Dynamic nonlinear simplified neutrosophic sets for multiple-attribute group decision making

Abstract

In this paper, the concept of a dynamic nonlinear simplified neutrosophic set (DNSNS) is proposed for describing the real-time changing expert preference information. Furthermore, the DNSNS aggregation model and decision algorithm are provided to solve the actual multiple-attribute group decision making (MAGDM) problems. The basic notions, the similarity measure, the entropy measure, and the index of distance of DNSNS are presented first. Secondly, the univariate time series of DNSNS are projected into dynamic nonlinear simplified neutrosophic curves in three-dimensional space. The areas of the surface enclosed by the curves represent the variance among the DNSNSs. Thus, the DNSNS aggregation model is established correctly without preprocessing the original data. Afterward, the aggregation algorithm extended from the plant growth simulation algorithm (PGSA) is proposed for calculating the optimal aggregation preference curve and constructing the collective matrix. Additionally, a novel corresponding decision algorithm based on TOPSIS and projection theory is proposed for obtaining the overall ranking of alternatives in the actual MAGDM problem. Finally, a typical example is presented to illustrate the feasibility and effectiveness of the proposed model and algorithm.

1. Introduction

The introduction by Zadeh on Fuzzy Sets (FS) [1] created a new research field in describing and analyzing uncertain data using linguistic variables. Additionally, Atanassov further developed this theory by proposing Intuitionistic Fuzzy Sets (IFS) as well as Interval-Valued Intuitionistic Fuzzy Sets (IVIFS) [2,3], significantly improving upon its accuracy in representing uncertain data. In recent years there has been an abundance of effective aggregation operators designed to solve Multiple Attribute Group Decision Making problems using FSs, IFSs or IVIFs [4,5]. However due to factors such as time constraints, limited domain knowledge or complex decision environments, experts often express their preferences with uncertainty. Traditional methods that rely solely on membership functions are inadequate in accurately capturing these nuances [6]. As such various extensions to classical Fuzzy Set Theory have emerged.

• (1)The concept of neutrosophic sets (NS) [7] extends intuitionistic fuzzy logic by incorporating truth-membership degree T, indeterminacy-membership degree I, and falsity-membership degree F. This extension eliminates the limitations on membership and non-membership degrees, thereby significantly enhancing the descriptive capability of uncertain fuzzy information. As subclasses of NS, simplified neutrosophic sets (SNS) [8,9], single-valued neutrosophic sets (SVNS) [10,11], interval-valued neutrosophic sets (INS) [[12], [13], [14]], multivalued neutrosophic sets (MNS) [15], and neutrosophic cubic sets (NCS) [[16], [17], [18], [19]] have been gradually introduced.
• (2)The concept of hesitant fuzzy sets (HFS) [20], which extends the notion of fuzzy sets, has been proposed to capture situations where multiple membership functions are considered plausible. Various operators have been introduced for aggregating dual hesitant fuzzy sets [[21], [22], [23]], multi-granulation dual hesitant fuzzy rough sets [24], and type-2 hesitant fuzzy sets [25,26] that are derived from HFS.
• (3)Triangular intuitionistic fuzzy sets [27] and trapezoidal intuitionistic fuzzy sets [28] are incorporated into linear intuitionistic fuzzy logic, simplifying the aggregation process of fuzzy information. Unlike traditional fuzzy sets, the membership degree and non-membership degree of triangular/trapezoidal intuitionistic fuzzy sets can effectively represent linear continuous preference information.
• (4)The concept of fuzzy linguistic term sets (FLTS) [29,30] is derived from the introduction of linguistic variables in Refs. [[31], [32], [33]], which are closer to human language than fuzzy sets. FLTS are defined as an ordered and consecutive subset of the linguistic term set for a given variable. To enhance accuracy in computing with words, various linguistic models have been proposed as generalizations of FLTS, including type-2 fuzzy sets-based models [34], the linguistic 2-tuple model [35,36], and the proportional 2-tuple model [37,38].

The suit problems of previous fuzzy tools can be divided into quantitative situations (FS, IFS, IVIFS, NS,and HFS), linear situations (triangular intuitionistic fuzzy sets and trapezoidal intuitionistic fuzzy sets), and vague human language situations. In such a situation, we focus on the description method and aggregation model of dynamic nonlinear expert preference information. In this paper, a dynamic nonlinear simplified neutrosophic set (DNSNS) is proposed first. Additionally, as the theoretical bases of the aggregation model, the similarity measure, the entropy measure, and the index of distance of DNSNS are introduced. Secondly, the multidimensional aggregation model of DNSNS and its aggregation algorithm are introduced for solving the actual MAGDM problem.

The whole paper is arranged as follows. The related work is introduced in Section 2. The concept of DNSNS is introduced in Section 3. In Section 4, the Euclidean 3-space aggregation model for calculating the optimal expert preference curve is proposed. Section 5 is devoted to the aggregation algorithm of the proposed aggregation model and the decision making algorithm for MAGDM problem with dynamic nonlinear expert preference information. In Section 6, a typical example is presented to illustrate the feasibility and effectiveness of the proposed model and algorithm. The main contributions of the work and the future research directions are pointed out in Section 7.

2. Related work

Singh [39] proposed a Generalized Interval-valued Neutrosophic Fuzzy Weighted Geometric (GIVNFWG) operator for the Multiple Attribute Group Decision Making (MAGDM) problem with interval-valued neutrosophic fuzzy information. Additionally, they presented a generalized framework for fuzzy-form tensors to address high-dimensional MCGDM problems. However, their approach employed principal component analysis (PCA) to reduce data dimensionality before solving the MAGDM problem, potentially compromising the connection between data attributes. Ghosh [40] investigated the Multi-Objective Waste Management (MOWM) problem under a neutrosophic hesitant fuzzy (NHF) environment and introduced a novel ranking approach for NHF data. The proposed model addresses three conflicting objective functions: maximizing profit for economic sustainability, minimizing workload deviation for social sustainability, and minimizing carbon emissions for environmental sustainability. Nevertheless, this method exhibits limited capability in processing real-time data. Gard [41] introduced an exponential-logarithm-based Single-Valued Neutrosophic Set (SVNS) to describe vagueness in information and defined weighted aggregation operators to solve the MAGDM problem. Ye [42] proposed a Multi-Criteria Decision-Making (MCDM) model based on trigonometric weighted average and geometric operators of single-valued neutrosophic credibility numbers (SvNCNs). However, both Gard's and Ye's operators involve numerous connected products. In case the $\epsilon \mathcal{l}-\text{SVNS}$ or SvNCN has any expert preference matrix containing a value of zero, the $\epsilon \mathcal{l}-\text{SVNS}$ or SvNCN in the collective matrix will be directly set to zero. Ullah [43] proposed a novel framework that integrates IVNSs with the entropy–MultiAtributive Ideal-Real Comparative Analysis (MAIRCA) framework, aiming to effectively handle both objective criteria with precise inputs and subjective criteria with ambiguous or uncertain information. Yang [44] introduced an enhanced TOPSIS method called NR-TOPSIS, which redefines the positive ideal solution and negative ideal solution. Additionally, this study introduces the concept of ranking stability and demonstrates that NR-TOPSIS satisfies ranking stability, thereby theoretically ensuring the absence of rank reversal phenomenon.

The above methods can deal with the MAGDM problem containing static fuzzy information efficiently. Nevertheless, with the increasing complexity of decision-making environment, expert preference information has some new characteristics.

• (1)Real-time performance, which means expert preferences are constantly in flux with the development of decision-making events, which is particularly prominent in the decision-making of emergency events and public opinion events. Because of the lack of information, time pressure,and event scale, the initial expert preference information is more subjective. Compared with traditional static fuzzy sets, dynamic fuzzy sets that describe real-time information can reflect expert preference information more accurately.
• (2)Nonlinear, which means internal meaning and changing trend of expert preference information are nonlinear. As preprocessing of membership/non-membership degree or processes of computing with words, the change trend between two preference terms is usually described as linear logic (Fig. 1, Fig. 2). Nevertheless, the density distributions of expert preference information between linguistic terms are uneven. Moreover, with the development of decision-making events, expert preference information also changes nonlinearly, even overturns.

Fig. 1.

Triangular intuitionistic fuzzy number.

Fig. 2.

A set of seven terms with its semantics.

The paper introduces DNSNS, a framework for capturing real-time preference information that is not suitable for preprocessing,and dynamic preference information that changes non-linearly over time. Additionally, an aggregation algorithm for the DNSNS is proposed to address practical MAGDM problems such as decision-making in future investment programs, natural disaster emergency programs, and network public opinion monitoring programs.

3. Preliminaries

Because our proposal is based on the NS [7], in this section, we review its main concepts and operations. Afterward, the concept, the operations, the distance measures and the similarity measures of DNSNS are introduced. Additionally, the plant growth simulation algorithm (PGSA) as the research basis of the proposed algorithm is introduced.

3.1. Neutrosophic sets and simplified neutrosophic sets

NS and its properties are discussed briefly in Ref. [7].

Definition 1

Let X be a space of points (objects), with a generic element in X denoted by x. An NS A in X is defined with the form (see Eq. (1)):

$$A=\left\{⟨x,T_A\left(x\right),I_A\left(x\right),F_A\left(x\right)⟩|x\in X\right\}\text{,}$$ (1)

where $T_A\left(x\right)$, $I_A\left(x\right)$, and $F_A\left(x\right)$ denotes the truth-membership function, the indeterminacy-membership function and the falsity-membership function of the element $x\in X$ to the set A respectively, which are real standard or nonstandard subsets of $]0^{-},1^{+}[$. Since the three functions are independent of each other, for each point x in X, we have $0^{-}\le T_A\left(x\right)+I_A\left(x\right)+F_A\left(x\right)\le 3^{+}$.

As a subclass of NSs, SNS and its properties are discussed briefly in Ref. [45] for preserving the operations of NSs properly.

Definition 2

Let X be a space of points (objects), with a generic element in X denoted by x. An SNS A in X is defined with the form (see Eq. (2)):

$$A=\left\{⟨x,T_A\left(x\right),I_A\left(x\right),F_A\left(x\right)⟩|x\in X\right\}\text{,}$$ (2)

where $T_A\left(x\right)$, $I_A\left(x\right)$, and $F_A\left(x\right)$ are singleton subsets in the real standard $\left[0,1\right]$. Since the three functions are independent of each other, for each point x in X, we have $0\le T_A\left(x\right)+I_A\left(x\right)+F_A\left(x\right)\le 3$.

The operations of NS and SNS are also introduced by Smarandache [7] and Ye [46].

3.2. Dynamic nonlinear simplified neutrosophic sets

Definition 3

Let X be a space of points (objects), with a generic element in X denoted by x. An DNSNS A in X is defined with the form (see Eq. (3)):

$$A=\left\{⟨x,T_A^{\mathcal{l}}\left(x\right),I_A^{\mathcal{l}}\left(x\right),F_A^{\mathcal{l}}\left(x\right)⟩|\mathcal{l}\ge 0,x\in X\right\}\text{,}$$ (3)

where $T_A^{\mathcal{l}}\left(x\right)={\left\{T_A\left(x^t\right)\right\}}_{t=1}^{\mathcal{l}}$, $I_A^{\mathcal{l}}\left(x\right)={\left\{I_A\left(x^t\right)\right\}}_{t=1}^{\mathcal{l}}$, and $F_A^{\mathcal{l}}\left(x\right)={\left\{F_A\left(x^t\right)\right\}}_{t=1}^{\mathcal{l}}$ are univariate time series of the nonlinear simplified truth-membership function, the nonlinear simplified indeterminacy-membership function and the nonlinear simplified falsity-membership function of the element $x\in X$ to the set A respectively. $T_A\left(x^t\right)$, $I_A\left(x^t\right)$ and $F_A\left(x^t\right)$ are singleton subsets in the real standard $\left[0,1\right]$. The sum of $T_A\left(x^t\right)$, $I_A\left(x^t\right)$ and $F_A\left(x^t\right)$ is singleton subsets in the real standard $\left[0,3\right]$.

The operations of DNSNS are introduced by definition 4.

Definition 4

Let A and B be two DNSNSs with same time series length ($\mathcal{l}\ge 0$, $x\in X$), then:

• (1)$A=B$, iff $A\subseteq B$ and $B\subseteq A$,
• (2)$A\subseteq B$, iff $T_A^{\mathcal{l}}\left(x\right)\le T_B^{\mathcal{l}}\left(x\right)$, $I_A^{\mathcal{l}}\left(x\right)\le I_B^{\mathcal{l}}\left(x\right)$, $F_A^{\mathcal{l}}\left(x\right)\le F_B^{\mathcal{l}}\left(x\right)$ for any x in X,
• (3)$A^c=\left\{⟨x,F_A^{\mathcal{l}}\left(x\right),{\left(1-I_A\left(x^t\right)\right)}_{t=1}^{\mathcal{l}},T_A^{\mathcal{l}}\left(x\right)⟩\right\}$,
• (4)$A\cup B=\{⟨x,{\left(\max \left\{T_A\left(x^t\right),T_B\left(x^t\right)\right\}\right)}_{t=1}^{\mathcal{l}}\text{,}{\left(\min \left\{I_A\left(x^t\right),I_B\left(x^t\right)\right\}\right)}_{t=1}^{\mathcal{l}},{\left(\min \left\{F_A\left(x^t\right),F_B\left(x^t\right)\right\}\right)}_{t=1}^{\mathcal{l}}⟩\}$,
• (5)$A\cap B=\{⟨x,{\left(\min \left\{T_A\left(x^t\right),T_B\left(x^t\right)\right\}\right)}_{t=1}^{\mathcal{l}}\text{,}{\left(\max \left\{I_A\left(x^t\right),I_B\left(x^t\right)\right\}\right)}_{t=1}^{\mathcal{l}},{\left(\max \left\{F_A\left(x^t\right),F_B\left(x^t\right)\right\}\right)}_{t=1}^{\mathcal{l}}⟩\}$.

The operations of dynamic nonlinear simplified neutrosophic time series (DNSNTSs) are introduced by definition 5.

Definition 5

Let $\alpha ={\left(a^t,b^t,c^t\right)}_{t=1}^{\mathcal{l}}$ and $\beta ={\left(d^t,e^t,f^t\right)}_{t=1}^{\mathcal{l}}$ be two DNSNTSs, then:

• (1)$\alpha \oplus \beta ={\left(a^t+d^t-a^t{d}^t,b^t{e}^t,c^t{f}^t\right)}_{t=1}^{\mathcal{l}}，$.
• (2)$\alpha \otimes \beta ={\left(a^t{d}^t,b^t+e^t-b^t{e}^t,c^t+f^t-c^t{f}^t\right)}_{t=1}^{\mathcal{l}}，$.
• (3)$\lambda \alpha ={\left(1-{\left(1-a^t\right)}^{\lambda },{\left(b^t\right)}^{\lambda },{\left(c^t\right)}^{\lambda }\right)}_{t=1}^{\mathcal{l}}，\lambda >0，$.
• (4)${\alpha }^{\lambda }={\left({\left(a^t\right)}^{\lambda },1-{\left(1-b^t\right)}^{\lambda },1-{\left(1-c^t\right)}^{\lambda }\right)}_{t=1}^{\mathcal{l}},\lambda >0\text{.}$.

The above operations with the following properties.

• (1)$\alpha \oplus \beta =\beta \oplus \alpha$,
• (2)$\alpha \otimes \beta =\beta \otimes \alpha$,
• (3)$\lambda \left(\alpha \oplus \beta \right)=\lambda \alpha \oplus \lambda \beta ，\lambda >0$,
• (4)${\left(\alpha \otimes \beta \right)}^{\lambda }={\alpha }^{\lambda }\oplus {\beta }^{\lambda }，\lambda >0$,
• (5)${\lambda }_1\alpha \oplus {\lambda }_2\alpha =\left({\lambda }_1+{\lambda }_2\right)\alpha ，{\lambda }_1>0,{\lambda }_2>0$,
• (6)${\alpha }^{{\lambda }_1}\otimes {\alpha }^{{\lambda }_2}={\alpha }^{{\lambda }_1+{\lambda }_2}，{\lambda }_1>0,{\lambda }_2>0$.

