z-numbers with the Best Worst Method to deal with the reliability of information in the decision making process. These studies simplified the operation rules and ranking methods of the Z-number considerably and are more conducive to applying the Z-number in practical problems. However, to some extent, converting the Z-number to other forms may lose and distort original information. Thus, such conversion cannot reflect adequately the characteristics of the Z-number.

The third aspect is the research on the basic theory and application of the Z-number in the framework proposed by Zadeh [11]. Aliev, Alizadeh and Huseynov [20] suggested several operations of discrete Z-number according to the definition of convolution, and also proposed the ranking method of Z-number based on the Pareto optimality principle. Aliev, Huseynov and Zeinalova [21] offered several basic arithmetic operations for continuous Z-number based on linear programming. Sharghi, Jabbarova and Aliyeva [22] used the Z-number valued weighted average aggregation operator to solve an optimal port choice problem. Aliev, Pedrycz, Huseynov and Eyupoglu [23] recommended a new approach to studying approximate reasoning with Z-rules as per linear interpolation and applied it to job satisfaction evaluation and evaluation problems pertaining to the educational achievement of students. Yang and Wang [24] constructed the linear programming model based on the minimum variance principle, and proposed a MCDM support model based on the discrete Z-number. Shen and Wang [25] defined the comprehensive weighted distance measure of Z-number which simultaneously considered the randomness and fuzziness of Z-number, and extended classic VIKOR method to the Z-information environment. The above literature preserved and maximized the information contained in the Z-number. Nonetheless, considering the mathematical programming and variation problems involved, calculations of the Z-number remain complicated.

Theoretical research on the Z-number continues to be scarce. The said studies on the Z-number focused on the situation wherein fuzzy restriction A is unidimensional. It can only deal with information as (about 1h, usually). However, because of the complexity of the real world, using only one aspect of information to describe uncertain events fully is difficult. People often describe things from various aspects. For example, should your body temperature be around 39 degrees and you have a bad headache, you are likely to have a cold. Accordingly, if we use a unidimensional Z-number to express the situation, we may use two unidimensional Z-numbers (bad headache, likely) and (around 39 degrees, likely). However, an expression using two unidimensional Z-numbers is not very accurate. A patient likely to be diagnosed with a cold may need to have two symptoms (“the body temperature is around 39 degrees” and “bad headache”) rather than just one of them. In this case, we may use a simple expression ((around 39 degrees, bad headache), likely) to express the situation. As evident from the example above, effective expression of information as a Z-number is well worth investigating.

Under the general framework for computations over Z-numbers proposed by Zadeh [11], the existing algebraic operations of the Z-numbers are presented in combination with convolution operations. These operations represent a good solution in cases wherein the description object obtained by operating two Z-numbers still has meaning. For example, when two Z-numbers describe the walking time from the dormitory to the canteen and the walking time from the canteen to the teaching building separately, we can use the addition of Z-numbers in [20] to obtain a Z-number on the walking time from the dormitory to the teaching building. However, these algebraic operations are inapplicable in the case where the description objects of two Z-numbers do not have quantitative relations. This circumstance is because in these algebraic operations, the convolution operation amalgamates cases that have same results. Nevertheless, such merger is hardly reasonable when two description objects cannot compensate for each other. We cannot claim that these two cases “height 180 cm, weight 80 kg” and “height 170 cm, weight 90 kg” are the same. Therefore, studying the operations between two Z-numbers for which description objects do not have a quantitative relationship under the general framework proposed by Zadeh [11] is crucial.

In this paper, we developed the concept of the multidimensional Z-number to describe multidimensional information. We also defined the basic operations of multidimensional Z-numbers to integrate unidimensional Z-numbers into multidimensional Z-number. The main contributions of this paper are as follows.

This paper introduced the multidimensional Z-number as an extension of the Z-number. The multidimensional Z-number not only inherits the advantages of Z-number to describe qualitative information and characterize the reliability of information, it can also be used to represent evaluation information with multidimensional features.

Based on the general computation principle of Z-number, this paper defined the basic operations of the multidimensional Z-number. The operations defined in this paper consider the relation between possibility distribution and probability distribution as well as the dimension of information.

This paper proposed a ranking method based on the centroid method, which can be applied to compare multidimensional Z-numbers with the same or similar restriction vectors. This method can simplify the operation process without losing information and rank more than two fuzzy numbers simultaneously.

The rest of this paper is organized as follows. Section 2 briefly reviews several concepts of Z-number. Section 3 defines the multidimensional Z-number and related concepts. Section 4 discusses four arithmetic operations of the multidimensional Z-number. Section 5 describes the method of ranking multidimensional Z-numbers. Finally, Section 6 presents the conclusions.

2 Preliminaries

Zadeh [11] introduced the concept of a Z-number to characterize uncertain, inaccurate, and incomplete information with restriction and reliability measure. This section reviews several concepts of the Z-number.

Definition 1. ([11] Z-number) A Z-number is composed of a pair of ordered fuzzy numbers, (A, B), which is used to describe a real-valued uncertain variable, X. Fuzzy number A is a fuzzy restriction to the possible value of variable X, and fuzzy number B is a restriction on the probability measure of A. The expression of a standard Z-number is “X is (A, B)”, or “X is Z = (A, B)”. The complement of Z is denoted by Z^c and defined as Z^c = (A^c, 1 - B), where A^c is the complement of A, and 1B is the antonym of B where μ_{A
^c} (x) = 1 - μ_A (x).

To facilitate the use of Z-number in daily reasoning and decision-making, Zadeh [11] introduced the concept of Z-valuation, which is denoted as an ordered triple (X, A, B). A Z-valuation is equivalent to an assignment statement, “X is Z = (A, B)”. Unlike the Z-number, the underlying probability density, p_X, in Z-valuation is known.

Definition 2. ([11] Z⁺-numbers) A Z⁺-number consists of a fuzzy number, A, and a random number, R, and is represented as Z⁺ = (A, R). Similar to the Z-number, A is also a fuzzy restriction. R is the probability distribution of a random number that can also be regarded as the underlying probability density in Z-valuation. Equivalently, the concept of Z⁺-valuation can be obtained according to the Z-valuation and is expressed as (X, A, R) or (X, A, p_X).

Note that a Z⁺-number incorporates more information than a Z-number, and computation with Z⁺-numbers is critical to computation with Z-numbers. Their relationship can be reflected in the following equality, $Z (A, B) = Z^{+} (A, μ_{A} \cdot p_{X} 1 pt 1 pt is 1 pt 1 pt B) .$

Definition 3. ([20] probability distribution) Let X be a continuous random variable, and x be all possible values that X can take. A continuous probability distribution for X that belongs to [a, b] can be defined as $P (a \leq X \leq b) = \int_{a}^{b} p (x) dx,$ where p (x) ≥ 0 is a probability density function, and $\int_{- \infty}^{+ \infty} p (x) dx = 1$ .

Let X be a discrete random variable, and {x₁, x₂, …, x_n } be all possible values that X can take. A discrete probability distribution for X can be defined as P (X = x_i) = p (x_i), where p (x_i) ∈ [0, 1] and $\sum_{i = 1}^{n} p (x_{i}) = 1$ .

Definition 4. ([26] probability measure) The probability measure of continuous random variable X can be expressed as $b = \int_{R} μ_{A} (x) p_{X} (x) dx,$ where μ_A (x) is the membership function of x.

The probability measure of discrete random variable X can be described as $b = \sum_{i = 1}^{n} μ_{A} (x_{i}) p (x_{i}),$ where μ_A (x) is the membership function of x.

3 Multidimensional Z-number

3.1 Concept of multidimensional Z-number

Given the complexity of decision-making problems, events in real life always contain multiple dimensions. However, the information represented by unidimensional Z-number is limited. We require a more effective means to reflect the information of all dimensions. Therefore, this paper considered extending the fuzzy restriction A in the Z-number to a multidimensional restriction vector (A₁, A₂, …, A_n). Below, we provide the definition of multidimensional Z-number.

