Normal distribution interval numbers-based multiple attribute decision-making with unknown attribute weights

Abstract

With the rapid development of the society, decision-making methods with single attribute are becoming increasingly difficult to satisfy the needs of practical problems. And in real life, it is usually only possible to obtain the range of attribute values because of the measurement error or the complexity and uncertainty of objective things, that is, the interval number values of decision information are easily obtainable. Interval number multi-attribute decision-making problems have gradually become a research hotspot for scholars. However, most current studies have not discussed the distribution on interval numbers, resulting in partial loss of decision information. Therefore, this paper considers the case of normal distribution interval numbers (NDINs) and proposes a multi-attribute decision-making method based on symmetric relative entropy with completely unknown attribute weights. Firstly, based on the idea of maximal deviation, a model for determining attribute weights is established by the symmetric relative entropy of NDINs. Secondly, the possibility degree formula for comparison between two NDINs is used to determine the positive ideal solution (PIS) and negative ideal solution (NIS) under each attribute. Thirdly, the total closeness degree between each scheme and the ideal point is calculated using the TOPSIS method and symmetric relative entropy. Then, the ranking of schemes is obtained based on the total closeness degree. Finally, a numerical example is offered to prove the feasibility and validity of the proposed method.

Keywords

Normal distribution interval number symmetrical relative entropy closeness degree TOPSIS method

1 Introduction

In real life, decision making commonly exists in various fields such as economy, education, management and military. With the development of society, decision-making methods with single attribute are becoming increasingly difficult to satisfy the needs of practical problems. Hence, multi-attribute decision-making emerges as time goes on [11, 37]. The multi-attribute decision-making problem is a decision-making problem that involves selecting the optimal scheme and ranking schemes while considering multiple attributes [17, 42]. However, in most cases, it is often difficult to accurately determine the parameters related to decision-making because of the measurement error or the complexity and uncertainty of objective things, as well as the finiteness and fuzziness of people’s cognition. Generally, only the upper and lower bounds of parameters can be given, that is, the interval values of parameters are easily obtainable. Therefore, using interval numbers to analyze practical problems is easier to reveal the complexity and uncertainty of objective things, and is more in line with people’s fuzzy thinking habits [7]. Interval number multi-attribute decision-making problems have gradually become a research hotspot for scholars.

At present, research on multi-attribute decision-making problems with interval numbers has attracted wide attention and achieved some results. Yu et al. [5] analyzed the evolutionary process in intuitionistic fuzzy set theory from different perspectives and identified important themes, showing the research topic has gradually translated from theorical construction to practical application vividly. Huang et al. [13] proposed a multi-attribute decision-making method based on dual reference points for multi-attribute decision-making problems where attribute weights are completely unknown. Jiang et al. [14] proposed a decision method based on TOPSIS method and weighted parameters for multi-attribute decision-making problems with completely unknown attribute weights. After the successful application of entropy theory in decision analysis, Feng et al. [32], Xiao [3], Zhao [21] and Meng [41] used entropy to determine attribute weights. Liu et al. [34] established an attribute weight determination model based on the principle of minimum relative entropy to solve the attribute weights. Yu et al. [6] explored the development trends of fuzzy theory by analyzing the geographic distribution of publications, international collaboration, research hot spot, subject categories and journals, and publication contributors.

In addition, the concept of interval numbers similarity was proposed to characterize the similarity between different interval numbers [22]. On this basis, Xu et al. [27] used the ratio of the intersection length and union length of two intervals to characterize the similarity of interval numbers. Chen et al. [28] provided a calculation formula for the similarity based on the mapping distance operator. Huang et al. [40] expressed the dispersion of a certain attribute by measuring the similarity between the interval values of each scheme, and then determined the attribute weights. Feng et al. [37] proposed a decision method based on the maximum deviation between interval numbers for multi-attribute group decision-making problems where experts’ preference information takes the form of interval numbers.

The TOPSIS method is a commonly used decision-making method for multi-attribute decision analysis. Its core is to determine the PIS and NIS and respectively calculate the distance between the schemes and the PIS, and the distance between the schemes and the NIS. Yu et al. [4] used a citation network technology named main path analysis to provide a historical perspective of TOPSIS. Currently, for the determination of PIS and NIS, Jahanshahloo et al. [10] classify interval number attributes into two categories: benefit type and cost type. For benefit attributes, PIS is defined by the maximum among the right endpoints of the interval numbers, and NIS is defined by the minimum among the left endpoints of the interval numbers. Ye et al. [8] considered that the PIS and NIS are still in the form of interval number. Ludmila et al. [2] treated an interval number as a whole and selected the PIS and NIS based on different interval number comparison methods. On this basis, Meng [41] applied the similarity theory of interval numbers to the evaluation of enterprise operational performance, and constructed a multi-attribute decision-making model based on TOPSIS method. Ling et al. [31] determined the attribute weights based on the idea of maximizing deviation and ranked the schemes based on TOPSIS. Yuan [19] presented a multi-attribute decision-making method to evaluate the risks of mandatory projects in intellectual property pledge financing. Li et al. [20] proposed an improved TOPSIS method with interval number for solving the decision-making problems.

There has been significant progress in the research on interval number decision-making problems, but most of the aforementioned methods have not taken into account the distribution of the interval numbers or simply assume that values are uniformly distributed within the interval number. However, in some problems, the values are concentrated in a certain area in the middle, while the values on both sides are less likely, that is, a situation similar to a normal distribution occurs, such as the distribution of students’ examination results, the life distribution of a species, and the height distribution of people. In recent years, the research on NDINs has attracted the attention of experts and scholars, and NDINs have been widely used in the field of multi-attribute decision-making problems. Therefore, Xu et al. [12] put forward the concept of NDINs. Then, Hu [33] proposed the corresponding multi-attribute economic evaluation method. Ding et al. [1] proposed a decision method based on inclusion degree and possibility degree for multi-attribute decision making with NDINs. Gong et al. [26] extended uniform distribution to general distribution and provided a definition of interval number similarity under general distribution, and then combined TOPSIS method to achieve ranking of schemes. Ding et al. [24] proposed a new similarity measure and distance measure based on the expectation and variance of NDINs. Fu et al. [29] proposed a risk decision-making method for the decision-making problems where the attribute values are NDINs. Gong et al. [25] used probability theory to construct a new calculation model, aiming at the non-uniform distribution of interval numbers to achieve the final ranking of interval numbers. Jiang et al. [15] used the probabilistic hesitant fuzzy set to adequately express the uncertainty of information, and proposed a decision-making method based on PHFS and the cloud model to tackle large group emergency decision-making. Liu et al. [32] used the line integral to define the distance between the intuitionistic fuzzy sets and introduced the accuracy function into the defined distance for evaluating the accuracy of distance. Chen et al. [39] proposed a multi-objective optimization-driven collective opinion generation approach that generalized the bi-objective optimization-based PDF aggregation paradigm. Chen et al. [38] designed a refined assessment system for the maturity measurement of BIM-based projects during the design and construction stages. Harish Garg [9] provided one of the extensions of the single-valued neutrosophic set by utilizing the features of the exponential and logarithmic parameters, and used exponential entropy measures to obtain the attribute weights.

