**Multi-Attribute Group Decision Making Based on Multigranulation Probabilistic Models with Interval-Valued Neutrosophic Information**

#### **Chao Zhang 1, Deyu Li 1,\*, Xiangping Kang 2,3, Yudong Liang 1, Said Broumi <sup>4</sup> and Arun Kumar Sangaiah <sup>5</sup>**


Received: 8 January 2020; Accepted: 4 February 2020; Published: 9 February 2020

**Abstract:** In plenty of realistic situations, multi-attribute group decision-making (MAGDM) is ubiquitous and significant in daily activities of individuals and organizations. Among diverse tools for coping with MAGDM, granular computing-based approaches constitute a series of viable and efficient theories by means of multi-view problem solving strategies. In this paper, in order to handle MAGDM issues with interval-valued neutrosophic (IN) information, we adopt one of the granular computing (GrC)-based approaches, known as multigranulation probabilistic models, to address IN MAGDM problems. More specifically, after revisiting the related fundamental knowledge, three types of IN multigranulation probabilistic models are designed at first. Then, some key properties of the developed theoretical models are explored. Afterwards, a MAGDM algorithm for merger and acquisition target selections (M&A TSs) with IN information is summed up. Finally, a real-life case study together with several detailed discussions is investigated to present the validity of the developed models.

**Keywords:** multi-attribute group decision-making; granular computing; interval-valued neutrosophic information; multigranulation probabilistic models; merger and acquisition target selections

#### **1. Introduction**

#### *1.1. A Brief Review of MAGDM*

By applying decision-making issues with multiple attributes to the setting of group decision-making, multi-attribute group decision-making (MAGDM) generally provides consistent group preferences by analyzing various alternatives expressed by individual preferences [1]. To date, many granular computing (GrC)-based approaches [2–6] have been utilized to solve numerous complicated MAGDM problems, which have started a new momentum of constant development of social economy.

In the process of solving a typical MAGDM problem, it is recognized that three key challenges need to be managed reasonably, i.e., MAGDM information representation, MAGDM information fusion, MAGDM information analysis. Among the above-stated key challenges, how to express MAGDM information, especially for a complicated uncertain real-world scenario, as a standard decision matrix via alternatives and attributes is the first step to address MAGDM problems.

#### *1.2. A Brief Review of Interval-valued Neutrosophic Information*

In order to meet the demands of describing fuzzy and indeterminate information at the same time from nature and society, Smarandache [7,8] founded the notion of neutrosophic sets (NSs), which can be regarded as many generalizations of extended fuzzy sets [9] and used in plenty of meaningful areas [10–12]. An NS contains three types of membership functions (the truth, indeterminacy and falsity ones), and all of them take values in ]0−, 1+[. In accordance with the mathematical formulation of NSs, using NSs directly to a range of realistic applications is relatively inconvenient because all membership functions are limited within ]0−, 1+[. Thus, it is necessary to update ]0−, 1+[ by virtue of standard sets and logic. Following the above-stated research route, Wang et al. [13] put forward the concept of IN sets (INSs) from the viewpoint of definitions, operational laws, and others. For each membership function in INSs, all of them take values in the power set of [0, 1] instead of ]0−, 1+[. Thus, INSs can tackle the first key challenge of solving typical MAGDM problems well [14–22].

#### *1.3. A Brief Review of Multigranulation Probabilistic Models*

To handle MAGDM information fusion and analysis with IN information effectively, GrC-based approaches own unique superiorities in constructing problem addressing approaches via multi-view problem solving tactics [23,24]. During the past several years, taking full advantage of GrC-based approaches, many scholars and practitioners have obtained fruitful results in merging NSs with rough sets [25–32], formal concept analysis [33,34], three-way decisions [35–37], and others [38,39]. In the current article, we plan to propose a new IN MAGDM method via multigranulation probabilistic models, which can provide a risk-based information synthesis scheme with the capability of error tolerance in light of GrC-based approaches. In particular, the notion of multigranulation rough sets (MGRSs) [40–42] and probabilistic rough sets (PRSs) [43–45] is scheduled to establish multigranulation probabilistic models, and the merits of MGRSs and PRSs can be reflected in the process of MAGDM problem addressing.