Definition 5. (Multidimensional Z-number) If X_i = (x_i1, x_i2 …, x_{im
_{k
_i}}) are random variables defined on sample space Ω and A_i ⊆ X_i, then (A₁, A₂ …, A_n) is called the n-dimension restriction vector, where Ω = { (x_1k₁, x_2k₂ …, x_{nk
_n}) |x_{ik
_i}∈ X_i, k_i = 1, 2 …, m_{k
_i}, i = 1, 2 …, n }!.

A multidimensional Z-number comprises multidimensional restriction vector, (A₁, A₂ …, A_n), and a fuzzy number, B, denoted as MZ = ((A₁, A₂ …, A_n), B). Let G = (A₁, A₂ …, A_n), a multidimensional Z-number can be expressed as MZ = (G, B). The complement of MZ is represented by MZ^c and defined as MZ^c = ((A₁, A₂ …, A_n) ^c, 1 - B), where (A₁, A₂, …, A_m) ^c is the complement of (A₁, A₂, …, A_m), and 1 - B is the antonym of B. Similar to Z-valuation, the multidimensional Z-valuation can be described as (X, (A₁, A₂ …, A_n), B) or (X, G, B). If n = 1, then the multidimensional Z-number MZ = (G, B) is reduced to a unidimensional Z-number. The unidimensional Z-number is the original Z-number proposed by Zadeh [11].

For example, in the case of a patient with body temperature of around 40 degrees and a very bad headache, the unidimensional Z-number cannot express fully the patient’s physical condition. Furthermore, we cannot judge what disease the patient has from just one aspect of body temperature or headache. It could be the flu, or the common cold. Hence, we express the information in the form of multidimensional Z-number, $\begin{matrix} (influenza, (temperature around 40^{\circ} C \\ and headache very bad), likely), \\ (common cold, (temperature above 40^{\circ} C \\ and headache very bad), very likely) . \end{matrix}$

Next, we proposed the definition of the probability distribution of a multidimensional Z-number.

Definition 6. (Probability density distribution) The probability density distribution of a multidimensional Z-number is expressed as p (x_1k₁, x_2k₂, …, x_{nk
_n}), where ∬ … ∫p (x_1k₁, …, x_{nk
_n}) dx_1k₁ … dx_{nk
_n} = 1 and p (x_1k₁, …, x_{nk
_n}) ≥ 0. This distribution is obtained by calculating p_{X
₁} (x_1k₁), p_{X
₂} (x_1k₂), …, p_{X
_n} (x_{nk
_n}), where p_{X
_i} (x_{ik
_i}) ≥ 0 and ∫p_{X
_i} (x_{ik
_i}) dx_{ik
_i} = 1, i = 1, 2, …, n. A probability distribution for X that belongs to the multidimensional restriction vector can be expressed as P { (x₁, …, x_m) ∈ G } = ∬_G … ∫p (x_1k₁, …, x_{nk
_n}) dx_1k₁ … dx_{nk
_n}, where G = (A₁, A₂ …, A_m).

The Z-number differs from previous fuzzy numbers because includes two parts: the fuzzy restriction and the reliability measure. The previous fuzzy numbers, such as intuitionistic fuzzy numbers, also includes two parts, but such parts are relatively independent and can be calculated separately. The two parts of Z-number are linked closely by the underlying probability distribution. If the two parts of Z-number are calculated separately, and loss of information may occur. Thus, existing operations of other fuzzy numbers cannot be used directly to handle Z-numbers.

In a complex decision environment, some problems occur that cannot be solved effectively by the algebraic operations proposed by [20]. For example, in the case of known Z-numbers, Z₁ = (A₁, B₁) and Z₂ = (A₂, B₂), we want to determine the reliability such that X satisfies A₁ or satisfies A₂. Therefore, this paper defined the basic operations of multidimensional Z-numbers by introducing the operations of probable events under the general framework in [11] to solve some special cases.

In the general framework by [11], the computation of the Z-number is divided into four steps. The first step is to compute the fuzzy restriction, A₁₂ = A₁ * A₂, of Z₁₂ = Z₁ * Z₂, where *∈ { +, -, ·, /}. The second step is to compute a underlying probability of Z₁₂. The third step is to compute the corresponding probability measure. The last step is to repeat the above three steps until the calculation completes the combination of all underlying probabilities to obtain the result Z₁₂. The operation steps of the multidimensional Z-number defined in this paper are the same as those of the Z-number. In view of this, this paper defined basic operations of restriction vectors, operations of probable events, and the operations of reliability measure of the multidimensional Z-number.

People are more used to expressing discrete information in their daily lives [27, 28], and hence, this paper focuses on the computation of discrete multidimensional Z-numbers. Furthermore, in some decision making problems, the criteria are often irrelevant [29, 30]. Hence, in the following definition, assume that the random variables are independent.

3.2 Basic operations of restriction vectors and operations of probable events

Let two multidimensional Z-numbers be MZ₁ = (G₁, B₁) and MZ₂ = (G₂, B₂) where G₁ = (A₁, A₂ …, A_m) and G₂ = (A_m+1, A_m+2 …, A_n). When n = m + 1 and m = 1, they are two unidimensional Z-numbers. According to the needs of different situations, we defined the basic operations of restriction vectors [31 –33] and operations of probable events [34 –36].

Definition 7. (Addition of restriction vectors) The addition of the multidimensional Z-number is applicable to the case wherein X satisfies G₁ or G₂. For restriction vectors G₁ and G₂, their addition (G₁, G₂) ₁₊₂ is defined as $\begin{matrix} {(G_{1}, G_{2})}_{1 + 2} = G_{1} + G_{2} = {g_{3} | g_{3} = {(g_{1}, g_{2})}_{+}, \\ g_{1} \in X_{G_{1}}, g_{2} \in X_{G_{2}}, g_{3} \in (G_{1} \times X_{G_{2}}) \cup (X_{G_{1}} \times G_{2})}, \end{matrix}$ where G₁ × X_{G
₂} is the Cartesian product of G₁ and X_{G
₂}, G_i ⊆ X_{G
_i}, and i = 1, 2. Moreover, the membership function is expressed as $μ_{G_{1} + G_{2}} (g_{3}) = \frac{μ_{G_{1}} (g_{1}) + μ_{G_{2}} (g_{2})}{1 + μ_{G_{1}} (g_{1}) μ_{G_{2}} (g_{2})}$ (1)

When both G₁ and G₂ are unidimensional, we have two unidimensional Z-numbers, Z₁ = (A₁, B₁) and Z₂ = (A₂, B₂). Similarly, for discrete fuzzy numbers A₁ and A₂, their addition (A₁, A₂) ₁₊₂ = A₁ + A₂ and membership function are described as $\begin{matrix} {(A_{1}, A_{2})}_{1 + 2} = A_{1} + A_{2} = {x_{3} | x_{3} = {(x_{1}, x_{2})}_{+}, \\ x_{1} \in X_{1}, x_{2} \in X_{2}, x_{3} \in (A_{1} \times X_{2}) \cup (X_{1} \times A_{2})}, \end{matrix}$ $μ_{A_{1} + A_{2}} (x_{3}) = \frac{μ_{A_{1}} (x_{1}) + μ_{A_{2}} (x_{2})}{1 + μ_{A_{1}} (x_{1}) μ_{A_{2}} (x_{2})} .$ (2)

Property 1. If μ_{G
₁} (g₁) = 1 or μ_{G
₂} (g₂) = 1, then μ_G₁+G₂ (g₃) = 1. If μ_{G
_i} (g_i) = 0, then μ_G₁+G₂ (g₃) = μ_{G
_j} (g_j) where i, j = 1, 2 and i ≠ j. If μ_{G
₁} (g₁) = μ_{G
₂} (g₂) = 0, then μ_G₁+G₂ (g₃) = 0.

Thus, when g₁ or g₂ is ascertained to belong to G₁ or G₂, g₃ = (g₁, g₂) ₊ is also verified to belong to (G₁, G₂) ₁₊₂. When the membership degree of one these values is zero, the membership degree of g₃ = (g₁, g₂) ₊ belongs to (G₁, G₂) ₁₊₂ is depend on the other. When both membership degrees are zero, g₃ = (g₁, g₂) ₊ is revealed as not belonging to (G₁, G₂) ₁₊₂. This property conforms to logical operators [37, 38]: A + 1 =1, A + 0 = A, and 0 + 0 =0, where the symbol “+” represents a logical addition (“OR” operation) and binary numbers 1 and 0 logically represent “true” and “false”.