However, most of the above literature have not been mature nor perfectly optimized. They simply use the expectation and variance of NDINs in the decision-making process, and seldom make full use of the information of normal distribution to obtain the ranking results of schemes. They do not fully utilize decision information, resulting in inaccurate final results. Therefore, this paper proposes a decision method based on the closeness degree for NDIN-based multiple attribute decision-making problems. The main contributions and innovations of this paper can be summarized as follows.

We propose a new symmetric relative entropy to describe the difference between NDINs.

We construct a model for determining attribute weights based on the symmetric relative entropy.

We propose a method to determine the PIS and NIS under each attribute.

The remainder is composed of six sections. Section 2 reviews some basic concepts of NDIN. Section 3 provides a new formula for measuring the similarity of NDINs. Section 4 provides a closeness degree based on TOPSIS method. In Section 5, an illustrative example is used to demonstrate the operability of the proposed method. Section 6 conducts a comparative analysis. Finally, conclusion is provided in Section 7.

2 Preliminaries

2.1 Interval number

Definition 1 [23] Let R be a set of real numbers, $\tilde{a} = [a^{L}, a^{U}]$ be an interval number, where a^L ≤ a^U, and a^L, a^U ∈ R. In addition, if a^L = a^U, $\tilde{a}$ will degenerate into a certain number.

Let $\tilde{a} = [a^{L}, a^{U}]$ and $\tilde{b} = [b^{L}, b^{U}]$ be two interval numbers, and the constant β ≥ 0, then

$\tilde{a} = \tilde{b}$ if and only if a^L = b^L and a^U = b^U;

If a^L ≥ b^U, the interval number $\tilde{a}$ is called to be better than $\tilde{b}$ , which is denoted as $\tilde{a} > \tilde{b}$ ;

$\tilde{a} + \tilde{b} = [a^{L} + b^{L}, a^{U} + b^{U}]$ ;

$β \tilde{a} = [β a^{L}, β a^{U}]$ , β ≥ 0. Especially, $β \tilde{a} = 0$ if β = 0.

2.2 Normal distribution interval number

However, in some problems, the values are concentrated in a certain area in the middle, while the values on both sides are less likely, that is, a situation similar to a normal distribution occurs, such as the distribution of test scores or the distribution of people’s height.

For general interval numbers, all values in [a^L, a^U] are equally possible to be obtained, that is, the values are uniformly distributed between a^L and a^U. However, not all problems follow a uniform situation. Sometimes, there may be situations where values are concentrated in a certain interval in the middle, while the values on both sides are less likely, similar to a normal distribution. Naturally, this paper considers the decision problems of NDINs, which can be denoted as η_a ∈ [a^L, a^U] and $η_{a} \sim N (μ_{a}, σ_{a}^{2})$ . In fact, there is a certain contradiction between interval numbers and normal distribution, that is, interval numbers are usually bounded, but the values of NDINs are all real numbers. In order to meet the characteristics of general interval numbers and combine the excellent properties of NDINs, this paper adopts the 3σ principle of normal distribution [18] to maximize the inclusion of the entire interval numbers, that is P {|η_a - μ_a|≤3σ_a} = P {μ_a - 3σ_a ≤ η_a ≤ 3σ_a + μ_a} ≈0.9973. Let a^L = μ_a - 3σ_a and a^U = μ_a + 3σ_a, so that the general interval number [a^L, a^U] and the binary group (μ_a, σ_a) can be uniquely determined from each other.

Definition 2 [16] Let $\tilde{a} = [a^{L}, a^{U}]$ be the general interval number on the real number set, then $N_{\tilde{a}} = (μ_{a}, σ_{a})$ is the NDIN, where $μ_{a} = (a^{L} + a^{U}) / 2, σ_{a} = (a^{U} - a^{L}) / 6$

2.3 Possibility degree for NDIN

Theorem 1 [12] Assume that the random variables A and B take values separately from the intervals [a^L, a^U] and [b^L, b^U], and they follow a normal distribution, i.e., $A \sim N (μ_{a}, σ_{a}^{2})$ , $B \sim N (μ_{b}, σ_{b}^{2})$ , then $P (A > B) = Φ (\frac{μ_{a} - μ_{b}}{\sqrt{σ_{a}^{2} + σ_{b}^{2}}})$ (1)

where Φ (·) is the distribution function of the standard normal distribution. Hence, P (A > B) is called as the possibility degree of A > B.

According to the above theorem, the possibility degree formula holds for the following conclusions:

0 ≤ P (A ≥ B) ≤1;

Complementarity: P (A ≥ B) + P (B ≥ A) =1. Especially, $P (A \geq A) = \frac{1}{2}$ ;

Transitivity: P (A ≥ C) >0.5 if P (A ≥ B) >0.5 and P (B ≥ C) >0.5.

3 Symmetric relative entropy of NDINs

Definition 3 [30] Let f (x) and g (x) be two probability density functions on R, then the relative entropy of f (x) relative to g (x) is defined as $D (f ∥ g) = \int_{- \infty}^{+ \infty} f (x) log \frac{f (x)}{g (x)} dx$

Remark: $0 log \frac{0}{0} = 0$ , $0 log \frac{0}{q} = 0$ and $p log \frac{p}{0} = \infty$ .

The relative entropy D (f ∥ g) satisfies the following properties:

Nonnegativity: D (f ∥ g) ≥0;

The relative entropy reaches the minimum value of 0 if and only if the two probability distributions are equal;

Asymmetry: D (f ∥ g) ≠ D (g ∥ f).