The score function and the accuracy function are defended by definition 6 for ranking the DNSNTSs.

Definition 6

Let $\alpha ={\left(a^t,b^t,c^t\right)}_{t=1}^{\mathcal{l}}$ be an DNSNTS, then the score function of α where $s\left(\alpha \right)\in \left[0,3\right]$ is (see Eq. (4)):

$$s\left(\alpha \right)=\sum _{t=1}^{\mathcal{l}}\frac{1}{3}\left(a^t+1-b^t+1-c^t\right)\text{,}$$ (4)

and the accuracy function of α where $s\left(\alpha \right)\in \left[0,3\right]$ is (see Eq. (5)):

$$h\left(A\right)=\sum _{t=1}^{\mathcal{l}}\left(a^t-c^t\right)\text{,}$$ (5)

A distance function or metric describes how far one element is away from another in fuzzy theory. The local distance, the global distance and the key time node distance of DNSNTS are proposed in definition 7.

Definition 7

Let A and B be two DNSNSs in the universe discourse (see Eq. (6)):

$$X=\left\{⟨x_1^1,\cdot \cdot \cdot ,x_1^t,\cdot \cdot \cdot ,x_1^{\mathcal{l}}⟩,\cdot \cdot \cdot ,⟨x_n^1,\cdot \cdot \cdot ,x_n^t,\cdot \cdot \cdot ,x_n^{\mathcal{l}}⟩\right\}\text{,}$$ (6)

then the distance measures between them can be defined as follows:

$$d_q\left(A,B\right)=\sum _{i=1}^n[\frac{1}{3}\sum _{t=k}^h({\left|T_A^t\left(x_i\right)-T_B^t\left(x_i\right)\right|}^q+{\left|I_A^t\left(x_i\right)-I_B^t\left(x_i\right)\right|}^q{+{\left|F_A^t\left(x_i\right)-F_B^t\left(x_i\right)\right|}^q)]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\text{,}$$ (7)

when $q=1,q=2,\text{and}\mathrm{q}\to \infty$, the corresponding $d_1\left(A,B\right)$, $d_2\left(A,B\right)$ and $d_{\infty }\left(A,B\right)$ represent the weighted Hamming, Euclidean, and Chebyshev distances, respectively. Additionally, the distance measures are divided into three cases according to the values of h and k.

• (1)The $d_q\left(A,B\right)$ is the local distance between the kth time node and the hth time node of A and B, when $1<k<h\le \mathcal{l}$ or $1\le k<h<\mathcal{l}$.
• (2)The $d_q\left(A,B\right)$ is the global distance of A and B, when $k=1\text{and}h=\mathcal{l}$.
• (3)The $d_q\left(A,B\right)$ is the key time node distance of A and B in kth time node, when $1\le k=h\le \mathcal{l}$.

Based on the extension of the Jaccard, Dice, and cosine similarity measures between two SNSs. the three similarity measures between DNSNSs A and B are proposed in definition 8.

Definition 8

Let A and B be two DNSNSs in the universe discourse

$$X=\left\{⟨x_1^1,\cdot \cdot \cdot ,x_1^t,\cdot \cdot \cdot ,x_1^{\mathcal{l}}⟩,\cdot \cdot \cdot ,⟨x_n^1,\cdot \cdot \cdot ,x_n^t,\cdot \cdot \cdot ,x_n^{\mathcal{l}}⟩\right\}\text{,}$$

then the Jaccard, Dice, and cosine similarity measures between them can be defined as follows:

$$S_{\mathrm{J}}^{\text{DNSNS}}\left(A,B\right)=\frac{C\left(A,B\right)}{n\left[E\left(A\right)+E\left(B\right)-C\left(A,B\right)\right]}\text{,}$$ (8)

$$S_{\mathrm{D}}^{\text{DNSNS}}\left(A,B\right)=\frac{2C\left(A,B\right)}{n\left[E\left(A\right)+E\left(B\right)\right]}\text{,}$$ (9)

$$S_{\mathrm{C}}^{\text{DNSNS}}\left(A,B\right)=\frac{C\left(A,B\right)}{n\sqrt{E\left(A\right)\cdot E\left(B\right)}}\text{,}$$ (10)

where the correlation of two INSs A and B is given by:

$$C\left(A,B\right)=\sum _{i=1}^n\sum _{t=1}^{\mathcal{l}}[T_A^t\left(x_i\right)\cdot T_B^t\left(x_i\right)+I_A^t\left(x_i\right)\cdot I_B^t\left(x_i\right)+F_A^t\left(x_i\right)\cdot F_B^t\left(x_i\right)]\text{,}$$ (11)

and the informational intuitional energies of A and B are defined as:

$$E\left(A\right)=\sum _{i=1}^n\sum _{t=1}^{\mathcal{l}}[{\left(T_A^t\left(x_i\right)\right)}^2+{\left(I_A^t\left(x_i\right)\right)}^2+{\left(F_A^t\left(x_i\right)\right)}^2]\text{,}$$ (12)

$$E\left(B\right)=\sum _{i=1}^n\sum _{t=1}^{\mathcal{l}}[{\left(T_B^t\left(x_i\right)\right)}^2+{\left(I_B^t\left(x_i\right)\right)}^2+{\left(F_B^t\left(x_i\right)\right)}^2]\text{.}$$ (13)

Furthermore, the differences of importance are considered in the elements in the universe. The weighted distance measures and the similarity measures between DNSNSs extended from Eqs. (7)–(13) are proposed as follows (see Eqs. (14), (15), (16), (17)):

$$d_q^{\omega }\left(A,B\right)=\sum _{i=1}^n{\omega }_i[\frac{1}{3}\sum _{t=k}^h({\left|T_A^t\left(x_i\right)-T_B^t\left(x_i\right)\right|}^q+{\left|I_A^t\left(x_i\right)-I_B^t\left(x_i\right)\right|}^q{+{\left|F_A^t\left(x_i\right)-F_B^t\left(x_i\right)\right|}^q)]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\text{,}$$ (14)

$$\begin{array}{c}{S}_{\mathrm{J}}^{\omega -\text{DNSNS}}\left(A,B\right)=\frac{C^{\omega }\left(A,B\right)}{E^{\omega }\left(A\right)+E^{\omega }\left(B\right)-C^{\omega }\left(A,B\right)}\text{,}\end{array}$$ (15)

$$S_{\mathrm{D}}^{\omega -\text{DNSNS}}\left(A,B\right)=\frac{2C^{\omega }\left(A,B\right)}{E^{\omega }\left(A\right)+E^{\omega }\left(B\right)}\text{,}$$ (16)

$$S_{\mathrm{C}}^{\omega -\text{DNSNS}}\left(A,B\right)=\frac{C^{\omega }\left(A,B\right)}{\sqrt{E^{\omega }\left(A\right)\cdot E^{\omega }\left(B\right)}}\text{,}$$ (17)

where ${\omega }_i$ represents the weight elements $x_i^t\in X$, which satisfy the normalized conditions ${\omega }_i\in \left[0，1\right]$ and $\sum _{i=1}^n{\omega }_i=1$. Additionally, the weighted correlation between A and B is calculated by Eq (18). The informational intuitional energies of A and B are defined by Eqs (19) and (20):

$$C^{\omega }\left(A,B\right)=\sum _{i=1}^n{\omega }_i\sum _{t=1}^{\mathcal{l}}[T_A^t\left(x_i\right)\cdot T_B^t\left(x_i\right)+I_A^t\left(x_i\right)\cdot I_B^t\left(x_i\right)+F_A^t\left(x_i\right)\cdot F_B^t\left(x_i\right)]\text{,}$$ (18)

$$E^{\omega }\left(A\right)=\sum _{i=1}^n{\omega }_i\sum _{t=1}^{\mathcal{l}}[{\left(T_A^t\left(x_i\right)\right)}^2+{\left(I_A^t\left(x_i\right)\right)}^2+{\left(F_A^t\left(x_i\right)\right)}^2]\text{,}$$ (19)

$$E^{\omega }\left(B\right)=\sum _{i=1}^n{\omega }_i\sum _{t=1}^{\mathcal{l}}[{\left(T_B^t\left(x_i\right)\right)}^2+{\left(I_B^t\left(x_i\right)\right)}^2+{\left(F_B^t\left(x_i\right)\right)}^2]\text{.}$$ (20)

According the proposed definitions, the distance measures and the similarity measures represent the differences and the correlations between DNSNSs respectively.

3.3. Plant growth simulation algorithm (PGSA)

The PGSA is a heuristic algorithm based on the plant growth mechanism first proposed by the Chinese

scholar, Li [47]. The advantages of the PGSA in solving the decision making problem, the site selection problem and the network planning problem are detailed in Refs. [[48], [49], [50]].

The majority of plants exhibit phototropism, whereby branches positioned towards the sun possess a higher concentration of morphactin, thereby affording them greater opportunities for branch proliferation. Consequently, a new branch will emerge from the seed with an appropriate angle relative to the original branch. Lindenmayer synthesized these principles and proposed a formal descriptive grammar, known as the 'L-system', to elucidate the growth patterns of plants. (see Fig. 3, $\theta ={45}^{\circ }$).

Fig. 3.

L-system.

Per this property, the PGSA for finding the global optimal solution of some problems can be designed as follows.

Step 1

The root $x^0$ is selected in the feasible region randomly, suppose there are $t$ nodes $S_{M_1},\cdot \cdot \cdot ,S_{M_t}$ on the trunk $M$ grows from $x^0$. The growth hormone concentration of each node is $C_{M_1},\cdot \cdot \cdot ,C_{M_t}$, and $C_{M_i}\left(1\le i\le t\right)$ can be calculated by Eq (21):

$$C_{M_i}=\frac{f\left(x^0\right)-f\left(S_{M_i}\right)}{\sum _{i=1}^t\left(f\left(x^0\right)-f\left(S_{M_i}\right)\right)}\text{,}$$ (21)

where $f\left(X\right)$ is the backlight function for describing the environment of the node $X$ in the plant, and it is inversely proportional with the distance between $X$ and the light source. The function value decreases as the illumination of the growth node increases.

Step (2)

We can derive $\sum _{i=1}^t{C}_{M_i}=1$ from Eq (21), and then a special roulette can be established per this feature for selecting a new growth node (Fig. 4).

The node $S_{M_i}$ occupies its own area on the roulette. A random number $\delta$ is selected in the interval $\left[0,1\right]$, a method that is similar to throwing a ball onto a state map. It will land in the area of one of $S_{M_1},\dots ,S_{M_t}$. The corresponding node $S_{M_k}$,which is the preferential growth node, will take priority to grow a new branch in the next step.

Step 3

It is assumed that a new branch $m$ grows from $S_{M_k}$, which has $r$ nodes, namely, $S_{m_1},\dots ,S_{m_r}$. The growth hormone concentration of each node is $C_{m_1},\cdot \cdot \cdot ,C_{m_r}$. The new growth node $S_{M_p}$ is selected from nodes on trunk $M$ and branch $m$, and then $C_{M_i}$ and $C_{m_j}$ can be calculated by Eq (22):

$$\{\begin{array}{c}{C}_{M_i}=\frac{f\left(x^0\right)-f\left(S_{M_i}\right)}{\sum _{i=1}^t\left(f\left(x^0\right)-f\left(S_{M_i}\right)\right)+\sum _{j=1}^r\left(f\left(x^0\right)-f\left(S_{m_j}\right)\right)}\\ C_{m_j}=\frac{f\left(x^0\right)-f\left(S_{m_j}\right)}{\sum _{i=1}^t\left(f\left(x^0\right)-f\left(S_{M_i}\right)\right)+\sum _{j=1}^r\left(f\left(x^0\right)-f\left(S_{m_j}\right)\right)}\end{array}$$ (22)

where $\left(1\le i\le t,1\le j\le r,i\ne k\right)$.

From Eq (22), $\sum _{i=1,i\ne k}^t{C}_{Mi}+\sum _{j=1}^r{C}_{mj}=1$ can be derived evidently. Then, the new preferential node $S_{M_p}$ will be selected in a similar way as $S_{M_k}$. A new branch grows from $S_{M_p}$ in the next step. The growth process is repeated until the new branch reaches the light source position, following which the plant stops growing.

Fig. 4.

Special roulette.

4. Aggregation model

The aggregation of the fuzzy sets is the key step for solving MAGDM problem with fuzzy information. Different from the traditional fuzzy aggregation operators generalized and extended from the weighted average (OA) operator and weighted geometric (OG) operator, the DNSNSs are projected as curves in the Euclidean 3-space by the proposed aggregation model. The aggregation of fuzzy sets is completed by finding the optimal aggregation curve in the aggregation model. The construction process and advantages of the proposed aggregation model are fully detailed in this section.

4.1. Optimal rally point and optimal rally curve

Definition 9

Let $\mathrm{P}=\left\{{\mathrm{P}}_1^{{\mathrm{\xi }}_1},\cdot \cdot \cdot ,{\mathrm{P}}_{\mathrm{m}}^{{\mathrm{\xi }}_{\mathrm{m}}}\right\}$ be a point set with m weighted points in a bounded closed box in the Euclidean n-space. The positive weighted of ${\mathrm{P}}_{\mathrm{i}}^{{\mathrm{\xi }}_{\mathrm{i}}}\left({\mathrm{a}}_{\mathrm{i}}^1,\cdot \cdot \cdot ,{\mathrm{a}}_{\mathrm{i}}^{\mathrm{n}}\right)$ is ${\mathrm{\xi }}_{\mathrm{i}}\in \left[1,0\right]$, and $\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{\xi }}_{\mathrm{i}}=1$. If a point ${\mathrm{P}}^{*}$ exists, whose Euclidean distances to ${\mathrm{P}}_{\mathrm{i}}^{{\mathrm{\xi }}_{\mathrm{i}}}$ meet the following condition (see Eq (23)):

$$d_E^{\xi }=\min \sum _{i=1}^m{\xi }_i\left|P^{*}{P}_i^{{\xi }_i}\right|\text{.}$$ (23)

Then, $P^{*}$ can be defined as the optimal rally point of $P$ in the Euclidean n-space. The situation of the Euclidean 3-space is depicted in Fig. 5.

Let $r\left(\mathcal{l}\right)$ be a regular space curve in a bounded closed box in the Euclidean 3-space ${\mathbb{E}}^3$ parameterized by its arc length $\mathcal{l}$. Denote the Frenet frame field along $r\left(\mathcal{l}\right)$ by $\left\{\overrightarrow{T},\overrightarrow{N},\overrightarrow{B}\right\}$, that is, $\overrightarrow{T}=r^{\prime }\left(\mathcal{l}\right)$ is the tangent vector field, $\overrightarrow{N}={\left|\overrightarrow{T^{\prime }}\right|}^{-1}\overrightarrow{T^{\prime }}$ is the normal vector field and $\overrightarrow{B}=\overrightarrow{T}\times \overrightarrow{N}$ is the binormal vector field of $r\left(\mathcal{l}\right)$. Then, the Frenet formulas of the curve $r\left(\mathcal{l}\right)$ are given by Eq (24):

$$\{\begin{array}{c}\frac{d\overrightarrow{T}}{d\mathcal{l}}=\kappa \overrightarrow{N}\text{,}\\ \frac{d\overrightarrow{N}}{d\mathcal{l}}=-\kappa \overrightarrow{T}+\tau \overrightarrow{N}\text{,}\\ \frac{d\overrightarrow{B}}{d\mathcal{l}}=-\tau \overrightarrow{N}\text{.}\end{array}$$ (24)

Where $\kappa$ and $\tau$ are the curvature function and torsion function of the curve $r\left(\mathcal{l}\right)$ in ${\mathbb{E}}^3$. Then, the definition of the optimal rally curve in the Euclidean 3-space is introduced by definition 10.