Definition 8. (Standard subtraction of restriction vectors) The subtraction of the multidimensional Z-number is suitable for computing the case in which X satisfies G₁ and does not satisfy G₂. For restriction vector G₁ and G₂, its standard subtraction (G₁, G₂) _1-2 can be defined as follows: $\begin{matrix} {(G_{1}, G_{2})}_{1 - 2} = & G_{1} - G_{2} \\ = & {g_{3} | g_{3} = {(g_{1}, g_{2})}_{-}, g_{1} \in G_{1}, \\ g_{2} \in G_{2}, g_{3} \in G_{1} \times {G_{2}}^{c}} . \end{matrix}$ The membership function is expressed as $μ_{G_{1} - G_{2}} (g_{3}) = \frac{μ_{G_{1}} (g_{1}) (1 - μ_{G_{2}} (g_{2}))}{1 + (1 - μ_{G_{1}} (g_{1})) μ_{G_{2}} (g_{2})} .$ (3)

Similarly, for discrete fuzzy numbers A₁ and A₂, their standard subtraction (A₁, A₂) _1-2 = A₁ - A₂ and membership function are described as $\begin{matrix} {(A_{1}, A_{2})}_{1 - 2} = A_{1} - A_{2} \\ = {x_{3} | x_{3} = {(x_{1}, x_{2})}_{-}, x_{1} \in A_{1}, x_{2} \in A_{2}, \\ x_{3} \in A_{1} \times {A_{2}}^{c}}, \\ μ_{A_{1} - A_{2}} (x_{3}) = \frac{μ_{A_{1}} (x_{1}) (1 - μ_{A_{2}} (x_{2}))}{1 + (1 - μ_{A_{1}} (x_{1})) μ_{A_{2}} (x_{2})} . \end{matrix}$ (4)

Property 2. If μ_{G
₁} (g₁) = 1, then μ_G₁-G₂ (g₃) = 1 - μ_{G
₂} (g₂). If μ_{G
₁} (g₁) = 0, then μ_G₁-G₂ (g₃) = 0. If μ_{G
₂} (g₂) = 0, then μ_G₁-G₂ (g₃) = μ_{G
₁} (g₁). If μ_{G
₂} (g₂) = 1, then μ_G₁-G₂ (g₃) = 0.

Similarly, we have a corresponding explanation that also conforms to common sense.

Definition 9. (Multiplication of restriction vectors) The multiplication of the multidimensional Z-number is appropriate for computing the case wherein X satisfies G₁ and G₂. For restriction vector G₁ and G₂, their multiplication (G₁, G₂) _1·2 is defined as $\begin{matrix} {(G_{1}, G_{2})}_{1 \cdot 2} = G_{1} \cdot G_{2} \\ = {g_{3} | g_{3} = {(g_{1}, g_{2})}_{\cdot}, g_{1} \in G_{1}, \\ g_{2} \in G_{2}, g_{3} \in G_{1} \times G_{2}} . \end{matrix}$ The membership function is described as $μ_{G_{1} \cdot G_{2}} (g_{3}) = \frac{μ_{G_{1}} (g_{1}) μ_{G_{2}} (g_{2})}{1 + (1 - μ_{G_{1}} (g_{1})) (1 - μ_{G_{2}} (g_{2}))} .$ (5)

Similarly, for discrete fuzzy numbers A₁ and A₂, their multiplication (A₁, A₂) _1·2 and membership function are explained as $\begin{matrix} {(A_{1}, A_{2})}_{1 \cdot 2} = A_{1} \cdot A_{2} \\ = {x_{3} | x_{3} = {(x_{1}, x_{2})}_{\cdot}, x_{1} \in A_{1}, x_{2} \in A_{2}, \\ x_{3} \in A_{1} \times A_{2}}, \\ μ_{ A_{1} \cdot A_{2}} (x_{3}) = \frac{μ_{A_{1}} (x_{1}) μ_{A_{2}} (x_{2})}{1 + (1 - μ_{A_{1}} (x_{1})) (1 - μ_{A_{2}} (x_{2}))} . \end{matrix}$ (6)

Property 3. If μ_{G
_i} (g_i) = 1, then μ_G₁·G₂ (g₃) = μ_{G
_j} (g_j) where i, j = 1, 2 and i ≠ j. If μ_{G
₁} (g₁) = 0 or μ_{G
₂} (g₂) = 0, then μ_G₁·G₂ (g₃) = 0.

Accordingly, only when g₁ and g₂ are respectively determined to belong to G₁ and G₂ can g₃ = (g₁, g₂) _· be ascertained to belong to (G₁, G₂) _1·2. When g₁ or g₂ are verified as not belonging to G₁ or G₂, g₃ = (g₁, g₂) _· is established as not belonging to (G₁, G₂) _1·2. This property conforms to logical operators: A × 0 =0, A × 1 = A and 1 × 1 =1, where the symbol “×” represents a logical multiplication (“AND” operation).

Definition 10. (Conditional division of restriction vectors) The conditional division of the multidimensional Z-number is applicable for calculating the case of the occurrence of A₁ under the premise of the occurrence of A₂. For restriction vector G₁ and G₂, their conditional division (G₁, G₂) _1/2 = G₁/G₂ can be defined as $\begin{matrix} {(G_{1}, G_{2})}_{1 / 2} = G_{1} / G_{2} = {g_{3} | g_{1} = {(g_{2}, g_{3},)}_{.} \\ g_{1} \in G_{1}, g_{2} \in G_{2}} . \end{matrix}$

The membership function can be described as $μ_{G_{1} / G_{2}} (g_{3}) = \frac{μ_{G_{1}} (g_{1}) (2 - μ_{G_{2}} (g_{2}))}{μ_{G_{1}} (g_{1}) + μ_{G_{2}} (g_{2}) - μ_{G_{1}} (g_{1}) μ_{G_{2}} (g_{2})} .$ (7)

Observe that the conditional division is a dimension reduction process.

Property 4. $μ_{G_{1}} (g_{1}) = \frac{μ_{G_{2}} (g_{2}) μ_{G_{1} / G_{2}} (g_{3})}{1 + (1 - μ_{G_{2}} (g_{2})) (1 - μ_{G_{1} / G_{2}} (g_{3}))} .$

This property indicates that the multiplication and conditional division of restriction vectors are reciprocal operations.

Definition 11. (Operations of probable events) We define the corresponding operations of probable events based on the basic operations of restriction vectors. When X_{G
₁} and X_{G
₂} are independent, the operation rules of probable events of multidimensional Z-number are given as follows: $\begin{matrix} p_{G_{1} + G_{2}} (g_{1}, g_{2}) = p_{G_{1}} (g_{1}) + p_{G_{2}} (g_{2}) \\ - p_{G_{1}} (g_{1}) p_{G_{2}} (g_{2}), \end{matrix}$ (8) $p_{G_{1} - G_{2}} (g_{1}, g_{2}^{c}) = p_{G_{1}} (g_{1}) (1 - p_{G_{2}} (g_{2})),$ (9) $p_{G_{1} \cdot G_{2}} (g_{1}, g_{2}) = p_{G_{1}} (g_{1}) p_{G_{2}} (g_{2}),$ (10) $p_{G_{1} / G_{2}} (g_{1}, g_{2}) = \frac{p_{G_{1}} (g_{2}, g_{3})}{p_{G_{2}} (g_{2})} = p_{G_{3}} (g_{3}),$ (11) where g_i ∈ G_i and i = 1, 2. Obviously, the multiplication and the conditional division of probable events are also reciprocal operations.

We can prove the basic operations of restriction vectors and operations of probable events satisfy the commutative law, associative law and distributive law.