Definition 4. Let f (x) and g (x) be two probability density functions on R, then the symmetric relative entropy of f (x) relative to g (x) is defined as $R (f, g) = \frac{1}{2} D (f ∥ g) + \frac{1}{2} D (g ∥ f)$ (2) Here, $D (f ∥ g) = \int_{- \infty}^{+ \infty} f (x) log \frac{f (x)}{g (x)} dx$ $D (g ∥ f) = \int_{- \infty}^{+ \infty} g (x) log \frac{g (x)}{f (x)} dx$

The symmetric relative entropy R (f, g) satisfies the following properties:

Nonnegativity: R (f, g) ≥0;

Identity: R (f, g) =0 if and only if for almost everywhere x ∈ R, f (x) = g (x), that is, the symmetric relative entropy reaches the minimum value of 0 if and only if the two probability distributions are equal;

Symmetry: R (f, g) = R (g, f).

Proof Property (3) can be easily proven by Definition 4. We prove properties (1) and (2) in the following.

(1) Let the probability density function of the random variable X be f (x), and then according to Jensen’s inequality:

$\begin{matrix} D (f ∥ g) & = \int_{- \infty}^{+ \infty} f (x) log \frac{f (x)}{g (x)} dx \\ = - E [log \frac{g (X)}{f (X)}] \\ \geq - log E [\frac{g (X)}{f (X)}] \\ = - log [\int_{- \infty}^{+ \infty} f (x) \frac{g (x)}{f (x)} dx] \\ = - log [\int_{- \infty}^{+ \infty} g (x)] dx \\ = 0 \end{matrix}$

Therefore, it can be concluded that $D (f ∥ g) \geq 0$

Similarly, it can be inferred that $D (g ∥ f) \geq 0$

Therefore, $R (f, g) = \frac{1}{2} D (f ∥ g) + \frac{1}{2} D (g ∥ f) \geq 0$

Proof is completed.

(2) Sufficiency is evident;

The following proves the necessity:

From property (1), if R (f, g) =0, we know: $D (f ∥ g) = D (f ∥ g) = 0$

Since D (f ∥ g) =0, it can be concluded that: $- E [log \frac{g (X)}{f (X)}] = - log [E (\frac{g (X)}{f (X)})]$

- log x is a strictly convex function, so for almost everywhere x ∈ R, $\frac{g (x)}{f (x)} \equiv C, a . s .$

Here, C is a constant.

Due to the fact that both f (x) and g (x) are probability density functions, we have: $\int_{- \infty}^{+ \infty} f (x) dx = 1$ $\begin{matrix} \int_{- \infty}^{+ \infty} g (x) dx & = \int_{- \infty}^{+ \infty} Cf (x) dx \\ = C \int_{- \infty}^{+ \infty} f (x) dx \\ = 1 \end{matrix}$

We obtain C = 1, that is, for almost everywhere x ∈ R, f (x) = g (x).

Therefore, $R (f, g) = 0 \Leftrightarrow f (x) = g (x), a . s .$

Proof is completed.

From the above properties, it can be seen that when x ∈ R, f (x) and g (x) are almost equal everywhere, their symmetric relative entropy reaches the minimum. Therefore, symmetric relative entropy can be used to measure the difference or degree of agreement between two probability distributions.

Proposition 1. Let f (x) and g (x) be probability density functions of normal distributions $N (μ_{1}, σ_{1}^{2})$ and $N (μ_{2}, σ_{2}^{2})$ respectively, then the symmetric relative entropy of f (x) relative to g (x) is $\begin{matrix} R (f, g) = \frac{1}{4} [{(\frac{σ_{1}}{σ_{2}} - \frac{σ_{2}}{σ_{1}})}^{2} \\ + (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}}) (μ_{1} - μ_{2})^{2}] \end{matrix}$ (3)

In the following, we give the detailed calculation about Equation (3):

$\begin{matrix} D (f ∥ g) = \int_{- \infty}^{+ \infty} f (x) \log \frac{f (x)}{g (x)} dx \\ = \int_{- \infty}^{+ \infty} \frac{1}{\sqrt{2 π} σ_{1}} e^{- \frac{(x - μ_{1})^{2}}{2 σ_{1}^{2}}} log \frac{\frac{1}{\sqrt{2 π} σ_{1}} e^{- \frac{(x - μ_{1})^{2}}{2 σ_{1}^{2}}}}{\frac{1}{\sqrt{2 π} σ_{2}} e^{- \frac{(x - μ_{2})^{2}}{2 σ_{2}^{2}}}} dx \\ = \int_{- \infty}^{+ \infty} \frac{1}{\sqrt{2 π} σ_{1}} e^{- \frac{(x - μ_{1})^{2}}{2 σ_{1}^{2}}} [log \frac{σ_{2}}{σ_{1}} - \frac{(x - μ_{1})^{2}}{2 σ_{1}^{2}} \\ + \frac{(x - μ_{2})^{2}}{2 σ_{2}^{2}}] dx \\ = log \frac{σ_{2}}{σ_{1}} \int_{- \infty}^{+ \infty} f (x) dx - \int_{- \infty}^{+ \infty} \frac{(x - μ_{1})^{2}}{2 σ_{1}^{2}} f (x) dx \\ + \int_{- \infty}^{+ \infty} \frac{(x - μ_{2})^{2}}{2 σ_{2}^{2}} f (x) dx \\ = log \frac{σ_{2}}{σ_{1}} - \frac{1}{2 σ_{1}^{2}} \int_{- \infty}^{+ \infty} (x - μ_{1})^{2} f (x) dx \\ + \frac{1}{2 σ_{2}^{2}} \int_{- \infty}^{+ \infty} (x - μ_{1} + μ_{1} - μ_{2})^{2} f (x) dx \\ = log \frac{σ_{2}}{σ_{1}} - \frac{1}{2 σ_{1}^{2}} \cdot σ_{1}^{2} + \frac{1}{2 σ_{2}^{2}} [\int_{- \infty}^{+ \infty} (x - μ_{1})^{2} f (x) dx \\ + \int_{- \infty}^{+ \infty} (μ_{1} - μ_{2})^{2} f (x) dx \\ + \int_{- \infty}^{+ \infty} 2 (x - μ_{1}) (μ_{1} - μ_{2}) f (x) dx] \\ = log \frac{σ_{2}}{σ_{1}} - \frac{1}{2} + \frac{1}{2 σ_{2}^{2}} [σ_{1}^{2} + (μ_{1} - μ_{2})^{2}] \end{matrix}$