#### *1.4. The Motivations of the Research*

MGRSs play a significant role in dealing with MAGDM problems in diverse backgrounds. One one hand, some scholars have made eminent contributions to the applications of MGRSs in MAGDM problems in recent years. For instance, Zhang et al. [46,47] developed various MGRSs in the context of hesitant fuzzy and interval-valued hesitant fuzzy sets for handling person-job fit and steam turbine fault diagnosis, respectively. Sun et al. [48,49] proposed several MGRSs with linguistic and heterogeneous preference information, then they further designed corresponding MAGDM approaches. Zhan et al. [50] and Zhang et al. [51] put forward two novel covering-based MGRSs with fuzzy and intuitionistic fuzzy information for addressing MAGDM problems. On the other hand, some scholars adopted PRSs to address MAGDM problems. For instance, Liang et al. [52–54] studied novel decision-theoretic rough sets in hesitant fuzzy, incomplete, and Pythagorean information systems. Xu and Guo [55] generalized MGRSs to double-quantitative and three-way decision frameworks. Zhang et al. [56,57] combined decision-theoretic rough sets with MGRSs in Pythagorean and hesitant fuzzy linguistic information systems. In this paper, we generalize MGRSs and PRSs to IN information and apply them to M&A TSs. Specifically, the following motivations of utilizing MGRSs and PRSs in IN MAGDM problems can be summed up:


#### *1.5. The Contributions of the Research*

In this work, we aim to utilize MGRSs and PRSs in solving complicated MAGDM issues with IN information. Specifically, several comprehensive risk-based models named IN multigranulation PRSs (MG-PRSs) over two universes are looked into. Then, we further present a MAGDM approach in the setting of M&A TSs in light of the developed theoretical models that can avoid an impact of three above-mentioned challenges. Finally, a real-world example is employed to prove the validity of the established decision-making rule. In addition, it is noteworthy that plenty of interesting nonlinear modeling approaches have been proved to be successful in various applications [58–67]. For instance, Medina and Ojeda-Aciego [58] applied multi-adjoint frameworks to general *t*-concept lattice, and some other works on fuzzy formal contexts based on GrC-based approaches were explored in succession [64–67]. Takacs et al. [59] put forward a brand-new oft tissue model for constructing telesurgical robot systems. Gil et al. [60] studied a surrogate model based optimization of traffic lights cycles and green period ratios by means of microscopic simulation and fuzzy rule interpolation. Smarandache et al. [61] explored word-level sentiment similarities in the context of NSs, and some other meaningful works on word-level sentiment analysis were also investigated recently [62,63,68,69].

Compared with existing popular nonlinear modeling approaches, the vital contributions of the work lie in the utilization of IN information and multigranulation probabilistic models. For one thing, the above-mentioned literature on nonlinear modeling approaches can not process various practical situations with indeterminate and incomplete information effectively, thus this work shows some merits in the representation of uncertain MAGDM information. For another, the majority of nonlinear modeling approaches can not fuse and analyze multi-source information with incorrect and noisy data reasonably, thus this work shows some merits in the fusion and analysis of MAGDM problems with IN information. Moreover, several specific key contributions of the work can be further concluded below:


#### *1.6. The Structure of the Research*

The rest of the work is arranged below. The next section intends to review several basic knowledge on INSs, MGRSs, and PRSs. Three types of theoretical models along with their key properties are explored in Section 3. In the next section, we develop an IN MAGDM approach via multigranulation probabilistic models in the context of M&A TSs. In Section 5, a practical illustrative case study is explored to highlight the validity of the presented IN MAGDM rule. Finally, Section 6 contains several conclusive results and future study options.

#### **2. Preliminaries**

The current section plans to revisit various preliminary knowledge in terms of INSs, MGRSs, and PRSs in a brief outline.

#### *2.1. INSs*

The concept of INSs was put forward by Wang et al. [13] by updating the formulation of ]0−, 1+[ from the scope of membership functions in NSs. With this update, INSs are equipped with the capability of expressing indeterminate and incomplete information simultaneously.