Definition 10

Let $\mathrm{R}=\left\{\mathrm{r}{\left(\mathcal{l}\right)}_1^{{\mathrm{\xi }}_1},\cdot \cdot \cdot ,\mathrm{r}{\left(\mathcal{l}\right)}_{\mathrm{m}}^{{\mathrm{\xi }}_{\mathrm{m}}}\right\}$ be a curve set with m weighted points in a bounded closed box in the Euclidean 3-space ${\mathbb{E}}^3$. The positive weighted of $\mathrm{r}{\left(\mathcal{l}\right)}_{\mathrm{i}}^{{\mathrm{\xi }}_{\mathrm{i}}}$ is ${\mathrm{\xi }}_{\mathrm{i}}\in \left[1,0\right]$, and $\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{\xi }}_{\mathrm{i}}=1$. If a curve $\mathrm{r}{\left(\mathcal{l}\right)}^{*}$ exists, the areas ${\int }_{\mathcal{l}}\left(\mathrm{r}{\left(\mathcal{l}\right)}^{*}，\mathrm{r}{\left(\mathcal{l}\right)}_{\mathrm{i}}^{{\mathrm{\xi }}_{\mathrm{i}}}\right)$ enclosed by $\mathrm{r}{\left(\mathcal{l}\right)}^{*}$ and $\mathrm{r}{\left(\mathcal{l}\right)}_{\mathrm{i}}^{{\mathrm{\xi }}_{\mathrm{i}}}$ meet the following condition (see Eq (25)):

$$s^{\xi }=\min \sum _{i=1}^m{\xi }_i{\int }_{\mathcal{l}}\left(r{\left(\mathcal{l}\right)}^{*},r{\left(\mathcal{l}\right)}_i^{{\xi }_i}\right)\text{.}$$ (25)

Then $r{\left(\mathcal{l}\right)}^{*}$ can be defined as the optimal rally curve of $R$. (Fig. 6).

Fig. 5.

The optimal rally point in the Euclidean 3-space.

Fig. 6.

The optimal rally curve in the Euclidean 3-space.

4.2. Euclidean 3-space aggregation model of DNSNS

For this research, we assume that each DNSNTS is considered as a regular space curve in Euclidean 3-space ${\mathbb{E}}^3$.

Let $A^1,\cdot \cdot \cdot ,A^M$ be family of sets of n DNSNSs with same time series length $\mathcal{l}$ in the universe discourse (see Eq (26)):

$$\begin{gathered} \begin{array}{c}{A}^1=\left\{\begin{array}{c}{A}_1^1=⟨T_{A^1}^{\mathcal{l}}\left(x_1\right),I_{A^1}^{\mathcal{l}}\left(x_1\right),F_{A^1}^{\mathcal{l}}\left(x_1\right)⟩\text{,}\\ \cdot \cdot \cdot \text{,}\\ A_n^1=⟨T_{A^1}^{\mathcal{l}}\left(x_n\right),I_{A^1}^{\mathcal{l}}\left(x_n\right),F_{A^1}^{\mathcal{l}}\left(x_n\right)⟩\end{array}\right\}\text{,} \\ \cdot \cdot \cdot \text{,}\\ A^M=\left\{\begin{array}{c}{A}_1^M=⟨T_{A^M}^{\mathcal{l}}\left(x_1\right),I_{A^M}^{\mathcal{l}}\left(x_1\right),F_{A^M}^{\mathcal{l}}\left(x_1\right)⟩\text{,}\\ \cdot \cdot \cdot \text{,}\\ A_n^M=⟨T_{A^M}^{\mathcal{l}}\left(x_n\right),I_{A^M}^{\mathcal{l}}\left(x_n\right),F_{A^M}^{\mathcal{l}}\left(x_n\right)⟩\end{array}\right\}\end{array}\text{.} \end{gathered}$$ (26)

The Euclidean 3-space ${\mathbb{E}}^3$ is established, and $T_A^{\mathcal{l}}\left(x\right)$, $I_A^{\mathcal{l}}\left(x\right)$, and $F_A^{\mathcal{l}}\left(x\right)$ are used as the x-axis, the y-axis and the z-axis. Thus, the DNSNTSs can be projected as n groups of curves in ${\mathbb{E}}^3$. The curves obtained from $A_h^k\left(1\le k\le M\text{and}1\le h\le n\right)$ in different DNSNSs are divided into the curve set $C_n$ (Fig. 7, n = 3, M = 10 and $\mathcal{l}=10$).

Fig. 7.

The regular space curves of the DNSNTSs.

The area of surface enclosed by $A_h^k$ and $A_h^j$ ($1\le k\le M,1\le j\le M$) indicates the similarity of the two DNSNTSs, the smaller the surface area is, the more similar the two DNSNTSs is. (Fig. 8).

Definition 11

Let $A_h^{*}$ be a DNSNS with time series length $\mathcal{l}$, the sum of areas of surfaces enclosed by $A_h^{*}$ and $A_h^k$ ($1\le k\le M$ and $1\le h\le n$) meet the following condition (see Eq (27)):

$$s\left(A_h^{*}\right)=\min \sum _{k=1}^M{\int }_{\mathcal{l}}\left(A_h^{*},A_h^k\right)\text{.}$$ (27)

Then $A_h^{*}=⟨T_{A^{*}}^{\mathcal{l}}\left(x_h\right),I_{A^{*}}^{\mathcal{l}}\left(x_h\right),F_{A^{*}}^{\mathcal{l}}\left(x_h\right)⟩$ can be defined as the collective DNSNS of $A_h$ (Fig. 9, M = 15 and $\mathcal{l}=20$).

Furthermore, the collective family of sets $A^{*}=\left\{A_1^{*},\cdot \cdot \cdot ,A_n^{*}\right\}$ of $A^k$ can be obtained by the proposed aggregation model. Thus, the Euclidean 3-space aggregation model of DNSNS is constructed. (Fig. 10, n = 3, M = 10 and $\mathcal{l}=10$).

Fig. 8.

The similarity of the DNSNTSs.

Fig. 9.

The collective DNSNS of A_h.

Fig. 10.

The Euclidean 3-space aggregation model of DNSNS

4.3. Advantages

When addressing practical MAGDM problems involving dynamic nonlinear simplified neutrosophic information, the proposed aggregation model offers several advantages as follows.

• (1)The distinct characteristics of alternatives in MAGDM problems are separately aggregated within the proposed model, enabling accurate calculation of collective expert preference information without preprocessing the original DNSNTSs.
• (2)The proposed model allows for aggregation of global collective expert preference, local collective expert preference, and key time node collective expert preference based on the actual situation.
• (3)Unlike operators generalized and extended from OA and OG, the differences between expert preferences are represented as areas enclosed by different DNSNTSs in the proposed model. This ensures that expert preference information is not averaged during aggregation, thereby preserving accuracy even when zero-valued or one-valued elements exist in expert preferences.

5. Key methods

The aggregation algorithm of the Euclidean 3-space aggregation model of DNSNS and its application in MAGDM is detailed in this section. Additionally, the scoring method for DNSNSs is proposed for selecting the best alternative to the MAGDM problem.

5.1. Aggregation idea

The areas of surfaces enclosed by nonlinear space curves are hard to calculate. Because the time complexity of correlation algorithms increases exponentially with the change of the time series length, it is inefficient to aggregate DNSNSs by finding the optimal rally DNSNTS directly. In such a situation, the differential idea is employed for solving this problem in the proposed aggregation algorithm.

Let A and B be two DNSNTSs with time series length $\mathcal{l}$. When $t_{h+1}-t_h=\Delta t$ ($1\le \mathrm{h}\le \mathcal{l}$, and $\Delta t$ is an infinitesimal time interval), There is a linear relationship between $A^{t_h}$ and $A^{t_{h+1}}$ in A (such as B). Thus, the surface enclosed by $A^h$, $A^{t_{h+1}}$, $B^{t_h}$ and $B^{t_{h+1}}$ is a convex quadrilateral (Fig. 11).

Fig. 11.

The area of surface enclosed by A and B.

The, $s_{h+1}$ can be calculated by Eqs (28)–(34):

$$s_h=\frac{1}{2}\left(ab\sqrt{1-{\cos }^2\alpha }+cd\sqrt{1-{\cos }^2\beta }\right)\text{,}$$ (28)

$$a=\left|A^{t_h}{B}^{t_h}\right|=[{\left(T_A^{t_h}\left(x\right)-T_B^{t_h}\left(x\right)\right)}^2+{\left(I_A^{t_h}\left(x\right)-I_B^{t_h}\left(x\right)\right)}^2{+{\left(F_A^{t_h}\left(x\right)-F_B^{t_h}\left(x\right)\right)}^2]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{,}$$ (29)

$$b=\left|A^{t_h}{A}^{t_{h+1}}\right|=[{\left(T_A^{t_h}\left(x\right)-T_A^{t_{h+1}}\left(x\right)\right)}^2+{\left(I_A^{t_h}\left(x\right)-I_A^{t_{h+1}}\left(x\right)\right)}^2{+{\left(F_A^{t_h}\left(x\right)-F_A^{t_{h+1}}\left(x\right)\right)}^2]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{,}$$ (30)

$$c=\left|A^{t_{h+1}}{B}^{t_{h+1}}\right|=[{\left(T_A^{t_{h+1}}\left(x\right)-T_B^{t_{h+1}}\left(x\right)\right)}^2+{\left(I_A^{t_{h+1}}\left(x\right)-I_B^{t_{h+1}}\left(x\right)\right)}^2{+{\left(F_A^{t_{h+1}}\left(x\right)-F_B^{t_{h+1}}\left(x\right)\right)}^2]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{,}$$ (31)

$$d=\left|B^{t_h}{B}^{t_{h+1}}\right|=[{\left(T_B^{t_h}\left(x\right)-T_B^{t_{h+1}}\left(x\right)\right)}^2+{\left(I_B^{t_h}\left(x\right)-I_B^{t_{h+1}}\left(x\right)\right)}^2{+{\left(F_B^{t_h}\left(x\right)-F_B^{t_{h+1}}\left(x\right)\right)}^2]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{,}$$ (32)

$$\cos \alpha =\cos ⟨\overrightarrow{A^{t_h}{B}^{t_h}},\overrightarrow{A^{t_h}{A}^{t_{h+1}}}⟩=\frac{\overrightarrow{A^{t_h}{B}^{t_h}}\cdot \overrightarrow{A^{t_h}{A}^{t_{h+1}}}}{\left|\overrightarrow{A^{t_h}{B}^{t_h}}\right|\left|\overrightarrow{A^{t_h}{A}^{t_{h+1}}}\right|}=\frac{\left[\begin{array}{c}{T}_B^{t_h}\left(x\right)-T_A^{t_h}\left(x\right),\\ I_B^{t_h}\left(x\right)-I_A^{t_h}\left(x\right),\\ F_B^{t_h}\left(x\right)-F_A^{t_h}\left(x\right)\end{array}\right]\cdot \left[\begin{array}{c}{T}_A^{t_{h+1}}\left(x\right)-T_A^{t_h}\left(x\right),\\ I_A^{t_{h+1}}\left(x\right)-I_A^{t_h}\left(x\right),\\ F_A^{t_{h+1}}\left(x\right)-F_A^{t_h}\left(x\right)\end{array}\right]}{\left|\left(\begin{array}{c}{T}_B^{t_h}\left(x\right)-T_A^{t_h}\left(x\right),\\ I_B^{t_h}\left(x\right)-I_A^{t_h}\left(x\right),\\ F_B^{t_h}\left(x\right)-F_A^{t_h}\left(x\right)\end{array}\right)\right|\left|\left(\begin{array}{c}{T}_A^{t_{h+1}}\left(x\right)-T_A^h\left(x\right),\\ I_A^{t_{h+1}}\left(x\right)-I_A^{t_h}\left(x\right),\\ F_A^{t_{h+1}}\left(x\right)-F_A^{t_h}\left(x\right)\end{array}\right)\right|}\text{,}$$ (33)

$$\cos \beta =\cos ⟨\overrightarrow{B^{t_{h+1}}{A}^{t_{h+1}}},\overrightarrow{B^{t_{h+1}}{B}^{t_h}}⟩=\frac{\overrightarrow{B^{t_{h+1}}{A}^{t_{h+1}}}\cdot \overrightarrow{B^{t_{h+1}}{B}^{t_h}}}{\left|\overrightarrow{B^{t_{h+1}}{A}^{t_{h+1}}}\right|\left|\overrightarrow{B^{t_{h+1}}{B}^{t_h}}\right|}=\frac{\left[\begin{array}{c}{T}_B^{t_{h+1}}\left(x\right)-T_A^{t_{h+1}}\left(x\right),\\ I_B^{t_{h+1}}\left(x\right)-I_A^{t_{h+1}}\left(x\right),\\ F_B^{t_{h+1}}\left(x\right)-F_A^{t_{h+1}}\left(x\right)\end{array}\right]\cdot \left[\begin{array}{c}{T}_B^{t_{h+1}}\left(x\right)-T_B^{t_h}\left(x\right),\\ I_B^{t_{h+1}}\left(x\right)-I_B^{t_h}\left(x\right),\\ F_B^{t_{h+1}}\left(x\right)-F_B^{t_h}\left(x\right)\end{array}\right]}{\left|\left(\begin{array}{c}{T}_B^{t_{h+1}}\left(x\right)-T_A^{t_{h+1}}\left(x\right),\\ I_B^{t_{h+1}}\left(x\right)-I_A^{t_{h+1}}\left(x\right),\\ F_B^{t_{h+1}}\left(x\right)-F_A^{t_{h+1}}\left(x\right)\end{array}\right)\right|\left|\left(\begin{array}{c}{T}_B^{t_{h+1}}\left(x\right)-T_B^{t_h}\left(x\right),\\ I_B^{t_{h+1}}\left(x\right)-I_B^{t_h}\left(x\right),\\ F_B^{t_{h+1}}\left(x\right)-F_B^{t_h}\left(x\right)\end{array}\right)\right|}\text{.}$$ (34)

Then, the area of surface enclosed by A and B can be calculated by Eq (35):

$$s\left(A,B\right)=\sum _{h=1}^{\mathcal{l}-1}{s}_h\text{.}$$ (35)

Because $b$ and $d$ are given in A and B, the smaller $a$ and $c$ are, the smaller $s\left(A,B\right)$ is. Thus, the optimal rally curve between A and B can be constructed by the optimal rally points of $A^{t_h}$ and $B^{t_h}$.