Therefore, $D (f ∥ g) = log \frac{σ_{2}}{σ_{1}} + \frac{σ_{1}^{2} + (μ_{1} - μ_{2})^{2}}{2 σ_{2}^{2}} - \frac{1}{2}$

Similarly, we also get $D (g ∥ f) = log \frac{σ_{1}}{σ_{2}} + \frac{σ_{2}^{2} + (μ_{1} - μ_{2})^{2}}{2 σ_{1}^{2}} - \frac{1}{2}$

As a result, $\begin{matrix} R (f, g) = \frac{1}{2} D (f ∥ g) + \frac{1}{2} D (g ∥ f) \\ = \frac{1}{4} [{(\frac{σ_{1}}{σ_{2}} - \frac{σ_{2}}{σ_{1}})}^{2} \\ + (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}}) (μ_{1} - μ_{2})^{2}] \end{matrix}$

Example 1: Using the TOPSIS method, we can calculate the closeness between interval number $\tilde{a} = [0.2, 0.725]$ and positive ideal point A⁺ = [0.6, 0.9], negative ideal point A^- = [0.1, 0.3], and the closeness between interval number $\tilde{b} = [0.35, 0.6]$ and positive ideal point A⁺ = [0.6, 0.9], negative ideal point A^- = [0.1, 0.3] respectively.

(1) If Euclidean distance is used, then $d_{E} (\tilde{a}, A^{+}) = 0.4366, d_{E} (\tilde{a}, A^{-}) = 0.4366$ $d_{E} (\tilde{b}, A^{+}) = 0.3905, d_{E} (\tilde{b}, A^{-}) = 0.3905$

Here, $d_{E} (\tilde{a}, A^{+})$ is the Euclidean distance between the interval number $\tilde{a}$ and the positive ideal point A⁺. Others are similar.

The closeness values between interval numbers $\tilde{a}$ and A⁺, A^-, and the closeness values between $\tilde{b}$ and A⁺, A^- are obtained as follows:

$T_{E} (\tilde{a}) = \frac{d_{E} (\tilde{a}, A^{-})}{d_{E} (\tilde{a}, A^{+}) + d_{E} (\tilde{a}, A^{-})} = \frac{1}{2}$ $T_{E} (\tilde{b}) = \frac{d_{E} (\tilde{b}, A^{-})}{d_{E} (\tilde{b}, A^{+}) + d_{E} (\tilde{b}, A^{-})} = \frac{1}{2}$

Here, T_E (·) denotes the closeness by Euclidean distance.

(2) If we use the symmetric relative entropy given in this paper, then $d_{R} (\tilde{a}, A^{+}) = 11.3119, d_{R} (\tilde{a}, A^{-}) = 19.0128$ $d_{R} (\tilde{b}, A^{+}) = 18.4861, d_{R} (\tilde{b}, A^{-}) = 27.95625$

Here, $d_{R} (\tilde{a}, A^{+})$ is the symmetric relative entropy between the interval number $\tilde{a}$ and positive ideal point A⁺. Others are similar.

The closeness values between interval numbers $\tilde{a}$ and A⁺, A^-, and the closeness values between $\tilde{b}$ and A⁺, A^- are obtained as follows:

$T_{R} (\tilde{a}) = \frac{d_{R} (\tilde{a}, A^{-})}{d_{R} (\tilde{a}, A^{+}) + d_{R} (\tilde{a}, A^{-})} = 0.6270$ $T_{R} (\tilde{b}) = \frac{d_{R} (\tilde{b}, A^{-})}{d_{R} (\tilde{b}, A^{+}) + d_{R} (\tilde{b}, A^{-})} = 0.6020$

Here, T_R (·) denotes the closeness by symmetric relative entropy.

Therefore, using symmetric relative entropy to measure the degree of difference between interval numbers has a higher degree of discrimination than that by Euclidean distance, which can make a better comparison between interval numbers.

4 NDIN-based decision-making method

The description on interval number multi-attribute decision-making problems with completely unknown attribute weights is as follows: let Y = {Y₁, Y₂, ⋯ , Y_m} be the scheme set, G = {G₁, G₂, ⋯ , G_n} be the attribute set, and $A = ({\tilde{a}}_{ij})_{m \times n}$ be the interval number decision matrix. The decision maker’s preference for the scheme set under the attribute set takes the form of NDIN. The attribute weight vector is ω = (ω₁, ω₂, ⋯ , ω_n). Here, ω_j ∈ [0, 1], j = 1, 2, ⋯ , n, and $\sum_{j = 1}^{n} ω_{j} = 1$ .

In this paper, we use the symmetric relative entropy of NDINs to characterize the difference values between interval numbers, and establish a decision model to determine attribute weights. Then, based on the TOPSIS method, the total closeness degree of each scheme to PIS and NIS is calculated to achieve the final ranking of the schemes.

4.1 Normalization method for NDINs

Decision matrix $A = ({\tilde{a}}_{ij})_{m \times n}$ can be transformed into a decision matrix M = ((μ_ij, σ_ij)) _m×n with NDIN using Definition 2. To eliminate the impact of different dimensions on decision results, it is necessary to normalize the decision information [1]. Let the normalized matrix be R = ((u_ij, v_ij)) _m×n. Here,

(1) If G_j (1 ≤ j ≤ n) is a benefit attribute, then $u_{ij} = \frac{μ_{ij}}{{max}_{1 \leq i \leq m} μ_{ij}}, v_{ij} = \frac{σ_{ij}}{{max}_{1 \leq i \leq m} σ_{ij}}$ (4)

(2) If G_j (1 ≤ j ≤ n) is a cost attribute, then $u_{ij} = \frac{{min}_{1 \leq i \leq m} μ_{ij}}{μ_{ij}}, v_{ij} = \frac{{min}_{1 \leq i \leq m} σ_{ij}}{σ_{ij}}$ (5)

Here, u_ij is the normalized expected value, and v_ij is the normalized standard deviation.

4.2 Determination of attribute weights

If the difference in attribute values between all decision schemes under attribute G_j is smaller, it indicates that the attribute has a smaller impact on the decision-making results; On the contrary, if there is a significant difference in the attribute values for attribute G_j, it implies that it will play an important role in decision-making results, and G_j should be assigned a larger weight [37]. Especially, if there is no difference in the attribute values for attribute G_j, then the attribute G_j will have no effect on the ranking of schemes, and its weight can be set to 0.