Note that the probability distribution of the multidimensional Z-number is $p_{1 + 2} = \frac{p_{G_{1} + G_{2}} (g_{1}, g_{2})}{m_{k_{1}} + m_{k_{2}} - 1},$ (12) $p_{1 - 2} = \frac{p_{G_{1} - G_{2}} (g_{1}, g_{2}^{c})}{m_{k_{2}} - 1},$ (13) $p_{1 \cdot 2} = p_{G_{1} \cdot G_{2}} (g_{1}, g_{2}),$ (14) $p_{1 / 2} = p_{G_{1} / G_{2}} (g_{1}, g_{2}),$ (15) where if X_{G
_i} is discrete, then m_{k
_i} is the number of sample points for X_{G
_i}; if X_{G
_i} is continuous, then m_{k
_i} = ∬_{G
_i} … ∫1dx_1idx_2i … dx_ni, i = 1, 2. In calculating the multidimensional Z-number, particularly in multiplication and subtraction, we should first calculate p₁ and p₂. If necessary, Equations (12) and (13) should be utilized to obtain p_{G
₁} and p_{G
₂}. The operations of probable events of the multidimensional Z-number are subsequently performed. Finally, we use Equations (12) and (13) to determine p₁₊₂ or p_1-2.

3.3 Reliability measure of the multidimensional Z-number

Definition 12. (Reliability measure) Assume that we must compute MZ_1*2 = MZ₁* MZ₂, * ∈ { +, 1pt1pt1pt -, 1pt1pt1pt · 1pt1pt, 1pt1pt1pt/}. First, calculate the corresponding Z⁺-numbers, i.e., ${MZ}_{1 * 2}^{+} = (G_{1} * G_{2}, p_{1} * p_{2})$ . The result can be obtained in accordance with Definitions 6–10. However, the true probability distributions p₁ and p₂ are unknown. We only know the fuzzy restrictions as follows $\iint \dots \int μ_{G_{1}} (g_{1}) p_{1} (g_{1}) d x_{1} d x_{2} \dots d x_{m} is B_{1},$ $\iint \dots \int μ_{G_{2}} (g_{2}) p_{2} (g_{2}) d x_{m + 1} d x_{m + 2} \dots d x_{n} is B_{2},$ which are represented in terms of membership functions defined as $μ_{B_{1}} (\iint \dots \int μ_{G_{1}} (g_{1}) p_{1} (g_{1}) d x_{1} d x_{2} \dots d x_{m}),$ $μ_{B_{2}} (\iint \dots \int μ_{G_{2}} (g_{2}) p_{2} (g_{2}) d x_{m + 1} d x_{m + 2} \dots d x_{n}),$ where g_i = (x_i1, x_i2, …, x_{im
_i}), i = 1, 2. The membership function of P_1*2 can written as $μ_{P_{1 * 2}} (P_{1 * 2}) = sup_{P_{1}, P_{2}} (min {μ_{P_{1}} (P_{1}), μ_{P_{2}} (P_{2})}),$ (16) subject to $μ_{P_{1}} (P_{1}) = μ_{B_{1}} (\int \int \dots \int μ_{G_{1}} (g_{1}) p_{1} (g_{1}) d x_{1} d x_{2} \dots d x_{m}),$ (17) $μ_{P_{2}} (P_{2}) = μ_{B_{2}} (\int \int \dots \int μ_{G_{2}} (g_{2}) p_{2} (g_{2}) d x_{m + 1} d x_{m + 2} \dots d x_{n}),$ (18) $\int \int \dots \int p_{1} (g_{1}) d x_{1} d x_{2} \dots d x_{m} = 1,$ (19) $\int \int \dots \int p_{2} (g_{2}) d x_{m + 1} d x_{m + 2} \dots d x_{n} = 1,$ (20) $\begin{matrix} \iint \dots \int x_{1} x_{2} \dots x_{m} p_{1} (g_{1}) d x_{1} \dots d x_{m} \\ = \frac{\iint \dots \int x_{1} x_{2} \dots x_{m} μ_{G_{1}} (g_{1}) d x_{1} \dots d x_{m}}{\iint \dots \int μ_{G_{1}} (g_{1}) d x_{1} \dots d x_{m}}, \end{matrix}$ (21) $\begin{matrix} \int \int \dots \int x_{m + 1} \dots x_{n} p_{2} (g_{2}) d x_{m + 1} \dots d x_{n} \\ = \frac{\int \int \dots \int x_{m + 1} \dots x_{n} μ_{G_{2}} (g_{2}) d x_{m + 1} \dots d x_{n}}{\iint \dots \int μ_{G_{2}} (g_{2}) d x_{m + 1} \dots d x_{n}} . \end{matrix}$ (22)

We can determine μ_{P
_1*2} (P_1*2) by calculating the nonlinear variational problem with Equations (14–20). Finally, the measure of reliability can be calculated as follows: $μ_{B_{1 * 2}} (b_{1 * 2}) = sup (μ_{P_{1 * 2}} (P_{1 * 2})),$ (23) subject to $b_{1 * 2} = \iint \dots \int p_{1 * 2} (g_{3}) μ_{G_{1 * 2}} (g_{3}) d x_{1} d x_{2} \dots d x_{n} .$ (24)

4 Basic operations of the multidimensional Z-number

The operation of the multidimensional Z-number includes primarily three situations. The first situation is the calculation of the two-dimensional Z-number by two unidimensional Z-numbers. The second is the calculation of a high dimensional multidimensional Z-number by computing a multidimensional Z-number and a unidimensional Z-number. The third one is the calculation of a multidimensional Z-number by two multidimensional Z-numbers. In practice, the second situation is the most common one. Therefore, this section takes the second case as an example to perform the addition, subtraction, multiplication, and division of the multidimensional Z-number.

4.1 Addition

Let MZ₁ = (G₁, B₁) be a discrete multidimensional Z-number and Z₂ = (A₂, B₂) be a discrete Z-number describing the different dimensions of the same random variable, X. Before calculating the multidimensional Z-numbers, we should compute the corresponding Z⁺-numbers, i.e., ${MZ}_{1}^{+} + Z_{2}^{+} = (G_{1} + A_{2}, R_{1} + R_{2})$ , where the probability distributions representing R₁ and R₂ are as follows: $p_{1} = p_{1} (g_{1}) ∖ g_{1} + p_{1} (g_{2}) ∖ g_{2} + \dots + p_{1} (g_{n}) ∖ g_{n},$ $p_{2} = p_{2} (x_{1}) ∖ x_{1} + p_{2} (x_{2}) ∖ x_{2} + \dots + p_{2} (x_{m}) ∖ x_{m},$ for which one necessarily has $\sum_{i = 1}^{n} p_{1} (g_{i}) = 1$ , $\sum_{j = 1}^{m} p_{2} (x_{j}) = 1$ . Addition G₁ + A₂ is defined in accordance with Definition 7 and R₁ + R₂, that is, p₁₊₂ (g₁, x₂) is defined according to Definition 11 as $\begin{matrix} p_{G_{1} + A_{2}} (g_{1}, A_{2}) = p_{G_{1}} (g_{1}) + p_{A_{2}} (x_{2}) \\ - p_{G_{1}} (g_{1}) p_{A_{2}} (x_{2}), \end{matrix}$ $p_{1 + 2} (g_{1}, x_{2}) = \frac{p_{G_{1} + A_{2}} (g_{1}, x_{2})}{n + m - 1} .$

We can determine the Z⁺-number by using the above calculation rules and completing the first step in the calculation of Z-number. $\sum_{i = 1}^{n} μ_{G_{1}} (g_{i}) p_{1} (g_{i}) is B_{1},$ $\sum_{j = 1}^{m} μ_{A_{2}} (x_{j}) p_{2} (x_{j}) is B_{2} .$

Following Definition 12, the membership function μ_B₁+B₂ is defined as follows: $μ_{B_{1 + 2}} (b_{1 + 2}) = sup (μ_{P_{1 + 2}} (P_{1 + 2})),$ subject to $b_{1 + 2} = \iint \dots \int p_{1 + 2} (g_{3}) μ_{G_{1 + 2}} (g_{3}) d x_{1} d x_{2} \dots d x_{m + n} .$

Consequently, MZ₁₊₂ = MZ₁ + Z₂ is obtained as MZ₁₊₂ = (G₁₊₂, B₁₊₂).