**Definition 1** ([13])**.** *Suppose U is an arbitrary universe of discourse. An INS E over U is provided as the following mathematical expression:*

$$E = \{ \langle \mathbf{x}, [\mu\_E^L(\mathbf{x}), \mu\_E^{\mathrm{II}}(\mathbf{x})], [\nu\_E^L(\mathbf{x}), \nu\_E^{\mathrm{II}}(\mathbf{x})], [\omega\_E^L(\mathbf{x}), \omega\_E^{\mathrm{II}}(\mathbf{x})] \rangle \, | \mathbf{x} \in \mathsf{U} \},$$

*where μ*, *ν*, *ω* : *U* → int [0, 1] *(*int [0, 1] *represents the set of all closed subintervals of* [0, 1]*). Similar with NSs, there also exists the restriction of* <sup>0</sup> <sup>≤</sup> *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*x*) + *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*x*) + *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*) ≤ 3*. Moreover, a set that includes all INSs over U is further named IN* (*U*)*.*

For an INS *E*, it is noticed that *E* may degenerate into two special forms, i.e., an INS *E* is named a full INS *U* when [*μ<sup>L</sup> <sup>E</sup>* (*x*), *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*x*)] = [1, 1], [*ν<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*x*)] = [0, 0] and [*ω<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*)] = [0, 0]; an INS *E* is named an empty INS ∅ when [*μ<sup>L</sup> <sup>E</sup>* (*x*), *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*x*)] = [0, 0], [*ν<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*x*)] = [1, 1] and [*ω<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*)] = [1, 1].

In IN MAGDM information analysis, it is common to compare the magnitude of IN numbers, thus a frequently-used method was developed in [15].

**Definition 2** ([15])**.** *Suppose <sup>x</sup>* <sup>=</sup> [*μ<sup>L</sup> <sup>E</sup>* (*x*), *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*x*)], [*ν<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*x*)], [*ω<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*)] *is an IN number. A score function with regard to x is provided as the following mathematical expression:*

$$\mathbf{s}\left(\mathbf{x}\right) = \left[\mu\_E^{\mathrm{L}}\left(\mathbf{x}\right) + \mathbf{1} - \nu\_E^{\mathrm{II}}\left(\mathbf{x}\right) + \mathbf{1} - \omega\_E^{\mathrm{II}}\left(\mathbf{x}\right), \mu\_E^{\mathrm{II}}\left(\mathbf{x}\right) + \mathbf{1} - \nu\_E^{\mathrm{L}}\left(\mathbf{x}\right) + \mathbf{1} - \omega\_E^{\mathrm{L}}\left(\mathbf{x}\right)\right]\_\mathbf{x}$$

*for arbitrary two IN numbers x and y, x* ≤ *y* ⇔ *s*(*x*) ≤ *s*(*y*) *is evident.*

It is noted that another significant issue in IN MAGDM information analysis is viable operational laws, which are used in constructing various IN MAGDM models. In what follows, we present several IN operational laws.

**Definition 3** ([13,15,22])**.** *Suppose E and F are arbitrary two INSs over U, then some common IN operational laws are provided as the following mathematical expressions:*


$$\begin{array}{ll} \text{4.} & E^{\lambda} = \{ \langle \mathbf{x}, \left[ \left( \mu\_{E}^{\mathrm{I}} \left( \mathbf{x} \right) \right)^{\lambda}, \left( \mu\_{E}^{\mathrm{II}} \left( \mathbf{x} \right) \right)^{\lambda} \right], \left[ 1 - \left( 1 - \nu\_{E}^{\mathrm{I}} \left( \mathbf{x} \right) \right)^{\lambda}, 1 - \left( 1 - \nu\_{E}^{\mathrm{II}} \left( \mathbf{x} \right) \right)^{\lambda} \right] \}, \\ & \left[ 1 - \left( 1 - \omega\_{E}^{\mathrm{I}} \left( \mathbf{x} \right) \right)^{\lambda}, 1 - \left( 1 - \omega\_{E}^{\mathrm{II}} \left( \mathbf{x} \right) \right)^{\lambda} \right] \rangle \mid \mathbf{x} \in \mathsf{U} \}; \end{array}$$