Therefore, we utilize the idea of maximizing deviation method [37], which implies that the determination of attribute weight vector ω should maximize the total deviation between all decision schemes under all attributes, and we use the symmetric relative entropy of NDINs to construct a model for calculating attribute weights: $\begin{matrix} max D (ω) & = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj}) ω_{j}^{*} \\ = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} \frac{1}{4} [{(\frac{v_{ij}}{v_{kj}} - \frac{v_{kj}}{v_{ij}})}^{2} \\ + (\frac{1}{(v_{ij})^{2}} + \frac{1}{(v_{kj})^{2}}) (u_{ij} - u_{kj})^{2}] ω_{j}^{*} \end{matrix}$ (6) $s . t . \sum_{j = 1}^{n} (ω_{j}^{*})^{2} = 1, ω_{j}^{*} \geq 0, j = 1, 2, \dots, n$ where d (r_ij, r_kj) represents the symmetric relative entropy between the NDINs r_ij = (u_ij, v_ij) and r_kj = (u_kj, v_kj), $\begin{matrix} d (r_{ij}, r_{kj}) = \frac{1}{4} [{(\frac{v_{ij}}{v_{kj}} - \frac{v_{kj}}{v_{ij}})}^{2} \\ + (\frac{1}{(v_{ij})^{2}} + \frac{1}{(v_{kj})^{2}}) (u_{ij} - u_{kj})^{2}] \end{matrix}$

Then we construct a Lagrange function: $\begin{matrix} L (ω, λ) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj}) ω_{j}^{*} \\ + \frac{λ}{2} (\sum_{j = 1}^{n} (ω_{j}^{*})^{2} - 1) \end{matrix}$ (7)

Take the partial derivative of $ω_{j}^{*}$ , j = 1, 2, ⋯ , n and λ separately, and set the derivative to 0, that is: $\frac{\partial L (ω, λ)}{\partial ω_{j}^{*}} = \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj}) + λ ω_{j}^{*} = 0$ $\frac{\partial L (ω, λ)}{\partial λ} = \sum_{j = 1}^{n} (ω_{j}^{*})^{2} - 1 = 0, j = 1, 2, \dots, n$

Then, $ω_{j}^{*} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj})}{\sqrt{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj}))^{2}}}, j = 1, 2, \dots, n$ After normalizing $ω_{j}^{*}$ , we can obtain $ω_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj})}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (r_{ij}, r_{kj})}, j = 1, 2, \dots, n$ (8)

4.3 Calculation of relative closeness degree

Ludmila et al. [2] treated an interval number as a whole and selected the optimal and worst interval numbers based on different interval numbers comparison methods, so that both PIS and NIS can be found in the initial decision matrix. Therefore, motivated by the idea of Ludmila et al. [2], we provide the following definition for determining PIS and NIS of NDINs in our paper.

Definition 5 Let $A^{+} = {r_{1}^{+}, r_{2}^{+}, \dots, r_{n}^{+}}$ be the PIS, and $A^{-} = {r_{1}^{-}, r_{2}^{-}, \dots, r_{n}^{-}}$ be the NIS, where $\begin{matrix} A^{+} & = {(u_{1}^{+}, v_{1}^{+}), (u_{2}^{+}, v_{2}^{+}), \dots, (u_{n}^{+}, v_{n}^{+})} \\ = {(max_{i} (u_{ij}, v_{ij}) | j \in K_{b}), (min_{i} (u_{ij}, v_{ij}) | j \in K_{c})} \end{matrix}$ (9) $\begin{matrix} A^{-} & = {(u_{1}^{-}, v_{1}^{-}), (u_{2}^{-}, v_{2}^{-}), \dots, (u_{n}^{-}, v_{n}^{-})} \\ = {(min_{i} (u_{ij}, v_{ij}) | j \in K_{b}), (max_{i} (u_{ij}, v_{ij}) | j \in K_{c})} \end{matrix}$ (10) Here, K_b represents the set of benefit attributes, and K_c represents the set of cost attributes. Specifically, the steps to obtain the PIS and NIS for the NDINs are as follows:

According to the possibility degree given in Equation (1), we construct a possibility matrix H^j by pairwise comparing the NDINs between different schemes under attribute G_j, $H_{pq}^{j} = Φ (\frac{μ_{pj} - μ_{qj}}{\sqrt{σ_{pj}^{2} + σ_{qj}^{2}}})$

Here, Φ (·) is the distribution function of the standard normal distribution. $H_{pq}^{j} \in H^{j}$ is the possibility degree of NDIN(μ_pj, σ_pj) > NDIN(μ_qj, σ_qj), where μ_pj and σ_pj respectively represent the mean and standard deviation of the NDIN (μ_pj, σ_pj).

We add each row of the possibility degree matrix H^j to obtain the total possibility degree, i.e., $H_{p \cdot}^{j} = \sum_{q} H_{pq}^{j}$

Finally, the PIS and NIS for the NDINs under attribute G_j are determined based on the total possibility degree $H_{p \cdot}^{j}$ . $PIS = (μ_{pj}, σ_{pj}) \Leftrightarrow max_{i \in [1, m]} H_{i \cdot}^{j} = H_{p \cdot}^{j}$ $NIS = (μ_{qj}, σ_{qj}) \Leftrightarrow min_{i \in [1, m]} H_{i \cdot}^{j} = H_{q \cdot}^{j}$

The difference value between scheme Y_i and PIS and the difference value between scheme Y_i and NIS under attribute G_j are calculated respectively using the symmetric relative entropy:

$\begin{matrix} d (r_{ij}, r_{j}^{+}) = \frac{1}{4} [{(\frac{v_{ij}}{v_{j}^{+}} - \frac{v_{j}^{+}}{v_{ij}})}^{2} \\ + (\frac{1}{(v_{ij})^{2}} + \frac{1}{(v_{j}^{+})^{2}}) {(u_{ij} - u_{j}^{+})}^{2}] \end{matrix}$

$\begin{matrix} d (r_{ij}, r_{j}^{-}) = \frac{1}{4} [{(\frac{v_{ij}}{v_{j}^{-}} - \frac{v_{j}^{-}}{v_{ij}})}^{2} \\ + (\frac{1}{(v_{ij})^{2}} + \frac{1}{(v_{j}^{-})^{2}}) {(u_{ij} - u_{j}^{-})}^{2}] \end{matrix}$

Here, r_ij is the NDIN (u_ij, v_ij), $r_{j}^{+}$ is the positive ideal point $(u_{j}^{+}, v_{j}^{+})$ under attribute G_j, and $r_{j}^{-}$ is the negative ideal point $(u_{j}^{-}, v_{j}^{-})$ under attribute G_j.