Example 1. Let us consider the addition MZ₁₊₂ = MZ₁ + Z₂ of a discrete multidimensional Z-number MZ₁ = (G₁, B₁) and a discrete Z-number Z₂ = (A₂, B₂) given as $\begin{matrix} G_{1} = 0 / (1, 1)_{\cdot} + 0.6 / (1, 2)_{\cdot} + 1 / (1, 3)_{\cdot} \\ + 1 / (2, 1)_{\cdot} + 0.5 / (2, 2)_{\cdot} + 0 / (2, 3)_{\cdot}, \end{matrix}$ $\begin{matrix} B_{1} = 0 / 0 + 0.5 / 0.1 + 0.8 / 0.2 + 1 / 0.3 + 0 . 8 / 0.4 \\ + 0 . 7 / 0.5 + 0.6 / 0.6 + 0.4 / 0.7 + 0.2 / 0.8 \\ + 0.1 / 0.9 + 0 / 1, A_{2} = 0 / 1 + 0.6 / 2 + 1 / 3 + 0.4 / 4 + 0 / 5, \end{matrix}$ $\begin{matrix} B_{2} = 0 / 0 + 0.3 / 0.1 + 0.5 / 0.2 + 0.6 / 0.3 \\ + 0 . 7 / 0.4 + 0 . 8 / 0.5 + 0.9 / 0.6 + 1 / 0.7 + 0.9 / 0.8 \\ + 0.8 / 0.9 + 0 / 1 . \end{matrix}$

To calculate MZ₁₊₂, we first compute ${MZ}_{1 + 2}^{+} = {MZ}_{1}^{+} + Z_{2}^{+}$ . A Z-number contains more than one underlying probability distribution, and thus, we only use an example to illustrate the steps of multidimensional Z-number operations. We can obtain the final result by repeating this process. Let us consider ${MZ}_{1}^{+} = (G_{1}, p_{1})$ and $Z_{2}^{+} = (A_{2}, p_{2})$ where p₁ and p₂ are as follows: $\begin{matrix} p_{1} = 0.39 ∖ {(1, 1)}_{\cdot} + 0.17 ∖ {(1, 2)}_{\cdot} + 0.05 ∖ {(1, 3)}_{\cdot} \\ + 0.08 ∖ {(2, 1)}_{\cdot} + 0.12 ∖ {(2, 2)}_{\cdot} + 0.18 ∖ {(2, 3)}_{\cdot}, \end{matrix}$ $p_{2} = 0.16 / 1 + 0.23 / 2 + 0.28 / 3 + 0.2 / 4 + 0.13 / 5 .$

According to Definition 7 we have $\begin{matrix} G_{1 + 2} = 0 / (1, 1, 1) + 0.6 / (1, 2, 1) + 1 / (1, 3, 1) + \dots \\ + 1 / (2, 1, 5) + 0.5 / (2, 2, 5) + 0 / (2, 3, 5) . \end{matrix}$

The membership degrees are shown in Table 1.

Table 1
Membership degrees of G₁₊₂

		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)
		0	0.6	1	1	0.5	0
1	0	0.00	0.60	1.00	1.00	0.50	0.00
2	0.6	0.60	0.88	1.00	1.00	0.85	0.60
3	1	1.00	1.00	1.00	1.00	1.00	1.00
4	0.4	0.40	0.81	1.00	1.00	0.75	0.40
5	0	0.00	0.60	1.00	1.00	0.50	0.00

Next we compute p₁ + p₂. In accordance with Definition 11, we have $\begin{matrix} p_{1 + 2} = 0.048 ∖ (1, 1, 1) + 0.03 ∖ (1, 2, 1) + \dots \\ + 0.024 ∖ (2, 2, 5) + 0.029 ∖ (2, 3, 5) . \end{matrix}$

The probability values of p₁₊₂ are shown in Table 2.

Table 2

Probability values of p₁₊₂

		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)
		0.39	0.17	0.05	0.08	0.12	0.18
1	0.16	0.048	0.030	0.020	0.023	0.026	0.031
2	0.24	0.053	0.037	0.028	0.030	0.033	0.038
3	0.27	0.056	0.040	0.032	0.034	0.037	0.041
4	0.20	0.051	0.033	0.024	0.027	0.030	0.035
5	0.13	0.046	0.027	0.017	0.020	0.024	0.029

Subsequently, ${MZ}_{1 + 2}^{+} = (G_{1 + 2}, p_{1 + 2})$ is obtained. To compute the membership degrees μ_{B
₁} (P (G₁)) and μ_{B
₂} (P (A₂)), we should determine the P (G₁) and P (A₂) according to Equations (2 and 3). Thus, we have $P (G_{1}) = \sum_{i = 1}^{6} μ_{G_{1}} (g_{1 i}) p_{1} (g_{1 i}) = 0.3,$ $P (A_{2}) = \sum_{j = 1}^{5} μ_{A_{2}} (x_{2 j}) p_{2} (x_{2 j}) = 0.5,$ then μ_{B
₁} (P (G₁)) = μ_{B
₁} (0.3) = 1 and μ_{B
₂} (P (A₂)) = μ_{B
₂} (0.5) = 0. 8. Following Equation (1), we have μ_{P
₁₂} (P₁₂) = μ_{P
₁} (P₁) ∧ μ_{P
₂} (P₂) = 1 ∧ 0.8 = 0.8. We determine the membership degrees for all considered p₁, p₂, and p₁₊₂. Next, we must compute the values of the probability measure: $P (G_{1 + 2}) = \sum_{i = 1}^{6} \sum_{j = 1}^{5} μ_{G_{1 + 2}} (g_{i}, x_{j}) p_{1 + 2} (g_{i}, x_{j}) = 0.69 .$

Thus far, we have ascertained one basic value of probability measure within the fuzzy restriction B₁₊₂, that is, b₁₂ = 0.69. Next, we obtain μ_{B
₁₊₂} (b₁₂ = 0.69) = 0.8. By repeating the above operations, we can construct B₁₊₂ as follows: $\begin{matrix} B_{1 + 2} = 0 / 0.57 + 0.3 / 0.6 + 0.5 / 0.62 + 0.6 / 0.64 + \\ 0.7 / 0.66 + 0.8 / 0.69 + 1 / 0.73 1 pt + 0.8 / 0.77 + 0 / 0.81 . \end{matrix}$

Therefore, we obtain the result of addition MZ₁₊₂ = (G₁₊₂, B₁₊₂).

4.2 Subtraction

Let MZ₁ = (G₁, B₁) be a discrete multidimensional Z-number and Z₂ = (A₂, B₂) be a discrete Z-number describing the different dimensions of the same random variable X. Before calculating the Z-number, we should compute the corresponding Z⁺-numbers, i.e., ${MZ}_{1}^{+} - Z_{2}^{+} = (G_{1} - A_{2}, R_{1} - R_{2})$ , where the probability distributions representing R₁ and R₂ are as follows: $p_{1} = p_{1} (g_{1}) ∖ g_{1} + p_{1} (g_{2}) ∖ g_{2} + \dots + p_{1} (g_{n}) ∖ g_{n},$ $p_{2} = p_{2} (x_{1}) ∖ x_{1} + p_{2} (x_{2}) ∖ x_{2} + \dots + p_{2} (x_{m}) ∖ x_{m},$ for which one necessarily has $\sum_{i = 1}^{n} p_{1} (g_{i}) = 1$ , $\sum_{j = 1}^{m} p_{2} (x_{j}) = 1$ . Subtraction G₁ - A₂ is defined in accordance with Definition 8 and R₁ - R₂, that is, p_1-2 (g₁, x₂) is defined in accordance with Definition 11 as $p_{G_{1} - A_{2}} (g_{1}, x_{2}^{c}) = p_{G_{1}} (g_{1}) (1 - p_{A_{2}} (x_{2})),$ $p_{1 - 2} = \frac{p_{G_{1} - A_{2}} (g_{1}, x_{2}^{c})}{m - 1} .$

We can ascertain the Z⁺-number by using the above calculation rules and then completing the first step in the calculation of Z-number. $\sum_{i = 1}^{n} μ_{G_{1}} (g_{i}) p_{1} (g_{i}) is B_{1},$ $\sum_{j = 1}^{m} μ_{A_{2}} (x_{j}) p_{2} (x_{j}) is B_{2} .$

According to Definition 12, the membership function μ_B₁-B₂ is defined as follows: $μ_{B_{1 - 2}} (b_{1 - 2}) = sup (μ_{P_{1 - 2}} (P_{1 - 2})),$ subject to $b_{1 - 2} = \iint \dots \int p_{1 - 2} (g_{3}) μ_{G_{1 - 2}} (g_{3}) d x_{1} d x_{2} \dots d x_{m + n} .$

Accordingly, MZ_1-2 = MZ₁ - Z₂ is obtained as MZ_1-2 = (G_1-2, B_1-2).