5.2. Aggregation algorithm of DNSNS in the MAGDM problem

In the MAGDM problem, assuming that there are M experts applying DNSNSs $d_{ij}^k$ ($1\le k\le M$, $1\le i\le m$ and $1\le j\le n$) to evaluate m alternatives with n characteristics. Additionally, the weighted vector of the experts is ${\left({\xi }^1,\cdot \cdot \cdot ,{\xi }^M\right)}^{\mathrm{T}}$ and the weighted vector of the characteristics is ${\left({\omega }^1,\cdot \cdot \cdot ,{\omega }^n\right)}^{\mathrm{T}}$, where ${\xi }^k\in \left[0，1\right]$ , $\sum _{k=1}^M{\xi }^k=1$, ${\omega }^j\in \left[0，1\right]$ and $\sum _{j=1}^n{\omega }^j=1$. The expert preference matrices ${\left[d_{ij}^k\right]}_{m\times n}$ are constructed as follows (see Eq (36)):

$$\begin{gathered} {\left[d_{ij}^k\right]}_{m\times n}=\left[\begin{array}{ccc}{A}_{1，1}^k& \cdot \cdot \cdot & A_{1,n}^k\\ ⋮& \ddots & ⋮\\ A_{m，1}^k& \cdot \cdot \cdot & A_{m,n}^k\end{array}\right]=\left[\begin{array}{ccc}{\left\{A_{1，1}^{t_1},\cdot \cdot \cdot ,A_{1，1}^{t_{\mathcal{l}}}\right\}}^k& \cdot \cdot \cdot & {\left\{A_{1,n}^{t_1},\cdot \cdot \cdot ,A_{1,n}^{t_{\mathcal{l}}}\right\}}^k\\ ⋮& \ddots & ⋮\\ {\left\{A_{m，1}^{t_1},\cdot \cdot \cdot ,A_{m，1}^{t_{\mathcal{l}}}\right\}}^k& \cdot \cdot \cdot & {\left\{A_{m,n}^{t_1},\cdot \cdot \cdot ,A_{m,n}^{t_{\mathcal{l}}}\right\}}^k\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{c}\{{⟨T_{d_1^1}^{t_1},I_{d_1^1}^{t_1},F_{d_1^1}^{t_1}⟩}^k,\\ \\ \cdot \cdot \cdot ,{⟨T_{d_1^1}^{t_{\mathcal{l}}},I_{d_1^1}^{t_{\mathcal{l}}},F_{d_1^1}^{t_{\mathcal{l}}}⟩}^k\}\end{array}& \cdot \cdot \cdot & \begin{array}{c}\{{⟨T_{d_n^1}^{t_1},I_{d_n^1}^{t_1},F_{d_n^1}^{t_1}⟩}^k,\\ \\ \cdot \cdot \cdot ,{⟨T_{d_n^1}^{t_{\mathcal{l}}},I_{d_n^1}^{t_{\mathcal{l}}},F_{d_n^1}^{t_{\mathcal{l}}}⟩}^k\}\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{c}\{{⟨T_{d_1^m}^{t_1},I_{d_1^m}^{t_1},F_{d_1^m}^{t_1}⟩}^k,\\ \\ \cdot \cdot \cdot ,{⟨T_{d_1^m}^{t_{\mathcal{l}}},I_{d_1^m}^{t_{\mathcal{l}}},F_{d_1^m}^{t_{\mathcal{l}}}⟩}^k\}\end{array}& \cdot \cdot \cdot & \begin{array}{c}\{{⟨T_{d_n^m}^{t_1},I_{d_n^m}^{t_1},F_{d_n^m}^{t_1}⟩}^k,\\ \\ \cdot \cdot \cdot ,{⟨T_{d_n^m}^{t_{\mathcal{l}}},I_{d_n^m}^{t_{\mathcal{l}}},F_{d_n^m}^{t_{\mathcal{l}}}⟩}^k\}\end{array}\end{array}\right]\text{.} \end{gathered}$$ (36)

Suppose there are M known DNSNTS points $A_{i.j}^{t_h^k}\in \mathrm{\Omega }$ in point set $A_{i.j}^{t_h}$, where $\mathrm{\Omega }$ is the bounded closed box in ${\mathbb{E}}^3$ and its length is $L$. To identify the optimal rally points $A_{i.j}^{t_h^{*}}$, the core steps extended from the PGSA are as follows.

step 1

Determine ${\gamma }^0\in \mathrm{\Omega }$ as the initial growing point and the step length $\lambda =L/200$. Set ${\mathrm{\Gamma }}_{\min }={\gamma }^0\text{,}$ $F_{\min }=f\left({\gamma }^0\right)$, where $f\left({\gamma }^0\right)$ is the backlight function of ${\gamma }^0$ ($f\left({\gamma }^0\right)=\sum _{k=1}^M{\xi }^k\left|{\gamma }^0{A}_{i,j}^{t_h^k}\right|$ is the sum of the Euclidean distances between ${\gamma }^0$ and $A_{i.j}^{t_h^k}$ in this research).

Step 2

Select g growth points ${\gamma }_z^0\in \mathrm{\Omega }$, whose growth hormone concentration $C_{{\gamma }_z^0}$ can be calculated as follows (see Eq (37)):

$$C_{{\gamma }_z^0}=\frac{f\left({\gamma }_z^0\right)}{\sum _{z=1}^{g}f\left({\gamma }_z^0\right)}\text{.}$$ (37)

Thus, the new growth point ${\gamma }^{0*}$ can be selected randomly by employing the special roulette established per the respective growth hormone concentrations for all alternative growth point. Set ${\gamma }^0={\gamma }^{0*}\text{,}$ ${\mathrm{\Gamma }}_{\min }={\gamma }^0\text{,}$ $F_{\min }=f\left({\gamma }^0\right)$.

Step 3

Establish L-system ($\theta =22.5^{\circ }$) with ${\gamma }^0$ as the thought center. This is the first layer of the branch. Then, g growth points ${\gamma }_z^1$ are selected from L-system randomly.

Step 4:

Update the growth hormone concentration of all “nodes”. If $f\left({\gamma }^0\right)\le f\left({\gamma }_z^1\right)$, then $C_{{\gamma }_z^1}=0$. Otherwise, the new growth hormone concentration of ${\gamma }_z^1$ is calculated as follows (see Eq (38)):

$$C_{{\gamma }_z^1}=\frac{f\left({\gamma }^0\right)-f\left({\gamma }_z^1\right)}{\sum _{z=1}^g\left(f\left({\gamma }^0\right)-f\left({\gamma }_z^1\right)\right)}\text{.}$$ (38)

Step 5

Establish the special roulette per the calculated growth hormone concentrations, and select ${\gamma }^{1*}$ as the new growth point randomly. Set ${\gamma }^1={\gamma }^{1*}\text{,}$ ${\mathrm{\Gamma }}_{\min }={\gamma }^1\text{,}$ $F_{\min }=f\left({\gamma }^1\right)$.

Step 6

Establish L-system ($\mathrm{\theta }=22.5^{\circ }$) with ${\mathrm{\gamma }}^1$ as the thought center. This is the second layer of the branch. Then, g growth points ${\mathrm{\gamma }}_{\mathrm{z}}^2$ are selected from L-system randomly.

Step 7

To avoid the global optimization of the algorithm, the growth hormone concentration of all “nodes” on the first and the second layers of “branch” are updated as follows.

• (1)if $f\left({\gamma }^0\right)\le f\left({\gamma }_z^1\right)$, then $C_{{\gamma }_z^1}=0$. Otherwise, $C_{{\gamma }_z^1}$ is updated by Eq (39):

$$C_{{\gamma }_z^1}=\frac{f\left({\gamma }^1\right)-f\left({\gamma }_z^1\right)}{\sum _{z=1}^g\left(f\left({\gamma }^1\right)-f\left({\gamma }_z^1\right)\right)+\sum _{z=1}^g\left(f\left({\gamma }^1\right)-f\left({\gamma }_z^2\right)\right)}\text{.}$$ (39)

• (2)if $f\left({\gamma }^1\right)\le f\left({\gamma }_z^2\right)$, then $C_{{\gamma }_z^2}=0$. Otherwise, $C_{{\gamma }_z^2}$ is updated by Eq (40):

$$C_{{\gamma }_z^2}=\frac{f\left({\gamma }^1\right)-f\left({\gamma }_z^2\right)}{\sum _{z=1}^g\left(f\left({\gamma }^1\right)-f\left({\gamma }_z^1\right)\right)+\sum _{z=1}^g\left(f\left({\gamma }^1\right)-f\left({\gamma }_z^2\right)\right)}\text{.}$$ (40)

Step 8

Select the new growth point ${\gamma }^{2*}$ by employing the special roulette. Set ${\gamma }^2={\gamma }^{2*}\text{,}$ ${\mathrm{\Gamma }}_{\min }={\gamma }^2\text{,}$ $F_{\min }=f\left({\gamma }^2\right)$.

Step 9

Repeat Steps 6 through 8, When the number of iterations reaches the default value $\phi$ ($\phi =2000$ in this research), or $F_{\min }$ remains unchanged, the algorithm ends. Then, $A_{i.j}^{t_h^{*}}={\mathrm{\Gamma }}_{\min }$ is the optimal rally point for $A_{i.j}^{t_h}$.

The collective expert preference of ith alternative with jth characteristic can be constructed as follows:

$$A_{i.j}^{*}=\left\{A_{i.j}^{t_1^{*}},\cdot \cdot \cdot ,A_{i.j}^{t_{\mathcal{l}}^{*}}\right\}=\left\{⟨T_{d_i^j}^{t_1^{*}},I_{d_i^j}^{t_1^{*}},F_{d_i^j}^{t_1^{*}}⟩,\cdot \cdot \cdot ,⟨T_{d_i^j}^{t_{\mathcal{l}}^{*}},I_{d_i^j}^{t_{\mathcal{l}}^{*}},F_{d_i^j}^{t_{\mathcal{l}}^{*}}⟩\right\}\text{.}$$ (41)

Then, the collective expert preference matrix can be constructed as follows:

$${\left[d_{ij}^{*}\right]}_{m\times n}=\left[\begin{array}{c}{A}_1^{*}\\ ⋮\\ A_m^{*}\end{array}\right]=\left[\begin{array}{ccc}{A}_{1，1}^{*}& \cdots & A_{1,n}^{*}\\ ⋮& \ddots & ⋮\\ A_{m，1}^{*}& \cdots & A_{m,n}^{*}\end{array}\right]=\left[\begin{array}{ccc}\left\{A_{1,1}^{t_1^{*}},\cdot \cdot \cdot ,A_{1,1}^{t_{\mathcal{l}}^{*}}\right\}& \cdots & \left\{A_{1,n}^{t_1^{*}},\cdot \cdot \cdot ,A_{1,n}^{t_{\mathcal{l}}^{*}}\right\}\\ ⋮& \ddots & ⋮\\ \left\{A_{m,1}^{t_1^{*}},\cdot \cdot \cdot ,A_{m,1}^{t_{\mathcal{l}}^{*}}\right\}& \cdots & \left\{A_{m,n}^{t_1^{*}},\cdot \cdot \cdot ,A_{m,n}^{t_{\mathcal{l}}^{*}}\right\}\end{array}\right]\text{.}$$ (42)

5.3. Scores of alternatives

Xu [5] introduced a number of definitions (Definition 12 - 14) for IVIFSs as follows.

Definition 12

Let $\alpha$ and $\beta$ be two vectors, then the projection of $\alpha$ onto $\beta$ is calculated by Eq (43):

$${\text{Prj}}_{\beta }\left(\alpha \right)=\left|\alpha \right|\cos \left(\alpha ,\beta \right)=\left|\alpha \right|\frac{\alpha \beta }{\left|\alpha \right|\left|\beta \right|}=\frac{\alpha \beta }{\left|\beta \right|}\text{,}$$ (43)

is defended as the projection of the vector $\alpha$ on the vector $\beta$.

Generally, ${\text{Prj}}_{\beta }\left(\alpha \right)$ shows the approaching degree of the vector $\alpha$ to the vector $\beta$, whose value rises with an increase in the two vectors approaching degree.

Definition 13

Let ${\dot{\mu }}_j^{*}=\left({\mu }_{j\min }^{\mathrm{L}*},{\mu }_{j\max }^{\mathrm{U}*}\right)$, then $\dot{\mu }*=\left({\dot{\mu }}_1^{*},\cdot \cdot \cdot ,{\dot{\mu }}_m^{*}\right)$ is called the positive idea vector of the membership degree values of m alternatives.

Definition 14

Let ${\dot{\nu }}_j^{*}=\left({\nu }_{j\max }^{\mathrm{L}*},{\nu }_{j\min }^{\mathrm{U}*}\right)$, then $\dot{\nu }*=\left({\dot{\nu }}_1^{*},\cdot \cdot \cdot ,{\dot{\nu }}_m^{*}\right)$ is called the positive idea vector of the non-membership degree values of m alternatives.

With reference to the above definitions, the projection method extended from the TOPSIS for scoring alternatives in the MAGDM problem with dynamic nonlinear simplified neutrosophic information.

Definition 15

Let ${\left[d_{ij}^{*}\right]}_{m\times n}$ be the collective expert preference matrix in the MAGDM problem, then $A_j^{*+}=\left\{A_j^{t_1^{*+}},\cdot \cdot \cdot ,A_j^{t_{\mathcal{l}}^{*+}}\right\}$ called the positive idea vector of jth characteristic, $A_j^{*-}=\left\{A_j^{t_1^{*-}},\cdot \cdot \cdot ,A_j^{t_{\mathcal{l}}^{*-}}\right\}$ called the negative idea vector of jth characteristic, $A_j^{t_h^{*+}}$ and $A_j^{t_h^{*-}}$ can be calculated by Eqs (44) and (45):

$$A_j^{t_h^{*+}}=⟨{\left(T_{d_i^j}^{t_h^{*}}\right)}_{\max }^k,{\left(I_{d_i^j}^{t_h^{*}}\right)}_{\min }^k,{\left(F_{d_i^j}^{t_h^{*}}\right)}_{\min }^k⟩\text{,}$$ (44)

$$A_j^{t_h^{*-}}=⟨{\left(T_{d_i^j}^{t_h^{*}}\right)}_{\min }^k,{\left(I_{d_i^j}^{t_h^{*}}\right)}_{\max }^k,{\left(F_{d_i^j}^{t_h^{*}}\right)}_{\max }^k⟩\text{.}$$ (45)

Then, $A^{*+}$ is the positive idea vector of ${\left[d_{ij}^{*}\right]}_{m\times n}$ and $A^{*-}$ is the negative idea vector of ${\left[d_{ij}^{*}\right]}_{m\times n}$ can be defended as follows (see Eq (46)- (47)):

$$A^{*+}=\left\{A_1^{*+},\cdot \cdot \cdot ,A_n^{*+}\right\}=\left\{⟨A_1^{t_1^{*+}},\cdot \cdot \cdot ,A_1^{t_{\mathcal{l}}^{*+}}⟩,\cdot \cdot \cdot ,⟨A_n^{t_1^{*+}},\cdot \cdot \cdot ,A_n^{t_{\mathcal{l}}^{*+}}⟩\right\}\text{,}$$ (46)

$$A^{*-}=\left\{A_1^{*-},\cdot \cdot \cdot ,A_n^{*-}\right\}=\left\{⟨A_1^{t_1^{*-}},\cdot \cdot \cdot ,A_1^{t_{\mathcal{l}}^{*-}}⟩,\cdot \cdot \cdot ,⟨A_n^{t_1^{*-}},\cdot \cdot \cdot ,A_n^{t_{\mathcal{l}}^{*-}}⟩\right\}\text{.}$$ (47)

Definition 16

Let ${\left[d_{ij}^{*}\right]}_{m\times n}$ be the collective expert preference matrix in the MAGDM problem, $A^{*+}$ and $A^{*-}$ be the positive idea vector and the negative idea vector of ${\left[d_{ij}^{*}\right]}_{m\times n}$, $A_i^{*}$ is the collective expert preference of ith alternative in ${\left[d_{ij}^{*}\right]}_{m\times n}$. Then the score of ith alternative can be calculated as follows.