Furthermore, under attribute G_j, the relative closeness degree between scheme Y_i and the ideal solutions is $T_{ij} = \frac{d (r_{ij}, r_{j}^{-})}{d (r_{ij}, r_{j}^{+}) + d (r_{ij}, r_{j}^{-})}$ (11)

Then the total closeness degree between scheme Y_i and the ideal solutions can be obtained, that is,

$T_{i} = \sum_{j = 1}^{n} T_{ij} ω_{j}$ (12)

The schemes can be ranked by T_i. Therefore, the larger the total closeness degree T_i, the better the scheme.

4.4 Decision steps

Based on the above analysis, a NDINs multi-attribute decision-making method with completely unknown attribute weights is proposed. The specific steps are as follows:

We transform the initial decision matrix $A = ({\tilde{a}}_{ij})_{m \times n}$ into a decision matrix M = ((μ_ij, σ_ij)) _m×n with NDIN using Definition 2, and then normalize decision matrix M = ((μ_ij, σ_ij)) _m×n to obtain the decision matrix R = ((u_ij, v_ij)) _m×n.

According to the Equations (6) and (7), we obtain the attribute weight vector ω.

According to Equation (1), we establish a possibility degree matrix, and then we determine the PIS and NIS under each attribute.

We calculate the total closeness degree of each scheme to the ideal solutions according to Equations (10) and (11). Then the ranking of the schemes can be determined.

5 An illustrative example

An example adopted from Meng [41] is used to evaluate the operational performance of five enterprises (Y_i, i = 1, 2, 3, 4, 5) from five attributes: financial management ability(G₁), operational ability(G₂), risk control ability(G₃), development ability(G₄) and environmental participation ability(G₅). The interval numbers evaluation matrix of five enterprises under five attributes is shown in Table 1.

Table 1
Initial Decision Matrix A

G ₁ G ₂ G ₃ G ₄ G ₅

Y ₁ [87,89] [60,63] [83,86] [64,68] [76,79]

Y ₂ [92,94] [70,73] [57,62] [62,66] [84,88]

Y ₃ [82,83] [69,72] [70,74] [59,63] [63,67]

Y ₄ [88,93] [87,89] [96,97] [50,52] [82,85]

Y ₅ [92,97] [81,85] [59,64] [73,76] [75,79]

	G ₁	G ₂	G ₃	G ₄	G ₅
Y ₁	[87,89]	[60,63]	[83,86]	[64,68]	[76,79]
Y ₂	[92,94]	[70,73]	[57,62]	[62,66]	[84,88]
Y ₃	[82,83]	[69,72]	[70,74]	[59,63]	[63,67]
Y ₄	[88,93]	[87,89]	[96,97]	[50,52]	[82,85]
Y ₅	[92,97]	[81,85]	[59,64]	[73,76]	[75,79]

Step 1. According to Definition 2, we convert the data in Table 1 into NDINs, and the results are shown in Table 2.

Table 2

Normal Distribution Interval Number Matrix M

	G ₁	G ₂	G ₃	G ₄	G ₅
Y ₁	(88,0.333)	(61.5,0.5)	(84.5,0.5)	(66,0.667)	(77.5,0.5)
Y ₂	(93,0.333)	(71.5,0.5)	(59.5,0.833)	(64,0.667)	(86,0.667)
Y ₃	(82.5,0.167)	(70.5,0.5)	(72,0.667)	(61,0.667)	(65,0.667)
Y ₄	(90.5,0.833)	(88,0.333)	(96.5,0.167)	(51,0.333)	(83.5,0.5)
Y ₅	(94.5,0.833)	(83,0.667)	(61.5,0.833)	(74.5,0.5)	(77,0.667)

Due to the fact that all attributes are benefit types, and according to Equation (4), we obtain the normalized matrix R shown in Table 3.

Table 3

Normalized Decision Matrix R

	G ₁	G ₂	G ₃	G ₄	G ₅
Y ₁	(0.931,0.4)	(0.699,0.75)	(0.876,0.6)	(0.886,1)	(0.901,0.75)
Y ₂	(0.984,0.4)	(0.8125,0.75)	(0.617,1)	(0.859,1)	(1,1)
Y ₃	(0.873,0.2)	(0.801,0.75)	(0.746,0.8)	(0.8188,1)	(0.756,1)
Y ₄	(0.958,1)	(1,0.5)	(1,0.2)	(0.685,0.5)	(0.971,0.75)
Y ₅	(1,1)	(0.943,1)	(0.637,1)	(1,0.75)	(0.895,1)

Step 2. According to the Equations (6) and (7), we obtain the attribute weight vector ω.

Firstly, the total difference values under each attribute can be obtained by using the symmetric relative entropy of NDINs. $\begin{matrix} \sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{i 1}, r_{k 1}) = 34.701 \\ \sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{i 2}, r_{k 2}) = 3.339 \\ \sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{i 3}, r_{k 3}) = 40.137 \\ \sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{i 4}, r_{k 4}) = 4.841 \\ \sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{i 5}, r_{k 5}) = 1.236 \end{matrix}$

According to Equation (7), we derive $ω_{j} = \frac{\sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{ij}, r_{kj})}{\sum_{j = 1}^{5} \sum_{i = 1}^{5} \sum_{k = 1}^{5} d (r_{ij}, r_{kj})}, j = 1, 2, 3, 4, 5$

As a result, we obtain ω = (0.41, 0.04, 0.48, linebreak0.06, 0.01).