Example 2. Let us consider subtraction MZ_1-2 = MZ₁ - Z₂ of a discrete multidimensional Z-number MZ₁ = (G₁, B₁) and a discrete Z-number Z₂ = (A₂, B₂) given in Example 1.

To calculate MZ_1-2, we compute ${MZ}_{1 - 2}^{+} = {MZ}_{1}^{+} - Z_{2}^{+} = (G_{1 - 2}, p_{1 - 2})$ . In accordance with Definition 8 we have $\begin{matrix} G_{1 - 2} = 0 / (1, 1, 1) + 0.6 / (1, 2, 1) + 1 / (1, 3, 1) + \dots \\ + 1 / (2, 1, 5) + 0.5 / (2, 2, 5) + 0 / (2, 3, 5) . \end{matrix}$

The membership degrees are shown in Table 3.

Table 3
Membership degrees of G_1-2

		(1,2)	(1,3)	(2,1)	(2,2)
		0.6	1	1	0.5
1	0	0.6	1	1	0.5
2	0.6	0.19	0.4	0.4	0.15
3	1	0	0	0	0
4	0.4	0.31	0.6	0.6	0.25
5	0	0.6	1	1	0.5

Next, we compute p₁ - p₂. Following Definition 11, we have $\begin{matrix} p_{1 - 2} = 0.08 ∖ (1, 1, 1) + 0.04 ∖ (1, 2, 1) + \dots \\ + 0.03 ∖ (2, 2, 5) + 0.04 ∖ (2, 3, 5) . \end{matrix}$

The probability values of p_1-2 are shown in Table 4.

Table 4

Probability values of p_1-2

		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)
		0.39	0.17	0.05	0.08	0.12	0.18
1	0.16	0.08	0.04	0.01	0.02	0.03	0.04
2	0.24	0.08	0.03	0.01	0.02	0.02	0.03
3	0.27	0.07	0.03	0.01	0.01	0.02	0.03
4	0.20	0.08	0.03	0.01	0.02	0.02	0.04
5	0.13	0.08	0.04	0.01	0.02	0.03	0.04

Subsequently, ${MZ}_{1 - 2}^{+} = (G_{1 - 2}, p_{1 - 2})$ is determined. As discussed in Section 4.1, we have μ_{P
₁₂} (P₁₂) = 0.6. We calculate the membership degrees for all considered p₁, p₂, and p_1-2. Next, we must compute the values of the probability measure: $P (G_{1 - 2}) = \sum_{i = 1}^{6} \sum_{j = 1}^{5} μ_{G_{1 - 2}} (g_{i}, x_{j}) p_{1 - 2} (g_{i}, x_{j}) = 0.18 .$

Thus far, we have identified one basic value of the probability measure within the fuzzy restriction B_1-2, that is, b₁₂ = 0.18. Subsequently, we obtain μ_{B
_1-2} (b₁₂ = 0.18) = 0.6. By repeating the above operations, we can construct B_1-2 as follows: $\begin{matrix} B_{1 - 2} = 0 / 0 + 0.8 / 0.01 + 1 / 0.06 + 0.7 / 0.15 \\ + 0.6 / 0.18 + 0.4 / 0.21 + 0.3 / 0.24 + 0 / 0.28 . \end{matrix}$

Thus, we obtain the result of addition MZ_1-2 = (G_1-2, B_1-2).

4.3 Multiplication

Let us consider the multiplication of MZ₁ = (G₁, B₁) and Z₂ = (A₂, B₂). First, we should calculate the corresponding Z+-numbers, i.e., ${MZ}_{1}^{+} \cdot Z_{2}^{+} = (G_{1} \cdot A_{2}, R_{1} \cdot R_{2})$ , where the probability distributions representing R₁ and R₂ are as follows: $p_{1} = p_{1} (g_{1}) ∖ g_{1} + p_{1} (g_{2}) ∖ g_{2} + \dots + p_{1} (g_{n}) ∖ g_{n},$ $p_{2} = p_{2} (x_{1}) ∖ x_{1} + p_{2} (x_{2}) ∖ x_{2} + \dots + p_{2} (x_{m}) ∖ x_{m} .$

The multiplication G₁ · A₂ is defined in accordance with Definition 9 and R₁ · R₂, that is, p_1·2 (g₁, x₂) is described in Definition 11 as $p_{1 \cdot 2} (g_{1}, x_{2}) = p_{G_{1} \cdot A_{2}} (g_{1}, x_{2}),$ $p_{G_{1} \cdot A_{2}} (g_{1}, x_{2}) = p_{G_{1}} (g_{1}) p_{A_{2}} (x_{2}) .$

We can identify the Z⁺-number by using the above calculation rules and completing the first step in the calculation of Z-number. $\sum_{i = 1}^{n} μ_{G_{1}} (g_{i}) p_{1} (g_{i}) is B_{1},$ $\sum_{j = 1}^{m} μ_{A_{2}} (x_{j}) p_{2} (x_{j}) is B_{2} .$

According to Definition 12, the membership function μ_B₁-B₂ is defined as follows: $μ_{B_{1 \cdot 2}} (b_{1 \cdot 2}) = sup (μ_{P_{1 \cdot 2}} (P_{1 \cdot 2})),$ subject to $b_{1 \cdot 2} = \iint \dots \int p_{1 \cdot 2} (g_{3}) μ_{G_{1 \cdot 2}} (g_{3}) d x_{1} d x_{2} \dots d x_{m + n} .$

Thus, MZ_1·2 = MZ₁ · Z₂ is obtained as MZ_1·2 = (G_1·2, B_1·2).

Example 3. Let us consider the multiplication MZ_1·2 = MZ₁ · Z₂ of a discrete multidimensional Z-number MZ₁ = (G₁, B₁) and a discrete Z-number Z₂ = (A₂, B₂) given in Example 1.

To calculate Z_1·2, we first compute $Z_{1 \cdot 2}^{+} = Z_{1}^{+} \cdot Z_{2}^{+}$ . In accordance with Definition 9: $\begin{matrix} G_{1 \cdot 2} = 0 / (1, 1, 1) + 0 / (1, 2, 1) + 0 / (1, 3, 1) + \dots \\ + 0 / (2, 1, 5) + 0 / (2, 2, 5) + 0 / (2, 3, 5) . \end{matrix}$

The membership degrees are shown in Table 5.

Table 5
Membership degrees of G_1·2

		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)
		0	0.6	1	1	0.5	0
1	0	0.00	0.00	0.00	0.00	0.00	0.00
2	0.6	0.00	0.31	0.60	0.60	0.25	0.00
3	1	0.00	0.60	1.00	1.00	0.50	0.00
4	0.4	0.00	0.19	0.40	0.40	0.15	0.00
5	0	0.00	0.00	0.00	0.00	0.00	0.00

Next, we compute p₁ · p₂. As per Definition 11, we have $\begin{matrix} p_{1 \cdot 2} = 0.06 ∖ (1, 1, 1) + 0.03 ∖ (1, 2, 1) + \dots \\ + 0.02 ∖ (2, 2, 5) + 0.02 ∖ (2, 3, 5) . \end{matrix}$

The probability values of p_1·2 are shown in Table 6.