$$S\left(A_i\right)=\text{Prj}{\left(A_i\right)}^{+}-\text{Prj}{\left(A_i\right)}^{-}\text{,}$$ (48)

$${\text{Prj}}^{+}\left(A_i\right)={\text{Prj}}_{T_{A^{*+}}}\left(T_{A_i^{*}}\right)+{\text{Prj}}_{I_{A^{*+}}}\left(T_{A_i^{*}}\right)-{\text{Prj}}_{F_{A^{*+}}}\left(T_{A_i^{*}}\right)\text{,}$$ (49)

$${\text{Prj}}^{-}\left(A_i\right)={\text{Prj}}_{T_{A^{*-}}}\left(T_{A_i^{*}}\right)+{\text{Prj}}_{I_{A^{*-}}}\left(T_{A_i^{*}}\right)-{\text{Prj}}_{F_{A^{*-}}}\left(T_{A_i^{*}}\right)\text{,}$$ (50)

$${\text{Prj}}_{T_{A^{*+}}}\left(T_{A_i^{*}}\right)=\sum _{j=1}^n\frac{\sum _{h=1}^{\mathcal{l}}\left(T_{d_i^j}^{t_h^{*}}\cdot {\left(T_{d_i^j}^{t_h^{*}}\right)}_{\max }^k\cdot {\omega }_j\right)}{\sqrt{{\left(\sum _{h=1}^{\mathcal{l}}{\left(T_{d_i^j}^{t_h^{*}}\right)}_{\max }^k\right)}^2\cdot {\omega }_j}}\text{,}$$ (51)

$${\text{Prj}}_{I_{A^{*+}}}\left(I_{A_i^{*}}\right)=\sum _{j=1}^n\frac{\sum _{h=1}^{\mathcal{l}}\left(I_{d_i^j}^{t_h^{*}}\cdot {\left(I_{d_i^j}^{t_h^{*}}\right)}_{\min }^k\cdot {\omega }_j\right)}{\sqrt{{\left(\sum _{h=1}^{\mathcal{l}}{\left(I_{d_i^j}^{t_h^{*}}\right)}_{\min }^k\right)}^2\cdot {\omega }_j}}\text{,}$$ (52)

$${\text{Prj}}_{F_{A^{*+}}}\left(F_{A_i^{*}}\right)=\sum _{j=1}^n\frac{\sum _{h=1}^{\mathcal{l}}\left(F_{d_i^j}^{t_h^{*}}\cdot {\left(F_{d_i^j}^{t_h^{*}}\right)}_{\min }^k\cdot {\omega }_j\right)}{\sqrt{{\left(\sum _{h=1}^{\mathcal{l}}{\left(F_{d_i^j}^{t_h^{*}}\right)}_{\min }^k\right)}^2\cdot {\omega }_j}}\text{,}$$ (53)

$${\text{Prj}}_{T_{A^{*-}}}\left(T_{A_i^{*}}\right)=\sum _{j=1}^n\frac{\sum _{h=1}^{\mathcal{l}}\left(T_{d_i^j}^{t_h^{*}}\cdot {\left(T_{d_i^j}^{t_h^{*}}\right)}_{\min }^k\cdot {\omega }_j\right)}{\sqrt{{\left(\sum _{h=1}^{\mathcal{l}}{\left(T_{d_i^j}^{t_h^{*}}\right)}_{\min }^k\right)}^2\cdot {\omega }_j}}\text{,}$$ (54)

$${\text{Prj}}_{I_{A^{*-}}}\left(I_{A_i^{*}}\right)=\sum _{j=1}^n\frac{\sum _{h=1}^{\mathcal{l}}\left(I_{d_i^j}^{t_h^{*}}\cdot {\left(I_{d_i^j}^{t_h^{*}}\right)}_{\max }^k\cdot {\omega }_j\right)}{\sqrt{{\left(\sum _{h=1}^{\mathcal{l}}{\left(I_{d_i^j}^{t_h^{*}}\right)}_{\max }^k\right)}^2\cdot {\omega }_j}}\text{,}$$ (55)

$${\text{Prj}}_{F_{A^{*-}}}\left(F_{A_i^{*}}\right)=\sum _{j=1}^n\frac{\sum _{h=1}^{\mathcal{l}}\left(F_{d_i^j}^{t_h^{*}}\cdot {\left(F_{d_i^j}^{t_h^{*}}\right)}_{\max }^k\cdot {\omega }_j\right)}{\sqrt{{\left(\sum _{h=1}^{\mathcal{l}}{\left(F_{d_i^j}^{t_h^{*}}\right)}_{\max }^k\right)}^2\cdot {\omega }_j}}\text{,}$$ (56)

6. Illustrative example and comparative study

6.1. Illustrative example

A management team with four experts is going to select the best among four investment projects after ten weeks of investigation.

An expert $e_k\left(1\le k\le 4\right)$ use an DNSNTS ${\left(A_{i,j}^{t_h}\right)}^k$ ($1\le i\le m,1\le j\le n$ and $1\le h\le \mathcal{l}$)to describe the characteristics $C_j$ ($C_1$: net present value, $C_2$: rate of return, $C_3$: benefit-cost analysis, $C_4$: payback period) of each alternative $A_i$ (A₁: corn futures, A₂: gold futures, A₃: soybean futures, A₄: wheat futures, A₅: petroleum futures). The weighted vector of the four experts is $\xi ={\left(0.25,0.3,0.3,0.15\right)}^{\mathrm{T}}$, the weighted vector of the four characteristics is $\omega ={\left(0.1,0.4,0.25,0.25\right)}^{\mathrm{T}}$. The decision matrices are established in Table 1, Table 2, Table 3, Table 4.

Table 1.

Decision matrice proposed by $e_1$.

A1	C1	t1:(0.28,0.46,0.28)	t2:(0.27,0.44,0.27)	t3:(0.30,0.47,0.31)	t4:(0.34,0.47,0.33)	t5:(0.33,0.42,0.28)
		t6:(0.30,0.45,0.29)	t7:(0.31,0.49,0.31)	t8:(0.27,0.48,0.29)	t9:(0.33,0.44,0.33)	t10:(0.30,0.46,0.31)
	C2	t1:(0.10,0.27,0.61)	t2:(0.08,0.28,0.59)	t3:(0.14,0.24,0.60)	t4:(0.09,0.23,0.64)	t5:(0.10,0.25,0.59)
		t6:(0.12,0.27,0.64)	t7:(0.14,0.26,0.61)	t8:(0.09,0.25,0.57)	t9:(0.09,0.23,0.61)	t10:(0.10,0.22,0.64)
	C3	t1:(0.65,0.18,0.22)	t2:(0.62,0.24,0.27)	t3:(0.68,0.23,0.22)	t4:(0.65,0.23,0.27)	t5:(0.67,0.23,0.23)
		t6:(0.64,0.20,0.24)	t7:(0.66,0.19,0.25)	t8:(0.66,0.20,0.21)	t9:(0.67,0.18,0.28)	t10:(0.68,0.23,0.23)
	C4	t1:(0.53,0.34,0.15)	t2:(0.56,0.33,0.18)	t3:(0.57,0.30,0.16)	t4:(0.56,0.33,0.14)	t5:(0.56,0.29,0.16)
		t6:(0.53,0.31,0.18)	t7:(0.24,0.29,0.15)	t8:(0.53,0.33,0.13)	t9:(0.53,0.30,0.14)	t10:(0.54,0.34,0.19)
A2	C1	t1:(0.24,0.45,0.56)	t2:(0.19,0.47,0.52)	t3:(0.19,0.45,0.57)	t4:(0.24,0.44,0.57)	t5:(0.23,0.46,0.57)
		t6:(0.22,0.50,0.56)	t7:(0.20,0.48,0.52)	t8:(0.20,0.47,0.58)	t9:(0.19,0.46,0.53)	t10:(0.21,0.44,0.53)
	C2	t1:(0.16,0.45,0.26)	t2:(0.19,0.46,0.29)	t3:(0.21,0.42,0.28)	t4:(0.19,0.46,0.29)	t5:(0.19,0.45,0.29)
		t6:(0.20,0.42,0.28)	t7:(0.20,0.47,0.26)	t8:(0.16,0.41,0.27)	t9:(0.17,0.45,0.29)	t10:(0.16,0.44,0.27)
	C3	t1:(0.48,0.09,0.16)	t2:(0.48,0.07,0.20)	t3:(0.44,0.14,0.20)	t4:(0.48,0.07,0.21)	t5:(0.46,0.14,0.17)
		t6:(0.49,0.13,0.21)	t7:(0.49,0.07,0.22)	t8:(0.43,0.09,0.21)	t9:(0.47,0.13,0.21)	t10:(0.45,0.10,0.22)
	C4	t1:(0.74,0.09,0.11)	t2:(0.72,0.05,0.10)	t3:(0.74,0.10,0.09)	t4:(0.75,0.06,0.09)	t5:(0.77,0.06,0.11)
		t6:(0.77,0.06,0.07)	t7:(0.73,0.06,0.12)	t8:(0.72,0.07,0.09)	t9:(0.77,0.04,0.08)	t10:(0.76,0.04,0.09)
A3	C1	t1:(0.16,0.33,0.68)	t2:(0.14,0.38,0.68)	t3:(0.20,0.36,0.73)	t4:(0.16,0.32,0.70)	t5:(0.18,0.35,0.70)
		t6:(0.17,0.31,0.72)	t7:(0.15,0.32,0.73)	t8:(0.18,0.34,0.71)	t9:(0.15,0.37,0.70)	t10:(0.15,0.32,0.73)
	C2	t1:(0.26,0.24,0.35)	t2:(0.29,0.26,0.34)	t3:(0.27,0.21,0.33)	t4:(0.26,0.21,0.33)	t5:(0.26,0.24,0.37)
		t6:(0.24,0.25,0.34)	t7:(0.27,0.27,0.35)	t8:(0.24,0.21,0.33)	t9:(0.25,0.24,0.37)	t10:(0.27,0.22,0.35)
	C3	t1:(0.17,0.05,0.86)	t2:(0.19,0.07,0.87)	t3:(0.23,0.12,0.87)	t4:(0.22,0.06,0.86)	t5:(0.20,0.09,0.84)
		t6:(0.21,0.11,0.88)	t7:(0.15,0.32,0.73)	t8:(0.18,0.34,0.71)	t9:(0.15,0.37,0.70)	t10:(0.23,0.07,0.86)
	C4	t1:(0.45,0.14,0.23)	t2:(0.46,0.11,0.22)	t3:(0.43,0.08,0.24)	t4:(0.42,0.09,0.24)	t5:(0.41,0.13,0.24)
		t6:(0.45,0.14,0.21)	t7:(0.42,0.15,0.21)	t8:(0.46,0.10,0.19)	t9:(0.45,0.10,0.22)	t10:(0.43,0.11,0.25)
A4	C1	t1:(0.34,0.10,0.07)	t2:(0.38,0.11,0.09)	t3:(0.37,0.09,0.08)	t4:(0.32,0.10,0.04)	t5:(0.35,0.08,0.05)
		t6:(0.36,0.08,0.02)	t7:(0.37,0.14,0.04)	t8:(0.36,0.08,0.03)	t9:(0.35,0.09,0.07)	t10:(0.32,0.10,0.03)
	C2	t1:(0.74,0.45,0.67)	t2:(0.73,0.43,0.69)	t3:(0.72,0.44,0.67)	t4:(0.76,0.49,0.69)	t5:(0.75,0.48,0.70)
		t6:(0.72,0.47,0.64)	t7:(0.73,0.48,0.69)	t8:(0.71,0.46,0.67)	t9:(0.73,0.48,0.67)	t10:(0.75,0.44,0.65)
	C3	t1:(0.31,0.57,0.07)	t2:(0.28,0.56,0.02)	t3:(0.31,0.54,0.03)	t4:(0.30,0.54,0.02)	t5:(0.33,0.55,0.07)
		t6:(0.28,0.57,0.07)	t7:(0.32,0.60,0.08)	t8:(0.32,0.56,0.03)	t9:(0.31,0.55,0.04)	t10:(0.28,0.55,0.08)
	C4	t1:(0.23,0.19,0.29)	t2:(0.20,0.13,0.28)	t3:(0.24,0.15,0.25)	t4:(0.19,0.18,0.28)	t5:(0.23,0.14,0.27)
		t6:(0.19,0.18,0.23)	t7:(0.19,0.12,0.27)	t8:(0.21,0.13,0.26)	t9:(0.23,0.18,0.26)	t10:(0.21,0.18,0.28)
A5	C1	t1:(0.07,0.58,0.44)	t2:(0.07,0.58,0.43)	t3:(0.05,0.60,0.39)	t4:(0.06,0.61,0.44)	t5:(0.02,0.61,0.44)
		t6:(0.04,0.59,0.45)	t7:(0.02,0.60,0.44)	t8:(0.07,0.64,0.39)	t9:(0.06,0.61,0.43)	t10:(0.02,0.59,0.38)
	C2	t1:(0.43,0.12,0.81)	t2:(0.38,0.13,0.82)	t3:(0.40,0.17,0.81)	t4:(0.39,0.14,0.80)	t5:(0.41,0.14,0.81)
		t6:(0.43,0.16,0.82)	t7:(0.43,0.14,0.81)	t8:(0.37,0.11,0.82)	t9:(0.38,0.18,0.84)	t10:(0.44,0.14,0.83)
	C3	t1:(0.24,0.04,0.69)	t2:(0.23,0.07,0.67)	t3:(0.22,0.06,0.67)	t4:(0.25,0.06,0.70)	t5:(0.24,0.10,0.72)
		t6:(0.26,0.05,0.67)	t7:(0.24,0.09,0.67)	t8:(0.27,0.04,0.72)	t9:(0.27,0.05,0.68)	t10:(0.27,0.09,0.69)
	C4	t1:(0.14,0.25,0.91)	t2:(0.18,0.27,0.94)	t3:(0.19,0.28,0.89)	t4:(0.13,0.27,0.94)	t5:(0.19,0.27,0.93)
		t6:(0.20,0.23,0.95)	t7:(0.19,0.26,0.91)	t8:(0.19,0.23,0.92)	t9:(0.13,0.24,0.91)	t10:(0.20,0.23,0.93)

Table 2.

Decision matrice proposed by $e_2$.