Step 3. We determine the positive and negative ideal solutions under each attribute. Firstly, we use Equation (1) to obtain the possibility degree matrixes. $H^{1} = [\begin{matrix} 0.5 & 0.4627402 & 0.5517728 & 0.4902018 & 0.4745395 \\ 0.5372598 & 0.5 & 0.5981076 & 0.5097982 & 0.4941207 \\ 0.4482272 & 0.4018924 & 0.5 & 0.466921 & 0.4504525 \\ 0.5097982 & 0.4902018 & 0.533079 & 0.5 & 0.4880613 \\ 0.5254605 & 0.5058793 & 0.5495475 & 0.5119387 & 0.5 \end{matrix}]$ $H^{2} = [\begin{matrix} 0.5 & 0.45734 & 0.4615921 & 0.3691593 & 0.4225186 \\ 0.54266 & 0.5 & 0.5042741 & 0.4176096 & 0.4583682 \\ 0.5384079 & 0.4957259 & 0.5 & 0.4126944 & 0.454763 \\ 0.6308407 & 0.5823904 & 0.5873056 & 0.5 & 0.5202654 \\ 0.5774814 & 0.5416318 & 0.545237 & 0.4797346 & 0.5 \end{matrix}]$ $H^{3} = [\begin{matrix} 0.5 & 0.5879008 & 0.5515323 & 0.4220631 & 0.5809704 \\ 0.4120992 & 0.5 & 0.4597162 & 0.3534681 & 0.4941537 \\ 0.4484677 & 0.5402838 & 0.5 & 0.3790861 & 0.5338554 \\ 0.5779369 & 0.6465319 & 0.6209139 & 0.5 & 0.638949 \\ 0.4190296 & 0.5058463 & 0.4661446 & 0.361051 & 0.5 \end{matrix}]$ $H^{4} = [\begin{matrix} 0.5 & 0.5075726 & 0.5189254 & 0.5714575 & 0.463637 \\ 0.4924274 & 0.5 & 0.511358 & 0.5620129 & 0.4551137 \\ 0.4810746 & 0.488642 & 0.5 & 0.5477811 & 0.4423687 \\ 0.4285425 & 0.4379871 & 0.4522189 & 0.5 & 0.36319 \\ 0.536363 & 0.5448863 & 0.5576313 & 0.63681 & 0.5 \end{matrix}]$ $H^{5} = [\begin{matrix} 0.5 & 0.4684886 & 0.5462843 & 0.4737775 & 0.5018555 \\ 0.5315114 & 0.5 & 0.5685429 & 0.5092769 & 0.5294946 \\ 0.4537157 & 0.4314571 & 0.5 & 0.4316822 & 0.4607017 \\ 0.5262225 & 0.4907231 & 0.5683178 & 0.5 & 0.5241074 \\ 0.4981445 & 0.4705054 & 0.5392983 & 0.4758926 & 0.5 \end{matrix}]$

Here, H^j (j = 1, 2, 3, 4, 5) represents the possibility degree matrix. Specifically, $H_{pq}^{j} \in H^{j}$ is the possibility degree of NDIN(μ_pj, σ_pj) > NDIN (μ_qj, σ_qj).

Then we calculate the sum of each row of the possibility degree matrix. For the financial management ability(G₁): $\begin{matrix} H_{1 \cdot}^{1} & = 2.4792543, H_{2 \cdot}^{1} = 2.6392863, \\ H_{3 \cdot}^{1} & = 2.2674931, H_{4 \cdot}^{1} = 2.5211403, \\ H_{5 \cdot}^{1} & = 2.592826 . \end{matrix}$

It can be obtained that the maximum value is $H_{2 \cdot}^{1}$ , and the minimum value is $H_{3 \cdot}^{1}$ . Therefore, for the financial management ability(G₁), the PIS is the NDIN (0.984, 0.4), and the NIS is the NDIN (0.873, 0.2). Similarly, PIS and NIS under other attributes can be obtained. Let A⁺ be the positive ideal solution, and let A^- be the negative ideal solution. A⁺ and A^- are as follows:

$A^{+} = {(0.984, 0.4), (1.0.5), (1, 0.2), (1, 0.75), (1, 1)}$ $\begin{matrix} A^{-} = { & (0.873, 0.2), (0.699, 0.75), (0.617, 1), (0.685, 0.5), \\ (0.756, 1)} \end{matrix}$ Step 4. We calculate the total closeness degree of each scheme and rank the enterprises.

Firstly, we use symmetric relative entropy to calculate the difference values matrix between each enterprise and PIS, and between each enterprise and NIS under each attribute.

$D^{+} = [\begin{matrix} 0.008748335 & 0.304597825 & 1.885163211 & 0.094109328 & 0.091853329 \\ 0 & 0.224392361 & 6.715569277 & 0.098863868 & 0 \\ 0.658950617 & 0.230734188 & 3.943667112 & 0.107872471 & 0.029813413 \\ 1.103768512 & 0 & 0 & 0.317333366 & 0.085656286 \\ 1.102956664 & 0.566535382 & 6.615056512 & 0 & 0.005475933 \end{matrix}]$ $D^{-} = [\begin{matrix} 0.588963775 & 0 & 0.347831679 & 0.613173393 & 0.099740475 \\ 0.658950617 & 0.011478421 & 0 & 0.098863868 & 0.029813413 \\ 0 & 0.009297521 & 0.06137403 & 0.585021508 & 0 \\ 1.103768512 & 0.304597825 & 6.715569277 & 0 & 0.117204871 \\ 5.864812295 & 0.126521787 & 0.000214771 & 0.317333366 & 0.009734992 \end{matrix}]$

Here, D⁺ is the difference values matrix between each scheme and the PIS, and D^- is the difference values matrix between each scheme and the NIS. The relative closeness degree matrix is obtained by using Equation (11).

$T = [\begin{matrix} 0.985363598 & 0 & 0.155769134 & 0.86694242 & 0.520582989 \\ 1 & 0.048664021 & 0 & 0.5 & 1 \\ 0 & 0.038734552 & 0.015324195 & 0.84431605 & 0 \\ 0.5 & 1 & 1 & 0 & 0.577759058 \\ 0.841705908 & 0.182556062 & 3.24659 \times 10^{- 5} & 1 & 0.64 \end{matrix}]$

Finally, the total closeness degree of each enterprise can be calculated according to Equation (12). $\begin{matrix} T_{1} & = 0.535990634, T_{2} = 0.451946561, \\ T_{3} & = 0.059563959, T_{4} = 0.730777591, \\ T_{5} & = 0.418817248 . \end{matrix}$

Therefore, there is T₄ > T₁ > T₂ > T₅ > T₃, that is, the optimal enterprise is Y₄.

6 Comparison and discussion

To illustrate the effectiveness of the proposed method, this section provides the comparative analyses with other methods. The final ranking results obtained are shown in Fig. 1, and the specific values are listed in Table 4.