Table 6

Probability values of p_1·2

		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)
		0.39	0.17	0.05	0.08	0.12	0.18
1	0.16	0.06	0.03	0.01	0.01	0.02	0.03
2	0.24	0.09	0.04	0.01	0.02	0.03	0.04
3	0.27	0.11	0.05	0.01	0.02	0.03	0.05
4	0.20	0.08	0.03	0.01	0.02	0.02	0.04
5	0.13	0.05	0.02	0.01	0.01	0.02	0.02

Subsequently, ${MZ}_{1 \cdot 2}^{+} = (G_{1 \cdot 2}, p_{1 \cdot 2})$ is obtained. From Section 4.1, we have μ_{P
₁₂} (P₁₂) = 0.8. We compute the membership degrees for all considered p₁, p₂, and p_1·2. Next, we must determine the values of the probability measure: $P (G_{1 \cdot 2}) = \sum_{i = 1}^{6} \sum_{j = 1}^{5} μ_{G_{1 \cdot 2}} (g_{1 i}, x_{2 j}) p_{1 \cdot 2} (g_{1 i}, x_{2 j}) = 0.14 .$

Thus far, we have identified one basic value of the probability measure within fuzzy restriction B_1·2, that is, b₁₂ = 0.14. Subsequently, we obtain μ_{B
_1·2} (b₁₂ = 0.14) = 0.8. By repeating the above operations, we can construct B_1·2 as follows: $\begin{matrix} B_{1 \cdot 2} = 0 / 0 + 0.3 / 0.03 + 0.5 / 0.06 + 0.6 / 0.08 + 0.7 / 0.11 \\ + 0.8 / 0.14 + 1 / 0.2 + 0.8 / 0.27 + 0 / 0.33 . \end{matrix}$

Hence, we obtain the result of addition MZ_1·2 = (G_1·2, B_1·2).

4.4 Division

Let us consider the division of MZ₁ = (G₁, B₁) and Z₂ = (A₁, B₂). First we should calculate the corresponding Z+-numbers, i.e., ${MZ}_{1}^{+} / Z_{2}^{+} = (G_{1} / A_{2}, R_{1} / R_{2})$ , where the probability distributions representing R₁ and R₂ are as follows: $p_{1} = p_{1} (g_{1}) ∖ g_{1} + p_{1} (g_{2}) ∖ g_{2} + \dots + p_{1} (g_{n}) ∖ g_{n},$ $p_{2} = p_{2} (x_{1}) ∖ x_{1} + p_{2} (x_{2}) ∖ x_{2} + \dots + p_{2} (x_{m}) ∖ x_{m} .$

The division G₁/A₂ is defined in accordance with Definition 10 and R₁/R₂, i.e., p_1/2 (g₁, x₂) is defined according to Definition 11 as $p_{1 / 2} (g_{1}, x_{2}) = p_{G_{1} / A_{2}} (g_{1}, x_{2}),$ $p_{G_{1} / A_{2}} (g_{1}, x_{2}) = \frac{p_{G_{1}} (x_{1}, x_{2})}{p_{A_{2}} (x_{2})} = p_{A_{1}} (x_{1}) .$

The Z⁺-number can be identified by using the above calculation rules and completing the first step in the calculation of the Z-number. $\sum_{i = 1}^{n} μ_{G_{1}} (g_{i}) p_{1} (g_{i}) is B_{1},$ $\sum_{j = 1}^{m} μ_{A_{2}} (x_{j}) p_{2} (x_{j}) is B_{2} .$

According to Definition 12, the membership function μ_B₁/B₂ can be defined as follows: $μ_{B_{1 / 2}} (b_{1 / 2}) = sup (μ_{P_{1 / 2}} (P_{1 / 2})),$ subject to $b_{1 / 2} = \iint \dots \int p_{1 / 2} (g_{3}) μ_{G_{1 / 2}} (g_{3}) d x_{1} d x_{2} \dots d x_{n / m} .$

As a result, MZ_1/2 = MZ₁/Z₂ is obtained as MZ_1/2 = (G_1/2, B_1/2).

Example 4. Let us consider the division MZ_3/4 = MZ₃/Z₄ of a discrete multidimensional Z-number MZ₃ = (G₃, B₃) and a discrete Z-number Z₄ = (A₄, B₄) is given as $G_{3} = 0 / {(1, 1)}_{\cdot} + 0 / {(1, 2)}_{\cdot} + \dots + 0 / {(6, 5)}_{\cdot},$ $\begin{matrix} B_{3} = 0 / 0.01 + 0 . 2 / 0.04 + 0 . 3 / 0.05 + 0 . 4 / 0.07 \\ + 0.8 / 0.1 + 1 / 0.12 + 0.5 / 0.13 + 0.3 / 0.15 + 0 / 0.16, \end{matrix}$ $A_{4} = 0 / 1 + 0.6 / 2 + 1 / 3 + 0.4 / 4 + 0 / 5,$ $\begin{matrix} B_{4} = 0 / 0 + 0.1 / 0.1 + 0.3 / 0.2 + 0.5 / 0.3 + 0 . 6 / 0.4 \\ + 1 / 0.5 + 0.7 / 0.6 + 0.5 / 0.7 + 0.3 / 0.8 + 0.1 / 0.9 + 0 / 1 . \end{matrix}$

The specific membership degrees of G₃ are shown in Table 7.

Table 7
Membership degrees of G₃

	2	3	4	5
1	0.00	0.00	0.00	0.00
2	0.09	0.25	0.60	0.31
3	0.20	0.50	1.00	0.60
4	0.05	0.15	0.40	0.19
5	0.00	0.00	0.00	0.00

Let us consider ${MZ}_{3}^{+} = (G_{3}, p_{3})$ and $Z_{4}^{+} = (A_{4}, p_{4})$ where p₃ and p₄ are as follows: $p_{3} = 0 ∖ {(1, 1)}_{\cdot} + 0 ∖ {(1, 2)}_{\cdot} . . . + 0.01 ∖ {(6, 5)}_{\cdot}$ $p_{4} = 0.12 / 1 + 0.14 / 2 + 0.33 / 3 + 0.22 / 4 + 0.19 / 5 .$

The specific probability values of p₃ are shown in Table 8.

Table 8

Probability values of p₃

	1	2	3	4	5	6
1	0.00	0.01	0.05	0.04	0.01	0.01
2	0.00	0.01	0.06	0.04	0.02	0.01
3	0.01	0.02	0.14	0.10	0.04	0.02
4	0.01	0.01	0.09	0.07	0.03	0.02
5	0.01	0.01	0.08	0.06	0.02	0.01

To calculate Z_3/4, we first compute $Z_{3 / 4}^{+} = Z_{3}^{+} / Z_{4}^{+}$ . In accordance to Definition 10, we have $G_{3 / 4} = 0 / 1 + 0.2 / 2 + 0.5 / 3 + 1 / 4 + 0.6 / 5 + 0 / 6 .$

Next, we calculate p₃/p₄. In line with Definition 11, we have $p_{3 / 4} = 0.03 ∖ 1 + 0.05 ∖ 2 + 0.41 ∖ 3 + 0.32 ∖ 4 + 0.12 ∖ 5 + 0.7 ∖ 6 .$

${MZ}_{3 / 4}^{+} = (G_{3 / 4}, p_{3 / 4})$ is obtained. As stated in Section 4.1, we have μ_{P
_3/4} (P₃₄) = 0.8. We compute the membership degrees for all considered p₃, p₄, and p_3/4. Next, the values of the probability measure must be calculated: $P (G_{3 / 4}) = \sum_{i = 1}^{6} \sum_{j = 1}^{5} μ_{G_{3 / 4}} (g_{i}, x_{j}) p_{3 / 4} (g_{i}, x_{j}) = 0.6 .$

At this point, we identify one basic value of the probability measure within the fuzzy restriction B_3/4, that is, b₃₄ = 0.6. We obtain μ_{B
_3/4} (b₃₄ = 0.6) = 0.8. By repeating the above operations, we can construct B_3/4 as follows: $\begin{matrix} B_{3 / 4} = 0 / 0 + 0.1 / 0.1 + 0.2 / 0.2 + 0.3 / 0.3 + 0.4 / 0.4 + \\ 0.6 / 0.5 + 0.8 / 0.6 + 1 / 0.7 + 0.5 / 0.8 + 0.3 / 0.9 + 0 / 1 . \end{matrix}$

Thus, we ascertain the result of the addition of MZ_3/4 = (G_3/4, B_3/4).