A1	C1	t1:(0.17,0.28,0.06)	t2:(0.15,0.22,0.56)	t3:(0.18,0.27,0.61)	t4:(0.17,0.24,0.54)	t5:(0.18,0.26,0.55)
		t6:(0.15,0.27,0.56)	t7:(0.15,0.21,0.61)	t8:(0.19,0.23,0.58)	t9:(0.19,0.25,0.56)	t10:(0.19,0.21,0.58)
	C2	t1:(0.29,0.48,0.12)	t2:(0.19,0.47,0.17)	t3:(0.31,0.50,0.18)	t4:(0.33,0.47,0.17)	t5:(0.29,0.53,0.17)
		t6:(0.30,0.51,0.18)	t7:(0.34,0.47,0.15)	t8:(0.29,0.49,0.17)	t9:(0.27,0.51,0.17)	t10:(0.30,0.50,0.12)
	C3	t1:(0.45,0.23,0.16)	t2:(0.46,0.27,0.09)	t3:(0.46,0.23,0.10)	t4:(0.50,0.23,0.13)	t5:(0.47,0.28,0.12)
		t6:(0.46,0.24,0.16)	t7:(0.51,0.22,0.10)	t8:(0.44,0.22,0.09)	t9:(0.44,0.26,0.11)	t10:(0.47,0.26,0.10)
	C4	t1:(0.68,0.20,0.16)	t2:(0.63,0.25,0.21)	t3:(0.61,0.24,0.21)	t4:(0.62,0.22,0.16)	t5:(0.61,0.21,0.21)
		t6:(0.64,0.24,0.16)	t7:(0.66,0.23,0.20)	t8:(0.62,0.21,0.19)	t9:(0.63,0.21,0.16)	t10:(0.64,0.21,0.19)
A2	C1	t1:(0.33,0.29,0.44)	t2:(0.37,0.31,0.43)	t3:(0.35,0.34,0.45)	t4:(0.36,0.32,0.44)	t5:(0.35,0.27,0.49)
		t6:(0.37,0.28,0.49)	t7:(0.32,0.27,0.44)	t8:(0.37,0.32,0.44)	t9:(0.37,0.32,0.46)	t10:(0.36,0.34,0.50)
	C2	t1:(0.23,0.39,0.38)	t2:(0.20,0.39,0.32)	t3:(0.24,0.33,0.33)	t4:(0.25,0.37,0.33)	t5:(0.25,0.38,0.36)
		t6:(0.22,0.33,0.32)	t7:(0.25,0.36,0.33)	t8:(0.21,0.38,0.34)	t9:(0.25,0.37,0.37)	t10:(0.20,0.35,0.37)
	C3	t1:(0.55,0.17,0.23)	t2:(0.57,0.23,0.26)	t3:(0.58,0.23,0.24)	t4:(0.52,0.23,0.25)	t5:(0.53,0.21,0.22)
		t6:(0.53,0.19,0.25)	t7:(0.56,0.17,0.23)	t8:(0.53,0.22,0.24)	t9:(0.58,0.19,0.25)	t10:(0.54,0.20,0.24)
	C4	t1:(0.64,0.15,0.19)	t2:(0.60,0.17,0.22)	t3:(0.66,0.14,0.21)	t4:(0.59,0.16,0.21)	t5:(0.62,0.17,0.23)
		t6:(0.60,0.18,0.21)	t7:(0.64,0.12,0.19)	t8:(0.61,0.13,0.21)	t9:(0.59,0.12,0.22)	t10:(0.62,0.14,0.21)
A3	C1	t1:(0.30,0.24,0.58)	t2:(0.27,0.26,0.58)	t3:(0.25,0.24,0.58)	t4:(0.24,0.21,0.60)	t5:(0.27,0.23,0.55)
		t6:(0.26,0.23,0.60)	t7:(0.25,0.20,0.59)	t8:(0.27,0.24,0.59)	t9:(0.24,0.21,0.55)	t10:(0.27,0.25,0.59)
	C2	t1:(0.31,0.23,0.42)	t2:(0.28,0.22,0.44)	t3:(0.31,0.21,0.49)	t4:(0.31,0.19,0.42)	t5:(0.34,0.21,0.42)
		t6:(0.35,0.23,0.47)	t7:(0.28,0.19,0.46)	t8:(0.28,0.23,0.43)	t9:(0.31,0.18,0.48)	t10:(0.29,0.18,0.48)
	C3	t1:(0.37,0.16,0.56)	t2:(0.35,0.16,0.59)	t3:(0.37,0.12,0.59)	t4:(0.36,0.15,0.58)	t5:(0.32,0.13,0.60)
		t6:(0.34,0.17,0.55)	t7:(0.33,0.14,0.57)	t8:(0.36,0.18,0.60)	t9:(0.35,0.14,0.54)	t10:(0.37,0.12,0.56)
	C4	t1:(0.29,0.20,0.24)	t2:(0.32,0.19,0.27)	t3:(0.29,0.23,0.26)	t4:(0.31,0.25,0.23)	t5:(0.30,0.26,0.22)
		t6:(0.28,0.19,0.23)	t7:(0.29,0.25,0.24)	t8:(0.27,0.25,0.23)	t9:(0.31,0.21,0.22)	t10:(0.29,0.23,0.25)
A4	C1	t1:(0.35,0.23,0.16)	t2:(0.23,0.20,0.19)	t3:(0.35,0.24,0.17)	t4:(0.31,0.18,0.20)	t5:(0.30,0.23,0.16)
		t6:(0.32,0.22,0.15)	t7:(0.32,0.19,0.21)	t8:(0.32,0.23,0.20)	t9:(0.31,0.22,0.21)	t10:(0.30,0.20,0.17)
	C2	t1:(0.81,0.40,0.25)	t2:(0.83,0.37,0.24)	t3:(0.82,0.37,0.26)	t4:(0.86,0.42,0.29)	t5:(0.81,0.39,0.25)
		t6:(0.85,0.39,0.26)	t7:(0.83,0.35,0.27)	t8:(0.86,0.41,0.24)	t9:(0.81,0.36,0.24)	t10:(0.86,0.37,0.24)
	C3	t1:(0.38,0.47,0.17)	t2:(0.38,0.48,0.13)	t3:(0.33,0.49,0.12)	t4:(0.37,0.46,0.17)	t5:(0.36,0.42,0.14)
		t6:(0.31,0.48,0.12)	t7:(0.32,0.46,0.19)	t8:(0.34,0.48,0.15)	t9:(0.34,0.43,0.19)	t10:(0.37,0.42,0.13)
	C4	t1:(0.23,0.16,0.35)	t2:(0.28,0.21,0.33)	t3:(0.28,0.17,0.34)	t4:(0.28,0.22,0.37)	t5:(0.23,0.20,0.37)
		t6:(0.25,0.19,0.32)	t7:(0.29,0.19,0.31)	t8:(0.27,0.23,0.32)	t9:(0.27,0.19,0.38)	t10:(0.24,0.17,0.32)
A5	C1	t1:(0.12,0.58,0.45)	t2:(0.16,0.57,0.44)	t3:(0.12,0.57,0.44)	t4:(0.14,0.59,0.41)	t5:(0.15,0.56,0.47)
		t6:(0.16,0.57,0.43)	t7:(0.13,0.58,0.41)	t8:(0.16,0.57,0.46)	t9:(0.15,0.53,0.47)	t10:(0.12,0.58,0.48)
	C2	t1:(0.36,0.24,0.66)	t2:(0.35,0.19,0.69)	t3:(0.38,0.20,0.69)	t4:(0.33,0.20,0.66)	t5:(0.36,0.20,0.68)
		t6:(0.33,0.25,0.67)	t7:(0.32,0.25,0.72)	t8:(0.34,0.21,0.68)	t9:(0.37,0.23,0.71)	t10:(0.36,0.21,0.71)
	C3	t1:(0.33,0.11,0.58)	t2:(0.30,0.11,0.58)	t3:(0.30,0.10,0.61)	t4:(0.31,0.11,0.59)	t5:(0.30,0.09,0.63)
		t6:(0.28,0.14,0.61)	t7:(0.29,0.12,0.60)	t8:(0.29,0.10,0.61)	t9:(0.32,0.11,0.59)	t10:(0.32,0.13,0.61)
	C4	t1:(0.24,0.16,0.76)	t2:(0.22,0.16,0.70)	t3:(0.18,0.19,0.74)	t4:(0.24,0.21,0.70)	t5:(0.22,0.15,0.72)
		t6:(0.20,0.20,0.76)	t7:(0.22,0.19,0.75)	t8:(0.18,0.18,0.72)	t9:(0.18,0.16,0.73)	t10:(0.20,0.17,0.71)

Table 3.

Decision matrice proposed by $e_3$.

A1	C1	t1:(0.22,0.21,0.63)	t2:(0.23,0.19,0.66)	t3:(0.28,0.17,0.66)	t4:(0.24,0.15,0.67)	t5:(0.25,0.19,0.65)
		t6:(0.23,0.18,0.63)	t7:(0.23,0.16,0.66)	t8:(0.23,0.18,0.65)	t9:(0.28,0.22,0.67)	t10:(0.22,0.22,0.68)
	C2	t1:(0.20,0.77,0.38)	t2:(0.19,0.78,0.34)	t3:(0.21,0.79,0.36)	t4:(0.21,0.77,0.38)	t5:(0.24,0.82,0.38)
		t6:(0.19,0.82,0.34)	t7:(0.22,0.77,0.38)	t8:(0.24,0.79,0.33)	t9:(0.24,0.83,0.37)	t10:(0.21,0.81,0.33)
	C3	t1:(0.37,0.44,0.25)	t2:(0.40,0.49,0.19)	t3:(0.42,0.45,0.19)	t4:(0.39,0.43,0.23)	t5:(0.39,0.49,0.21)
		t6:(0.39,0.43,0.22)	t7:(0.42,0.47,0.20)	t8:(0.44,0.44,0.25)	t9:(0.42,0.46,0.18)	t10:(0.42,0.47,0.21)
	C4	t1:(0.49,0.23,0.29)	t2:(0.51,0.25,0.34)	t3:(0.49,0.25,0.29)	t4:(0.51,0.26,0.34)	t5:(0.52,0.25,0.28)
		t6:(0.51,0.24,0.30)	t7:(0.48,0.24,0.32)	t8:(0.51,0.27,0.34)	t9:(0.52,0.27,0.31)	t10:(0.53,0.29,0.32)
A2	C1	t1:(0.44,0.22,0.56)	t2:(0.46,0.24,0.54)	t3:(0.45,0.21,0.59)	t4:(0.46,0.20,0.54)	t5:(0.43,0.23,0.53)
		t6:(0.47,0.23,0.56)	t7:(0.47,0.21,0.58)	t8:(0.47,0.22,0.59)	t9:(0.45,0.20,0.53)	t10:(0.43,0.21,0.58)
	C2	t1:(0.20,0.39,0.34)	t2:(0.19,0.38,0.29)	t3:(0.20,0.38,0.31)	t4:(0.17,0.34,0.29)	t5:(0.19,0.34,0.31)
		t6:(0.18,0.40,0.29)	t7:(0.20,0.34,0.27)	t8:(0.21,0.38,0.33)	t9:(0.18,0.40,0.31)	t10:(0.17,0.33,0.28)
	C3	t1:(0.54,0.27,0.31)	t2:(0.50,0.28,0.35)	t3:(0.50,0.26,0.35)	t4:(0.47,0.26,0.31)	t5:(0.49,0.25,0.37)
		t6:(0.49,0.23,0.33)	t7:(0.50,0.27,0.35)	t8:(0.51,0.26,0.31)	t9:(0.50,0.24,0.36)	t10:(0.51,0.24,0.35)
	C4	t1:(0.55,0.24,0.16)	t2:(0.49,0.23,0.19)	t3:(0.55,0.23,0.14)	t4:(0.49,0.23,0.13)	t5:(0.52,0.25,0.14)
		t6:(0.53,0.23,0.16)	t7:(0.53,0.27,0.14)	t8:(0.48,0.24,0.16)	t9:(0.52,0.30,0.15)	t10:(0.49,0.26,0.18)
A3	C1	t1:(0.32,0.31,0.63)	t2:(0.27,0.33,0.67)	t3:(0.32,0.34,0.66)	t4:(0.33,0.34,0.63)	t5:(0.30,0.28,0.67)
		t6:(0.34,0.29,0.62)	t7:(0.33,0.32,0.65)	t8:(0.32,0.32,0.64)	t9:(0.33,0.35,0.66)	t10:(0.34,0.31,0.63)
	C2	t1:(0.24,0.35,0.44)	t2:(0.24,0.39,0.42)	t3:(0.26,0.36,0.42)	t4:(0.27,0.40,0.45)	t5:(0.24,0.39,0.41)
		t6:(0.25,0.39,0.43)	t7:(0.23,0.35,0.46)	t8:(0.23,0.38,0.44)	t9:(0.28,0.39,0.42)	t10:(0.27,0.35,0.40)
	C3	t1:(0.47,0.32,0.71)	t2:(0.46,0.38,0.69)	t3:(0.48,0.33,0.69)	t4:(0.48,0.33,0.73)	t5:(0.47,0.39,0.71)
		t6:(0.47,0.37,0.68)	t7:(0.44,0.34,0.73)	t8:(0.48,0.38,0.68)	t9:(0.44,0.37,0.71)	t10:(0.47,0.33,0.71)
	C4	t1:(0.35,0.30,0.47)	t2:(0.32,0.26,0.50)	t3:(0.38,0.25,0.48)	t4:(0.33,0.29,0.50)	t5:(0.37,0.26,0.50)
		t6:(0.33,0.28,0.49)	t7:(0.36,0.30,0.53)	t8:(0.35,0.28,0.52)	t9:(0.35,0.29,0.52)	t10:(0.33,0.29,0.50)
A4	C1	t1:(0.29,0.33,0.42)	t2:(0.29,0.27,0.40)	t3:(0.32,0.32,0.42)	t4:(0.30,0.27,0.42)	t5:(0.26,0.27,0.39)
		t6:(0.31,0.31,0.43)	t7:(0.31,0.32,0.39)	t8:(0.28,0.30,0.42)	t9:(0.26,0.30,0.39)	t10:(0.25,0.31,0.36)
	C2	t1:(0.18,0.24,0.57)	t2:(0.23,0.23,0.54)	t3:(0.19,0.23,0.57)	t4:(0.18,0.25,0.55)	t5:(0.19,0.26,0.55)
		t6:(0.23,0.27,0.53)	t7:(0.22,0.28,0.53)	t8:(0.20,0.28,0.52)	t9:(0.21,0.25,0.55)	t10:(0.20,0.28,0.56)
	C3	t1:(0.41,0.27,0.11)	t2:(0.47,0.28,0.11)	t3:(0.44,0.25,0.10)	t4:(0.44,0.28,0.12)	t5:(0.44,0.29,0.12)
		t6:(0.43,0.25,0.10)	t7:(0.42,0.28,0.11)	t8:(0.41,0.27,0.08)	t9:(0.41,0.26,0.11)	t10:(0.41,0.28,0.09)
	C4	t1:(0.20,0.34,0.20)	t2:(0.21,0.36,0.22)	t3:(0.24,0.34,0.22)	t4:(0.18,0.33,0.23)	t5:(0.20,0.36,0.24)
		t6:(0.20,0.37,0.23)	t7:(0.23,0.35,0.20)	t8:(0.21,0.38,0.19)	t9:(0.20,0.34,0.20)	t10:(0.18,0.36,0.22)
A5	C1	t1:(0.19,0.15,0.27)	t2:(0.20,0.18,0.29)	t3:(0.22,0.13,0.26)	t4:(0.22,0.12,0.31)	t5:(0.11,0.18,0.28)
		t6:(0.21,0.15,0.31)	t7:(0.21,0.14,0.26)	t8:(0.18,0.17,0.32)	t9:(0.23,0.13,0.30)	t10:(0.21,0.12,0.26)
	C2	t1:(0.33,0.39,0.15)	t2:(0.34,0.36,0.18)	t3:(0.28,0.39,0.13)	t4:(0.32,0.40,0.17)	t5:(0.28,0.34,0.15)
		t6:(0.30,0.35,0.14)	t7:(0.28,0.36,0.15)	t8:(0.34,0.36,0.13)	t9:(0.30,0.38,0.14)	t10:(0.34,0.41,0.12)
	C3	t1:(0.27,0.19,0.24)	t2:(0.27,0.26,0.23)	t3:(0.27,0.25,0.26)	t4:(0.28,0.22,0.26)	t5:(0.24,0.26,0.24)
		t6:(0.27,0.22,0.26)	t7:(0.25,0.24,0.23)	t8:(0.27,0.23,0.28)	t9:(0.21,0.22,0.25)	t10:(0.24,0.24,0.29)
	C4	t1:(0.37,0.18,0.53)	t2:(0.35,0.17,0.47)	t3:(0.37,0.21,0.47)	t4:(0.39,0.18,0.47)	t5:(0.35,0.16,0.51)
		t6:(0.40,0.19,0.47)	t7:(0.38,0.18,0.52)	t8:(0.38,0.21,0.48)	t9:(0.38,0.19,0.50)	t10:(0.39,0.19,0.47)

Table 4.

Decision matrice proposed by $e_4$.