Fig. 1

Comparison of results of multiple methods.

Table 4

Decision results of different methods

Methods	Y ₁	Y ₂	Y ₃	Y ₄	Y ₅	Ranking results
Meng’s method [38]	0.673	0.712	0.601	0.808	0.502	Y₄ > Y₂ > Y₁ > Y₃ > Y₅
Xu’s method [9]	0.228	0.137	0.113	0.3	0.222	Y₄ > Y₁ > Y₅ > Y₂ > Y₃
Gong’s method [23]	0	0.334	0	0.879	0.345	Y₄ > Y₅ > Y₂ > Y₁ = Y₃
The proposed method	0.536	0.452	0.060	0.731	0.419	Y₄ > Y₁ > Y₂ > Y₅ > Y₃

From Fig. 1 and Table 4, it can be seen that among different methods, enterprise 4 ranks first in operational performance, and among most methods, enterprise 3 ranks last in operational performance. In addition, the ranking results of different methods are slightly different. By comparison, we find the reasons for the differences are as follows:

In terms of determining attribute weights, Meng [41] and Gong et al. [26] first used ordered weighting operators to convert interval numbers to real numbers, and then used entropy weight method to determine attribute weights. The idea used by Xu et al. [12] to solve attribute weights is the same as that by Meng [41], but in the calculation process, it is not necessary to convert interval numbers to real numbers. The attribute weights can be directly obtained using the initial interval number decision matrix. This paper uses the symmetric relative entropy between different schemes to establish an attribute weight determination model based on the idea of maximizing deviation. On the one hand, it continues the excellent property of higher resolution ratio of general relative entropy in comparison operations. On the other hand, it overcomes the shortcomings of relative entropy not being satisfied with symmetry and makes the obtained results more accurate.

In terms of the interval numbers’ distribution, Meng [41] did not consider the distribution of interval numbers and converted interval numbers to real numbers while calculating attribute weights. Gong et al. [26] considered the interval numbers follow a triangular distribution and used the triangular distribution to describe skewness and kurtosis of the distribution to present information about the decision makers’ risk preference. Xu et al. [12] considered the interval numbers follow a normal distribution, but only in the calculation of the possibility degree matrix, the information of the NDINs was utilized. This paper also considers that interval numbers follow a normal distribution, and utilizes the information of normal distribution in the decision-making process. Therefore, the utilization of interval information is more comprehensive, and the results obtained are more scientific and objective.

In terms of calculating interval numbers similarity, Meng [41] used the left and right endpoints of interval numbers to provide a formula for calculating interval number similarity, but its definition is not intuitive. Gong et al. [26] defined the similarity between two interval numbers as the probability that two random variables independently take values at the intersection of two interval numbers. This paper uses symmetric relative entropy to extend the calculation of interval number similarity to continuous variables, and provides a specific formula for calculating interval number similarity under normal distribution to improve the practicality of the method.

In addition, in our paper, we propose explicit expressions for solving attribute weights and comparing interval numbers. Therefore, when solving practical problems, the calculation is simple and there are no restrictions on initial and operational conditions. The method has a wide range of applications.

7 Conclusions

In practical economic and management problems, using interval numbers to characterize the uncertainty is an effective method that conforms to statistical principles and people’s cognitive fuzziness. The attribute features of objective things usually follow a certain statistical distribution, which contains the preferences and risk attitudes of decision-makers, as well as the uncertainty of objective things. The decision problems described by the NDINs is closer to real life and more widely used. Therefore, the research on multi-attribute decision-making methods based on the NDINs is particularly significant in decision-making. In this paper, we study the multi-attribute decision-making problem where the attribute values are interval numbers and the attribute weights are completely unknown, and propose a decision-making method based on interval numbers closeness degree. Firstly, based on the principle of 3σ, the correspondence between the general interval numbers and the NDINs is established. Then, based on the idea of maximizing deviation, a concise formula for determining attribute weights is given using the symmetric relative entropy. This method fully utilizes the distribution of the information on the interval numbers and is easy to calculate, which improves the reliability and objectivity of decision-making. Furthermore, the possibility degree formula for comparing the NDINs is used to determine the positive and negative ideal solutions, and the total closeness degree between each scheme and the ideal solutions is calculated based on the TOPSIS method, so as to obtain the final ranking of the schemes. The given example shows that the proposed method has achieved good evaluation results for decision-making problems.

However, there are some limitations for the proposed method. Firstly, the proposed method only considers the case where interval numbers follow a normal distribution. Secondly, this paper only provides a measure of interval number difference based on relative entropy. In the future, we will study the case where interval numbers follow other distributions. Additionally, we will explore and provide other forms about difference measurement of interval number. Moreover, we will consider the situation where there are multi-time points in the decision-making process, and extend the interval number multi-attribute decision-making method in a single time period to dynamic situations.

Footnotes

Acknowledgments

This work was supported by the National Social Science Foundation of China (No. 22BGL211), the Anhui Provincial Natural Science Foundation (No. 2108085MG240), the National Natural Science Foundation of China (No. 62072044) and Beijing Natural Science Foundation (No. 1202001).

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Normal distribution interval numbers-based multiple attribute decision-making with unknown attribute weights

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Interval number

2.2 Normal distribution interval number

2.3 Possibility degree for NDIN

4.1 Normalization method for NDINs

5 An illustrative example

Table 1 Initial Decision Matrix A G 1 G 2 G 3 G 4 G 5 Y 1 [87,89] [60,63] [83,86] [64,68] [76,79] Y 2 [92,94] [70,73] [57,62] [62,66] [84,88] Y 3 [82,83] [69,72] [70,74] [59,63] [63,67] Y 4 [88,93] [87,89] [96,97] [50,52] [82,85] Y 5 [92,97] [81,85] [59,64] [73,76] [75,79]

Footnotes

Acknowledgments

References

Table 1
Initial Decision Matrix A

G ₁ G ₂ G ₃ G ₄ G ₅

Y ₁ [87,89] [60,63] [83,86] [64,68] [76,79]

Y ₂ [92,94] [70,73] [57,62] [62,66] [84,88]

Y ₃ [82,83] [69,72] [70,74] [59,63] [63,67]

Y ₄ [88,93] [87,89] [96,97] [50,52] [82,85]

Y ₅ [92,97] [81,85] [59,64] [73,76] [75,79]