5 Ranking of the multidimensional Z-number

Ranking plays a crucial part in the theoretical study of the Z-number. Two kinds of sorting methods are applied for unidimensional Z-numbers in existing research. One is to convert the Z-number to real number or classical fuzzy number, then compare and sort them [17, 39], which is accompanied by the information loss in the conversion process. The other technique is to consider the two parts of Z-number simultaneously [40, 41] which is too complicated to calculate. To better compare unidimensional Z-numbers, a restriction is necessary for their comparison, thus, A₁ and A₂ must have the same measurement unit. In other words, we can only compare two Z-numbers (about 20 miles, sure) and (about 15 miles, quite sure). However, because the dimensions of description objects differ, we cannot compare (about 20 miles, sure) and (about 15 min, quite sure). Furthermore, the decision-maker usually describes real problems from multiple dimensions. Hence, a unidimensional sorting cannot reflect the actual situation fully. The multidimensional Z-number proposed in this paper can integrate information of each dimension. We can achieve a comprehensive evaluation by using the operational rules mentioned in Section 3, and subsequently comparing the multidimensional Z-numbers. For example, we can compare two multidimensional Z-numbers such as $\begin{matrix} (influenza, (temperature around 40^{\circ} C \\ and headache very bad), likely), \\ (cold, (temperature above 40^{\circ} C \\ and headache very bad), very likely) . \end{matrix}$

In this section, we propose a method to rank multidimensional Z-numbers.

Let MZ₁ = (G₁, B₁) and MZ₂ = (G₂, B₂) be two multidimensional Z-numbers. In real life, if G₁∩ G₂ = ∅, comparing MZ₁ and MZ₂ is meaningless. Hence, the multidimensional restriction vectors cannot be disconnected completely. The comparison of multidimensional Z-numbers focuses on comparing B₁ with B₂. Therefore, we define the comparison method of multidimensional Z-numbers based on the centroid comparison method [42 –44] as follows.

Calculate the centroid point $(x_{B}^{*}, y_{B}^{*})$ for the fuzzy number B. If B is a continuous fuzzy number, $(x_{B}^{*}, y_{B}^{*})$ is defined as $x_{B}^{*} = \frac{\int x_{B} μ_{B} (x_{B}) d x_{B}}{\int μ_{B} (x_{B}) d x_{B}}, y_{B}^{*} = \frac{\int y_{B} {μ_{B}}^{- 1} (y_{B}) d y_{B}}{\int {μ_{B}}^{- 1} (y_{B}) d y_{B}},$ where x_B = ∫_Rμ_A (x) p_X (x) dx, μ_B (x_B) = y_B and $μ_{B}^{- 1} (μ_{B} (x_{B})) \equiv x_{B}$ . If B is a discrete fuzzy number, $(x_{B}^{*}, y_{B}^{*})$ is defined as $x_{B}^{*} = \frac{\sum x_{B} μ_{B} (x_{B})}{\sum μ_{B} (x_{B})}, y_{B}^{*} = \frac{\sum y_{B} {μ_{B}}^{- 1} (y_{B})}{\sum {μ_{B}}^{- 1} (y_{B})} .$

Calculate the ranking function.

When G₁ = G₂, the ranking fuzzy number has the following properties:

If D (B₁) > D (B₂), then MZ₁ > MZ₂,

if D (B₁) = D (B₂), then MZ₁ = MZ₂, and

if D (B₁) < D (B₂), then MZ₁ < MZ₂.

(D) When G₁ ≠ G₂ and G₁∩ G₂ = G₀ ≠ ∅, the ranking fuzzy number has the following properties.

If D (B₀₁) < D (B₀₂) and D (B₁) < D (B₂), then MZ₁ < MZ₂,

if D (B₀₁) = D (B₀₂) and D (B₁) < D (B₂), then MZ₁ ≺ MZ₂,

if D (B₀₁) = D (B₀₂) and D (B₁) = D (B₂), then MZ₁ = MZ₂,

if D (B₀₁) = D (B₀₂) and D (B₁) > D (B₂), then MZ₁ ≻ MZ₂,

if D (B₀₁) > D (B₀₂) and D (B₁) > D (B₂), then MZ₁ > MZ₂, and

if otherwise, MZ₁ and MZ₂ cannot be compared.

Example 5. Mr. Li is looking for a job. He is 178 centimeters (cm) tall and weighs 68 kilograms (kg). He has three years working experience. He has good comprehension skills. Company A needs a person who is about 175 cm height, weighs less than 70 kg, and has over two years working experience. By contrast, company B needs a person who is about 180 cm height, weighs less than 65 kg, and has very good comprehension skills. We can use the multidimensional Z-numbers to indicate the possibility of Mr. Li being employed by both companies, i.e., MZ₁ = (G₁, B₁) and MZ₂ = (G₂, B₂), where $\begin{matrix} G_{1} = (Height of about 178 cm, weighs \\ about 68 kg about 3 years working experience), \end{matrix}$ $\begin{matrix} G_{2} = (Height of about 178 cm, weighs about 68 kg, \\ comprehension skills is good), \end{matrix}$ $\begin{matrix} B_{1} = 0 / 0.85 + 0.5 / 0.875 + 1 / 0.9 + 0.5 / 0.925 + 0 / 0.95, \end{matrix}$ $\begin{matrix} B_{2} = 0 / 0.75 + 0.5 / 0.775 + 1 / 0.8 + 0.5 / 0.825 + 0 / 0.85 . \end{matrix}$

Clearly, G₁ ≠ G₂ and G₁∩ G₂ = G₀ ≠ ∅, where $G_{0} = (Height of about 178 cm, weight of about 68 kg),$ $B_{01} = 0 / 0.75 + 0.5 / 0.775 + 1 / 0.8 + 0.5 / 0.825 + 0 / 0.85,$ $B_{02} = 0 / 0.7 + 0.5 / 0.725 + 1 / 0.75 + 0.5 / 0.775 + 0 / 0.8 .$

In accordance with Eqs. (25)–(27), we have $x_{B_{1}}^{*} = 0.9, y_{B_{1}}^{*} = 0.4, x_{B_{2}}^{*} = 0.8, x_{B_{2}}^{*} = 0.4,$ $x_{B_{01}}^{*} = 0.8, y_{B_{01}}^{*} = 0.4, x_{B_{02}}^{*} = 0.75, x_{B_{02}}^{*} = 0.4 .$

Then following Equations (25–27), we have D (B₁) = 0.985, D (B₂) = 0.894, D (B₀₁) = 0.894, and D (B₀₂) = 0.85. Obviously, D (B₀₁) > D (B₀₂) and D (B₁) > D (B₂), so MZ₁ > MZ₂.

6 Conclusions

This paper proposed the concept of multidimensional Z-number that considers the multiple dimensions of real world problems. The multidimensional Z-number can be used to evaluate a certain phenomenon from multiple dimensions and provide reliability measure of comprehensive evaluation. To gather numerous unidimensional Z-number information into multidimensional Z-number information, we defined the basic operations of multidimensional Z-number, based on concepts of set theory and probability theory. The defined operations can be used to handle information collection under different requirements and obtain the desired multidimensional Z-number through repeated calculation. We also suggested a feasible comparison method that is applicable to multidimensional Z-numbers with the same or similar constraint vectors for ranking multidimensional Z-numbers. The proposed ranking method is more effective and convenient because it only needs to consider the reliability measure. In this framework, we can deal with more complex information, and we extended the practical application of Z-number considerably.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 71871228).

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The arithmetic of multidimensional Z-number

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Multidimensional Z-number

3.1 Concept of multidimensional Z-number

3.2 Basic operations of restriction vectors and operations of probable events

4.1 Addition

Table 1 Membership degrees of G1+2

Table 3 Membership degrees of G1-2

Table 5 Membership degrees of G1·2

Table 7 Membership degrees of G3

6 Conclusions

Conflict of interests

Footnotes

Acknowledgments

References

Table 1
Membership degrees of G₁₊₂

Table 3
Membership degrees of G_1-2

Table 5
Membership degrees of G_1·2

Table 7
Membership degrees of G₃