A1	C1	t1:(0.38,0.19,0.61)	t2:(0.36,0.20,0.61)	t3:(0.36,0.22,0.62)	t4:(0.34,0.18,0.61)	t5:(0.36,0.24,0.55)
		t6:(0.35,0.20,0.61)	t7:(0.34,0.18,0.61)	t8:(0.34,0.23,0.61)	t9:(0.37,0.22,0.56)	t10:(0.35,0.21,0.60)
	C2	t1:(0.41,0.61,0.20)	t2:(0.37,0.58,0.17)	t3:(0.39,0.63,0.21)	t4:(0.41,0.65,0.23)	t5:(0.39,0.62,0.18)
		t6:(0.36,0.60,0.19)	t7:(0.42,0.63,0.18)	t8:(0.36,0.61,0.20)	t9:(0.36,0.63,0.17)	t10:(0.38,0.62,0.24)
	C3	t1:(0.42,0.34,0.16)	t2:(0.46,0.36,0.13)	t3:(0.46,0.35,0.16)	t4:(0.46,0.37,0.16)	t5:(0.48,0.35,0.15)
		t6:(0.44,0.33,0.18)	t7:(0.42,0.40,0.16)	t8:(0.46,0.38,0.15)	t9:(0.45,0.34,0.14)	t10:(0.47,0.35,0.16)
	C4	t1:(0.60,0.36,0.13)	t2:(0.60,0.35,0.15)	t3:(0.55,0.34,0.16)	t4:(0.59,0.38,0.12)	t5:(0.58,0.32,0.12)
		t6:(0.57,0.37,0.12)	t7:(0.59,0.37,0.11)	t8:(0.61,0.34,0.13)	t9:(0.56,0.35,0.17)	t10:(0.58,0.36,0.13)
A2	C1	t1:(0.31,0.38,0.68)	t2:(0.30,0.34,0.72)	t3:(0.28,0.38,0.67)	t4:(0.28,0.34,0.69)	t5:(0.33,0.34,0.71)
		t6:(0.32,0.39,0.72)	t7:(0.30,0.38,0.71)	t8:(0.31,0.34,0.66)	t9:(0.32,0.37,0.66)	t10:(0.29,0.40,0.66)
	C2	t1:(0.10,0.47,0.68)	t2:(0.09,0.46,0.65)	t3:(0.10,0.45,0.67)	t4:(0.12,0.46,0.63)	t5:(0.14,0.48,0.63)
		t6:(0.09,0.45,0.67)	t7:(0.13,0.50,0.68)	t8:(0.11,0.47,0.69)	t9:(0.08,0.46,0.64)	t10:(0.10,0.45,0.64)
	C3	t1:(0.56,0.35,0.20)	t2:(0.61,0.35,0.14)	t3:(0.59,0.34,0.19)	t4:(0.61,0.36,0.13)	t5:(0.60,0.33,0.14)
		t6:(0.58,0.34,0.20)	t7:(0.59,0.33,0.16)	t8:(0.60,0.37,0.14)	t9:(0.57,0.34,0.17)	t10:(0.60,0.36,0.16)
	C4	t1:(0.39,0.20,0.40)	t2:(0.41,0.19,0.42)	t3:(0.41,0.18,0.42)	t4:(0.41,0.20,0.38)	t5:(0.41,0.20,0.37)
		t6:(0.43,0.24,0.40)	t7:(0.42,0.18,0.35)	t8:(0.41,0.22,0.36)	t9:(0.39,0.23,0.37)	t10:(0.37,0.24,0.39)
A3	C1	t1:(0.34,0.28,0.55)	t2:(0.33,0.24,0.58)	t3:(0.33,0.21,0.57)	t4:(0.37,0.26,0.57)	t5:(0.37,0.25,0.56)
		t6:(0.34,0.27,0.59)	t7:(0.34,0.26,0.58)	t8:(0.37,0.26,0.53)	t9:(0.36,0.23,0.57)	t10:(0.37,0.24,0.54)
	C2	t1:(0.51,0.22,0.20)	t2:(0.49,0.25,0.18)	t3:(0.46,0.25,0.24)	t4:(0.48,0.24,0.24)	t5:(0.47,0.25,0.20)
		t6:(0.52,0.22,0.21)	t7:(0.45,0.24,0.24)	t8:(0.52,0.24,0.24)	t9:(0.45,0.28,0.19)	t10:(0.49,0.24,0.20)
	C3	t1:(0.13,0.30,0.92)	t2:(0.13,0.29,0.97)	t3:(0.16,0.29,0.98)	t4:(0.16,0.32,0.97)	t5:(0.12,0.29,0.96)
		t6:(0.14,0.28,0.93)	t7:(0.14,0.27,0.95)	t8:(0.16,0.31,0.96)	t9:(0.17,0.26,0.95)	t10:(0.11,0.31,0.94)
	C4	t1:(0.06,0.61,0.98)	t2:(0.06,0.55,0.96)	t3:(0.01,0.61,0.96)	t4:(0.02,0.06,0.96)	t5:(0.06,0.62,0.93)
		t6:(0.02,0.56,0.98)	t7:(0.02,0.60,0.96)	t8:(0.07,0.60,0.95)	t9:(0.04,0.56,0.93)	t10:(0.06,0.59,0.93)
A4	C1	t1:(0.24,0.52,0.03)	t2:(0.27,0.54,0.06)	t3:(0.25,0.53,0.08)	t4:(0.26,0.48,0.03)	t5:(0.25,0.51,0.07)
		t6:(0.26,0.54,0.08)	t7:(0.26,0.49,0.03)	t8:(0.26,0.51,0.06)	t9:(0.24,0.52,0.03)	t10:(0.27,0.49,0.09)
	C2	t1:(0.29,0.06,1.00)	t2:(0.32,0.03,0.98)	t3:(0.34,0.03,0.94)	t4:(0.34,0.02,0.99)	t5:(0.31,0.01,0.97)
		t6:(0.29,0.03,0.94)	t7:(0.34,0.03,0.98)	t8:(0.31,0.03,0.96)	t9:(0.34,0.03,0.97)	t10:(0.35,0.06,0.94)
	C3	t1:(0.17,0.49,0.29)	t2:(0.18,0.49,0.30)	t3:(0.23,0.46,0.28)	t4:(0.19,0.50,0.26)	t5:(0.20,0.50,0.26)
		t6:(0.22,0.51,0.29)	t7:(0.21,0.51,0.26)	t8:(0.19,0.47,0.27)	t9:(0.20,0.51,0.24)	t10:(0.20,0.47,0.24)
	C4	t1:(0.42,0.18,0.59)	t2:(0.46,0.12,0.54)	t3:(0.48,0.14,0.54)	t4:(0.45,0.14,0.53)	t5:(0.48,0.13,0.53)
		t6:(0.43,0.14,0.52)	t7:(0.47,0.15,0.52)	t8:(0.45,0.12,0.54)	t9:(0.46,0.16,0.58)	t10:(0.47,0.11,0.55)
A5	C1	t1:(0.35,0.04,0.45)	t2:(0.38,0.06,0.45)	t3:(0.34,0.06,0.44)	t4:(0.34,0.07,0.41)	t5:(0.34,0.08,0.45)
		t6:(0.32,0.08,0.41)	t7:(0.35,0.05,0.46)	t8:(0.34,0.06,0.44)	t9:(0.38,0.03,0.43)	t10:(0.35,0.05,0.44)
	C2	t1:(0.06,0.23,0.87)	t2:(0.06,0.22,0.87)	t3:(0.05,0.20,0.81)	t4:(0.07,0.19,0.86)	t5:(0.06,0.17,0.87)
		t6:(0.07,0.21,0.83)	t7:(0.08,0.22,0.83)	t8:(0.05,0.18,0.82)	t9:(0.03,0.17,0.83)	t10:(0.03,0.17,0.82)
	C3	t1:(0.77,0.08,0.13)	t2:(0.80,0.06,0.09)	t3:(0.78,0.06,0.09)	t4:(0.82,0.06,0.11)	t5:(0.78,0.04,0.07)
		t6:(0.83,0.08,0.10)	t7:(0.84,0.03,0.07)	t8:(0.84,0.06,0.12)	t9:(0.84,0.04,0.10)	t10:(0.78,0.07,0.11)
	C4	t1:(0.58,0.27,0.11)	t2:(0.56,0.26,0.11)	t3:(0.58,0.24,0.09)	t4:(0.57,0.28,0.13)	t5:(0.59,0.28,0.14)
		t6:(0.58,0.26,0.11)	t7:(0.61,0.26,0.09)	t8:(0.54,0.29,0.07)	t9:(0.60,0.28,0.14)	t10:(0.59,0.29,0.13)

Map the DNSNTS into the Euclidean 3-space aggregation model of DNSNS, and the collective expert preference ${\left[d_{ij}^{*}\right]}_{5\times 4}$ can be established by the proposed aggregation algorithm as follows (see Eq (57)):

$${\left[d_{ij}^{*}\right]}_{5\times 4}=\left[\begin{array}{c}{A}_1^{*}\\ ⋮\\ A_m^{*}\end{array}\right]=\left[\begin{array}{ccc}{A}_{1，1}^{*}& \cdots & A_{1,4}^{*}\\ ⋮& \ddots & ⋮\\ A_{5，1}^{*}& \cdots & A_{5,n}^{*}\end{array}\right]=\left[\begin{array}{ccc}\left\{A_{1,1}^{t_1^{*}},\cdot \cdot \cdot ,A_{1,1}^{t_{10}^{*}}\right\}& \cdots & \left\{A_{1,4}^{t_1^{*}},\cdot \cdot \cdot ,A_{1,4}^{t_{10}^{*}}\right\}\\ ⋮& \ddots & ⋮\\ \left\{A_{5,1}^{t_1^{*}},\cdot \cdot \cdot ,A_{5,1}^{t_{10}^{*}}\right\}& \cdots & \left\{A_{5,4}^{t_1^{*}},\cdot \cdot \cdot ,A_{5,4}^{t_{10}^{*}}\right\}\end{array}\right]$$ (57)

$$A_1^{*}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.2074,0.2517,0.6054\right)& \{\left(0.3063,0.5351,0.1762\right)& \{\left(0.4348,0.2976,0.1718\right)& \{\left(0.6192,0.2541,0.1720\right)\\ \left(0.1989,0.2148,0.5847\right)& \left(0.2926,0.4901,0.1799\right)& \left(0.4600,0.3600,0.1300\right)& \left(0.6273,0.2526,0.2103\right)\\ \left(0.2470,0.2373,0.6185\right)& \left(0.3147,0.5383,0.2041\right)& \left(0.4600,0.3500,0.1600\right)& \left(0.5881,0.2576,0.2080\right)\\ \left(0.2164,0.2186,0.5712\right)& \left(0.3315,0.5449,0.2204\right)& \left(0.4664,0.3384,0.1656\right)& \left(0.6047,0.2467,0.1684\right)\\ \left(0.2076,0.2524,0.5575\right)& \left(0.2935,0.5392,0.1768\right)& \left(0.4800,0.3500,0.1500\right)& \left(0.5963,0.2269,0.2070\right)\\ \left(0.2089,0.2366,0.5821\right)& \left(0.3030,0.5330,0.1920\right)& \left(0.4400,0.3300,0.1800\right)& \left(0.6076,0.2639,0.1709\right)\\ \left(0.2061,0.1939,0.6216\right)& \left(0.3428,0.5232,0.1841\right)& \left(0.4426,0.3755,0.1593\right)& \left(0.6272,0.2513,0.1973\right)\\ \left(0.1955,0.2289,0.5824\right)& \left(0.3012,0.5439,0.1993\right)& \left(0.4589,0.3512,0.1536\right)& \left(0.6104,0.2273,0.1919\right)\\ \left(0.2339,0.2451,0.5739\right)& \left(0.2781,0.5367,0.1842\right)& \left(0.4500,0.3400,0.1400\right)& \left(0.5829,0.2658,0.1842\right)\\ \left(0.1952,0.2113,0.5830\right)\}& \left(0.3169,0.5781,0.2041\right)\}& \left(0.4700,0.3500,0.1600\right)\}& \left(0.6010,0.2644,0.1996\right)\}\end{array}\right]$$

$$A_2^{*}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.3405,0.2957,0.4878\right)& \{\left(0.2300,0.3900,0.3800\right)& \{\left(0.5500,0.1700,0.2300\right)& \{\left(0.6400,0.1500,0.1900\right)\\ \left(0.3723,0.3072,0.4667\right)& \left(0.2000,0.3900,0.3200\right)& \left(0.5700,0.2300,0.2600\right)& \left(0.6000,0.1700,0.2200\right)\\ \left(0.3474,0.3346,0.4910\right)& \left(0.2260,0.3484,0.3327\right)& \left(0.5800,0.2300,0.2400\right)& \left(0.6600,0.1400,0.2100\right)\\ \left(0.3603,0.3143,0.4679\right)& \left(0.2500,0.3700,0.3300\right)& \left(0.5200,0.2300,0.2500\right)& \left(0.5900,0.1600,0.2100\right)\\ \left(0.3500,0.2700,0.4900\right)& \left(0.2500,0.3800,0.3600\right)& \left(0.5300,0.2100,0.2200\right)& \left(0.6200,0.1700,0.2300\right)\\ \left(0.3700,0.2800,0.4901\right)& \left(0.2097,0.3479,0.3229\right)& \left(0.5300,0.1900,0.2500\right)& \left(0.6000,0.1800,0.2100\right)\\ \left(0.3315,0.2797,0.4819\right)& \left(0.2500,0.3600,0.3300\right)& \left(\mathrm{0.5600.0.1700.0.2300}\right)& \left(0.6400,0.1200,0.1900\right)\\ \left(0.3696,0.3136,0.5021\right)& \left(0.2100,0.3800,0.3400\right)& \left(0.5300,0.2200,0.2400\right)& \left(0.6100,0.1300,0.2100\right)\\ \left(0.3700,0.3200,0.4600\right)& \left(0.2410,0.3750,0.3695\right)& \left(0.5800,0.1900,0.2500\right)& \left(0.5900,0.1200,0.2200\right)\\ \left(0.3600,0.3400,0.5000\right)\}& \left(0.2000,0.3500,0.3700\right)\}& \left(0.5400,0.2000,0.2400\right)\}& \left(0.6200,0.1400,0.2100\right)\}\end{array}\right]$$

$$A_3^{*}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.3032,0.2487,0.5816\right)& \{\left(0.3100,0.2300,0.4200\right)& \{\left(0.3581,0.2029,0.6415\right)& \{\left(0.3107,0.2550,0.3648\right)\\ \left(0.2700,0.2600,0.5800\right)& \left(0.2872,0.2416,0.4173\right)& \left(0.3416,0.1970,0.6451\right)& \left(0.3137,0.2314,0.3908\right)\\ \left(0.2506,0.2403,0.5803\right)& \left(0.3121,0.2391,0.4439\right)& \left(0.3626,0.1783,0.6652\right)& \left(0.3020,0.2393,0.3212\right)\\ \left(0.2785,0.2487,0.6041\right)& \left(0.3200,0.1900,0.4200\right)& \left(0.3560,0.2013,0.6702\right)& \left(0.3067,0.2742,0.3703\right)\\ \left(0.2840,0.2403,0.5668\right)& \left(0.3394,0.2121,0.4179\right)& \left(0.3133,0.1766,0.6633\right)& \left(0.3120,0.2687,0.3245\right)\\ \left(0.3077,0.2629,0.6055\right)& \left(0.3498,0.2306,0.4691\right)& \left(0.3359,0.2143,0.6292\right)& \left(0.2938,0.2392,0.3619\right)\\ \left(0.2890,0.2448,0.6029\right)& \left(0.2839,0.2122,0.4423\right)& \left(\mathrm{0.3192.0.1985.0.6741}\right)& \left(0.2984,0.2682,0.3225\right)\\ \left(0.2708,0.2408,0.5902\right)& \left(0.2800,0.2300,0.4300\right)& \left(0.3565,0.2102,0.6400\right)& \left(0.2906,0.2744,0.3538\right)\\ \left(0.2770,0.2389,0.5746\right)& \left(0.3105,0.2385,0.4223\right)& \left(0.3371,0.2046,0.6584\right)& \left(0.3139,0.2476,0.3531\right)\\ \left(0.2700,0.2500,0.5900\right)\}& \left(0.3017,0.2275,0.4179\right)\}& \left(0.3548,0.1850,0.6553\right)\}& \left(0.2992,0.2649,0.3795\right)\}\end{array}\right]$$

$$A_4^{*}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.3500,0.2300,0.1600\right)& \{\left(0.5255,0.3002,0.5340\right)& \{\left(0.3800,0.4700,0.1700\right)& \{\left(0.2300,0.1600,0.3500\right)\\ \left(0.3200,0.2000,0.1900\right)& \left(0.5370,0.2738,0.5244\right)& \left(0.3800,0.4800,0.1300\right)& \left(0.2800,0.2100,0.3300\right)\\ \left(0.3500,0.2400,0.1700\right)& \left(0.5391,0.2760,0.5338\right)& \left(0.3300,0.4900,0.1200\right)& \left(0.2800,0.1700,0.3400\right)\\ \left(0.3100,0.1800,0.2000\right)& \left(0.5852,0.3182,0.5330\right)& \left(0.3700,0.4600,0.1700\right)& \left(0.2800,0.2200,0.3700\right)\\ \left(0.3000,0.2300,0.1600\right)& \left(0.5318,0.2955,0.5196\right)& \left(0.3600,0.4200,0.1400\right)& \left(0.2300,0.2000,0.3700\right)\\ \left(0.3200,0.2200,0.1500\right)& \left(0.5515,0.3049,0.5101\right)& \left(0.3100,0.4800,0.1200\right)& \left(0.2500,0.1900,0.3200\right)\\ \left(0.3200,0.1900,0.2100\right)& \left(0.5621,0.2929,0.5164\right)& \left(\mathrm{0.3200.0.4600.0.1900}\right)& \left(0.2900,0.1900,0.3100\right)\\ \left(0.3200,0.2300,0.2000\right)& \left(0.5506,0.3137,0.5119\right)& \left(0.3400,0.4800,0.1500\right)& \left(0.2700,0.2300,0.3200\right)\\ \left(0.3100,0.2200,0.2100\right)& \left(0.5387,0.2840,0.5151\right)& \left(0.3400,0.4300,0.1900\right)& \left(0.2700,0.1900,0.3800\right)\\ \left(0.3000,0.2000,0.1700\right)\}& \left(0.5693,0.2978,0.5206\right)\}& \left(0.3700,0.4200,0.1300\right)\}& \left(0.2400,0.1700,0.3200\right)\}\end{array}\right]$$

$$A_5^{*}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.1304,0.5216,0.4363\right)& \{\left(0.3600,0.2400,0.6600\right)& \{\left(0.3300,0.1100,0.5800\right)& \{\left(0.3393,0.1831,0.5800\right)\\ \left(0.1709,0.4922,0.4239\right)& \left(0.3500,0.1900,0.6900\right)& \left(0.3000,0.1100,0.5800\right)& \left(0.3204,0.1774,0.5291\right)\\ \left(0.1402,0.4889,0.4173\right)& \left(0.3800,0.2000,0.6900\right)& \left(0.3000,0.1000,0.6100\right)& \left(0.3700,0.2100,0.4700\right)\\ \left(0.1640,0.4723,0.4001\right)& \left(0.3300,0.2000,0.6600\right)& \left(0.3100,0.1100,0.5900\right)& \left(0.2485,0.2107,0.6870\right)\\ \left(0.1654,0.4745,0.4433\right)& \left(0.3600,0.2000,0.6800\right)& \left(0.3000,0.0900,0.6300\right)& \left(0.3155,0.1697,0.5735\right)\\ \left(0.1807,0.4238,0.4070\right)& \left(0.3300,0.2500,0.6700\right)& \left(0.2800,0.1400,0.6100\right)& \left(0.3252,0.2038,0.5790\right)\\ \left(0.1455,0.4944,0.4015\right)& \left(0.3200,0.2500,0.7200\right)& \left(\mathrm{0.2900.0.1200.0.6000}\right)& \left(0.3106,0.1957,0.6190\right)\\ \left(0.1722,0.4741,0.4335\right)& \left(0.3400,0.2100,0.6800\right)& \left(0.2900,0.1000,0.6100\right)& \left(0.3057,0.2066,0.5604\right)\\ \left(0.1658,0.4674,0.4495\right)& \left(0.3700,0.2300,0.7100\right)& \left(0.3200,0.1100,0.5900\right)& \left(0.3286,0.1909,0.5551\right)\\ \left(0.1546,0.4350,0.4279\right)\}& \left(0.3600,0.2100,0.7100\right)\}& \left(0.3200,0.1300,0.6100\right)\}& \left(0.3485,0.1947,0.5231\right)\}\end{array}\right]$$

Then, the positive idea vector $A^{*+}$ and the negative idea vector $A^{*-}$ can be selected by Eq (41) and (42).

$$A^{*+}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.3500,0.2300,0.1600\right)& \{\left(0.5255,0.2300,0.1762\right)& \{\left(0.5500,0.1100,0.1700\right)& \{\left(0.6400,0.1500,0.1720\right)\\ \left(0.3723,0.2000,0.1900\right)& \left(0.5370,0.1900,0.1799\right)& \left(0.5700,0.1100,0.1300\right)& \left(0.6273,0.1700,0.2103\right)\\ \left(0.3500,0.2373,0.1700\right)& \left(0.5391,0.2000,0.2041\right)& \left(0.5800,0.1000,0.1200\right)& \left(0.6600,0.1400,0.2080\right)\\ \left(0.3603,0.1800,0.2000\right)& \left(0.5852,0.1900,0.2204\right)& \left(0.5200,0.1100,0.1656\right)& \left(0.6047,0.1600,0.1684\right)\\ \left(0.3500,0.2300,0.1600\right)& \left(0.5318,0.2000,0.1768\right)& \left(0.5300,0.0900,0.1400\right)& \left(0.6200,0.1697,0.2070\right)\\ \left(0.3700,0.2200,0.1500\right)& \left(0.5515,0.2306,0.1920\right)& \left(0.5300,0.1400,0.1200\right)& \left(0.6076,0.1800,0.1709\right)\\ \left(0.3315,0.1900,0.2100\right)& \left(0.5621,0.2122,0.1841\right)& \left(\mathrm{0.5600.0.1200.0.1593}\right)& \left(0.6400,0.1200,0.1900\right)\\ \left(0.3696,0.2289,0.2000\right)& \left(0.5506,0.2100,0.1993\right)& \left(0.5300,0.1000,0.1500\right)& \left(0.6104,0.1300,0.1919\right)\\ \left(0.3700,0.2200,0.2100\right)& \left(0.5387,0.2300,0.1842\right)& \left(0.5800,0.1100,0.1400\right)& \left(0.5900,0.1200,0.1842\right)\\ \left(0.3600,0.2000,0.1700\right)\}& \left(0.5693,0.2100,0.2041\right)\}& \left(0.5400,0.1300,0.1300\right)\}& \left(0.6200,0.1400,0.1996\right)\}\end{array}\right]$$

$$A^{*-}=\left[\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ \{\left(0.1304,0.5216,0.6054\right)& \{\left(0.2300,0.5351,0.6600\right)& \{\left(0.3300,0.4700,0.6415\right)& \{\left(0.2300,0.2550,0.5800\right)\\ \left(0.1709,0.4922,0.5847\right)& \left(0.2000,0.4901,0.6900\right)& \left(0.3000,0.4800,0.6451\right)& \left(0.2800,0.2526,0.5291\right)\\ \left(0.1402,0.4889,0.6185\right)& \left(0.2260,0.5383,0.6900\right)& \left(0.3000,0.4900,0.6652\right)& \left(0.2800,0.2576,0.4700\right)\\ \left(0.1640,0.4723,0.6041\right)& \left(0.2500,0.5449,0.6600\right)& \left(0.3100,0.4600,0.6702\right)& \left(0.2485,0.2742,0.6870\right)\\ \left(0.1654,0.4745,0.5668\right)& \left(0.2500,0.5392,0.6800\right)& \left(0.3000,0.4200,0.6633\right)& \left(0.2300,0.2687,0.5735\right)\\ \left(0.1807,0.4238,0.6055\right)& \left(0.2097,0.5330,0.6700\right)& \left(0.2800,0.4800,0.6296\right)& \left(0.2500,0.2639,0.5790\right)\\ \left(0.1455,0.4944,0.6216\right)& \left(0.2500,0.5232,0.7200\right)& \left(\mathrm{0.2900.0.4600.0.6741}\right)& \left(0.2900,0.2682,0.6190\right)\\ \left(0.1722,0.4741,0.5902\right)& \left(0.2100,0.5439,0.6800\right)& \left(0.2900,0.4800,0.6440\right)& \left(0.2700,0.2744,0.5604\right)\\ \left(0.1658,0.4674,0.5746\right)& \left(0.2410,0.5367,0.7100\right)& \left(0.3200,0.4300,0.6584\right)& \left(0.2700,0.2658,0.5551\right)\\ \left(0.1546,0.4350,0.5900\right)\}& \left(0.2000,0.5781,0.7100\right)\}& \left(0.3200,0.4200,0.6553\right)\}& \left(0.2400,0.2649,0.5231\right)\}\end{array}\right]$$

The positive projection of $A_i^{*}$ on $A^{*+}$ and the negative projection of $A_i^{*}$ on $A^{*-}$ can be calculated by Eqs. (48)–(56).

$$\begin{array}{c}\text{Prj}{\left(A_1\right)}^{+}=1.02013787,\text{Prj}{\left(A_1\right)}^{-}=1.02199371\text{;}\\ \text{Prj}{\left(A_2\right)}^{+}=0.75052065,\text{Prj}{\left(A_2\right)}^{-}=0.74997139\text{;}\\ \text{Prj}{\left(A_3\right)}^{+}=0.09328418,\text{Prj}{\left(A_3\right)}^{-}=0.09355276\text{;}\\ \text{Prj}{\left(A_4\right)}^{+}=0.70071218,\text{Prj}{\left(A_4\right)}^{-}=0.70321114\text{;}\\ \text{Prj}{\left(A_5\right)}^{+}=-0.124323,\text{Prj}{\left(A_5\right)}^{-}=-0.1296633\text{.}\end{array}$$

Finally, we obtain the alternative scores $S\left(A_i\right)$:

$$S\left(A_1\right)=-0.00186,S\left(A_2\right)=0.000549,S\left(A_3\right)=-0.00027,S\left(A_4\right)=-0.0025,S\left(A_5\right)=0.00534\text{.}$$

Thus, $A_5\succ A_2\succ A_3\succ A_1\succ A_4$ (Fig. 12).

Fig. 12.

The ranking of alternatives.

6.2. Comparative study

To confirm the efficiency of the proposed method and the importance of credibility levels in the MAGDM with nonlinear information application, we make a comparison with existing related methods in this actual example.

the Hamming distance between the collective matrix $\left[d_{ij}^{*}\right]$ and the preference matrix $\left[d_{ij}^{e_k}\right]$ proposed by expert $e_k$ (${\hat{d}}_H\left(d_{ij}^{*},d_{ij}^{e_k}\right)$), the correlation of $\left[d_{ij}^{*}\right]$ and $\left[d_{ij}^{e_k}\right]$ ($\hat{C}\left(d_{ij}^{*},d_{ij}^{e_k}\right)$), the informational intuitional energy of $\left[d_{ij}^{*}\right]$ and $\left[d_{ij}^{e_k}\right]$ ($E\left(d_{ij}^{*}\right)$ and $E\left(d_{ij}^{e_k}\right)$). and the correlation coefficient of $\left[d_{ij}^{*}\right]$ and $\left[d_{ij}^{e_k}\right]$ ($\hat{K}\left(d_{ij}^{*},d_{ij}^{e_k}\right)$) can describe the quality of the collective matrix accurately. Smaller value of ${\hat{d}}_H\left(d_{ij}^{*},d_{ij}^{e_k}\right)$ and greater values of $\hat{C}\left(d_{ij}^{*},d_{ij}^{e_k}\right)$ and $\hat{K}\left(d_{ij}^{*},d_{ij}^{e_k}\right)$ indicate a higher quality of the collective matrix $\left[d_{ij}^{*}\right]$. A smaller value of $E\left(d_{ij}^{*}\right)$ indicates a less fuzzy degree of $\left[d_{ij}^{*}\right]$. The calculation processes are shown in Eqs. (58)–(61):

$${\hat{d}}_H\left(d_{ij}^{*},d_{ij}^{e_k}\right)=\sum _{k=1}^4{d}_H\left(d_{ij}^{*},d_{ij}^{e_k}\right)=\sum _{k=1}^4\frac{1}{3}\sum _{i=1}^5\sum _{j=1}^4\left[\left|T_{d_{ij}^{*}}\left(x_{ij}\right)-T_{d_{ij}^{e_k}}\left(x_{ij}\right)\right|+\left|I_{d_{ij}^{*}}\left(x_{ij}\right)-I_{d_{ij}^{e_k}}\left(x_{ij}\right)\right|+\left|F_{d_{ij}^{*}}\left(x_{ij}\right)-F_{d_{ij}^{e_k}}\left(x_{ij}\right)\right|\right]$$ (58)

$$\hat{C}\left(d_{ij}^{*},d_{ij}^{e_k}\right)=\sum _{k=1}^{4}C\left(d_{ij}^{*},d_{ij}^{e_k}\right)=\frac{1}{2}\sum _{i=1}^5\sum _{j=1}^4\left[T_{d_{ij}^{*}}\left(x_{ij}\right)\cdot T_{d_{ij}^{e_k}}\left(x_{ij}\right)+I_{d_{ij}^{*}}\left(x_{ij}\right)\cdot I_{d_{ij}^{e_k}}\left(x_{ij}\right)+F_{d_{ij}^{*}}\left(x_{ij}\right)\cdot F_{d_{ij}^{e_k}}\left(x_{ij}\right)\right]$$ (59)

$$E\left(d_{ij}^{*}/d_{ij}^{e_k}\right)=\sum _{i=1}^5\sum _{j=1}^4\left[{\left(T_{d_{ij}^{*}/d_{ij}^{e_k}}\left(x_{ij}\right)\right)}^2+{\left(I_{d_{ij}^{*}/d_{ij}^{e_k}}\left(x_{ij}\right)\right)}^2+{\left(F_{d_{ij}^{*}/d_{ij}^{e_k}}\left(x_{ij}\right)\right)}^2\right]$$ (60)

$$\hat{K}\left(d_{ij}^{*},d_{ij}^{e_k}\right)=\sum _{k=1}^{4}K\left(d_{ij}^{*},d_{ij}^{e_k}\right)=\sum _{k=1}^4\frac{C\left(\left(d_{ij}^{*},d_{ij}^{e_k}\right)\right)}{\sqrt{E\left(d_{ij}^{*}\right)\cdot E\left(d_{ij}^{e_k}\right)}}$$ (61)

The proposed method, the exponential-logarithm SVNS [41], the SvNCNTWA and the SvNCNTWG [42] were respectively employed to solve the same example. Then, different results were obtained.

• (1)The proposed method: $A_5\succ A_2\succ A_3\succ A_1\succ A_4$;
• (2)The exponential-logarithm SVNS: $A_5\succ A_2\succ A_3\succ A_4\succ A_1$;
• (3)The SvNCNTWA: $A_5\succ A_2\succ A_3\succ A_1\succ A_4$;
• (4)The SvNCNTWG: $A_5\succ A_3\succ A_2\succ A_1\succ A_4$.

The indexes of the collective matrix constructed by different algorithms are shown in Table 5.

Table 5.

Accuracy comparison.

	${\hat{d}}_H\left(d_{ij}^{*},d_{ij}^{e_k}\right)$	$\hat{C}\left(d_{ij}^{*},d_{ij}^{e_k}\right)$	$E\left(d_{ij}^{*}\right)$	$\hat{K}\left(d_{ij}^{*},d_{ij}^{e_k}\right)$
PGSA	82.7795	166.6943	84.0071	3.7176
EL- SVNS	84.5682	158.5288	85.2588	3.5521
SvNCNTWA	2.967163	162.2254	86.2458	3.6628
SvNCNTWG	3.48762	161.5345	87.0126	3.5872

Obviously, the result obtained by the proposed method is more accurate.

7. Conclusions and suggestions for future research

The DNSNS and its aggregation model are introduced in this paper to address the MAGDM problem. The main contributions of this study are as follows.

• (1)The DNSNS enables the representation of changes in solution attributes over time and variations in the external environment, thereby facilitating the description of real-time fuzzy preference information provided by experts.
• (2)The proposed aggregation model maps nonlinear expert fuzzy preference information onto a three-dimensional curve, visually capturing the dynamics of preference information without compromising the relationships between scheme attributes caused by data preprocessing. Furthermore, it accurately describes differences in preference information using the area enclosed between space curves.
• (3)In this paper, we employ the PGSA algorithm to identify an optimal aggregation curve with a minimal sum of Euclidean distances from all preference curves. The obtained results adhere to Pareto optimality and exhibit higher accuracy compared to TOPSIS which directly employs positive and negative ideal solutions for preference aggregation. Moreover, during algorithm execution, PGSA updates morphactin concentration for all nodes in the solution space while dynamically adjusting their growth direction through a selection of new growth points from them, thus avoiding local optima pitfalls when compared with nearest neighbor search (NNS).

However, this study also reveals certain limitations.

• (1)The integration of the traditional TOPSIS algorithm for aggregating nonlinear fuzzy preference information does not effectively address the issue of rank reversal.
• (2)There is currently no efficient and accurate aggregation method available for handling high-dimensional nonlinear fuzzy preference information.

In future research, we will explore the use of NR-TOPSIS as an alternative to traditional TOPSIS in order to enhance the accuracy of aggregating nonlinear preference information. Additionally, we will investigate the Euclidean n-space Aggregation Model (n > 3) and its corresponding aggregation algorithm for addressing the Multi-Attribute Group Decision Making (MAGDM) problem involving dynamic nonlinear hesitant fuzzy linguistic information.

Data availability statement

The data of the paper has not been deposited into a publicly available repository. Data included in article/supp. material/referenced in article.

CRediT authorship contribution statement

Junda Qiu: Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Linjia Jiang: Formal analysis, Data curation, Conceptualization. Honghui Fan: Funding acquisition, Formal analysis, Data curation. Peng Li: Supervision, Software, Resources. Congzhe You: Validation, Supervision, Software, Resources.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.