*Mathematics* **2020**, *8*, 223

*5. E F* = {*x*, [ *μL E*(*x*)−*μ<sup>L</sup> <sup>F</sup>*(*x*) <sup>1</sup>−*μ<sup>L</sup> <sup>F</sup>*(*x*) , *μ<sup>U</sup> <sup>E</sup>* (*x*)−*μ<sup>U</sup> <sup>F</sup>* (*x*) <sup>1</sup>−*μ<sup>U</sup> <sup>F</sup>* (*x*) ], [ *νL <sup>E</sup>*(*x*) *νL <sup>F</sup>* (*x*) , *νU <sup>E</sup>* (*x*) *νU <sup>F</sup>* (*x*) ], [ *ω<sup>L</sup> <sup>E</sup>*(*x*) *ω<sup>L</sup> <sup>F</sup>* (*x*) , *ω<sup>U</sup> <sup>E</sup>* (*x*) *ω<sup>U</sup> <sup>F</sup>* (*x*) ]|*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* }*, if <sup>E</sup>* <sup>⊇</sup> *F, <sup>ν</sup><sup>L</sup> <sup>E</sup>* (*x*) ≤ *νU <sup>E</sup>* (*x*)*ν<sup>L</sup> <sup>F</sup>* (*x*) *νU <sup>F</sup>* (*x*) *and <sup>ω</sup><sup>L</sup> <sup>E</sup>* (*x*) <sup>≤</sup> *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*)*ω<sup>L</sup> <sup>F</sup>* (*x*) *ω<sup>U</sup> <sup>F</sup>* (*x*) *; 6. E F* = {*x*, [ *μL <sup>E</sup>*(*x*) *μL <sup>F</sup>*(*x*) , *μ<sup>U</sup> <sup>E</sup>* (*x*) *μ<sup>U</sup> <sup>F</sup>* (*x*) ], [ *νL E*(*x*)−*ν<sup>L</sup> <sup>F</sup>* (*x*) <sup>1</sup>−*ν<sup>L</sup> <sup>F</sup>* (*x*) , *νU <sup>E</sup>* (*x*)−*ν<sup>U</sup> <sup>F</sup>* (*x*) <sup>1</sup>−*ν<sup>U</sup> <sup>F</sup>* (*x*) ], [ *ω<sup>L</sup> E*(*x*)−*ω<sup>L</sup> <sup>F</sup>* (*x*) <sup>1</sup>−*ω<sup>L</sup> <sup>F</sup>* (*x*) , *ω<sup>U</sup> <sup>E</sup>* (*x*)−*ω<sup>U</sup> <sup>F</sup>* (*x*) <sup>1</sup>−*ω<sup>U</sup> <sup>F</sup>* (*x*) ]|*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* }*, if <sup>F</sup>* <sup>⊇</sup> *<sup>E</sup> and ω<sup>L</sup> <sup>E</sup>* (*x*) <sup>≤</sup> *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*)*ω<sup>L</sup> <sup>F</sup>* (*x*) *ω<sup>U</sup> <sup>F</sup>* (*x*) *; 7. E<sup>c</sup>* <sup>=</sup> {*x*, [*ω<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*x*)], [<sup>1</sup> <sup>−</sup> *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*x*), 1 <sup>−</sup> *<sup>ν</sup><sup>L</sup> <sup>E</sup>* (*x*)], [*μ<sup>L</sup> <sup>E</sup>* (*x*), *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*x*)]|*x* ∈ *U* }*; 8. <sup>E</sup>* <sup>∪</sup> *<sup>F</sup>* <sup>=</sup> {*x*, / max  *μL <sup>E</sup>* (*x*), *<sup>μ</sup><sup>L</sup> <sup>F</sup>* (*x*) , max  *μ<sup>U</sup> <sup>E</sup>* (*x*), *<sup>μ</sup><sup>U</sup> <sup>F</sup>* (*x*) 0 , / min  *νL <sup>E</sup>* (*x*), *<sup>ν</sup><sup>L</sup> <sup>F</sup>* (*x*) , min  *νU <sup>E</sup>* (*x*), *<sup>ν</sup><sup>U</sup> <sup>F</sup>* (*x*) 0 , / min  *ω<sup>L</sup> <sup>E</sup>* (*x*), *<sup>ω</sup><sup>L</sup> <sup>F</sup>* (*x*) , min  *ω<sup>U</sup> <sup>E</sup>* (*x*), *<sup>ω</sup><sup>U</sup> <sup>F</sup>* (*x*) 0- <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* . ;

$$\begin{array}{ll} \mathfrak{g}. & E \cap F = \{ \langle \mathbf{x}, \left[ \min \left( \mu\_{E}^{\mathrm{L}}(\mathbf{x}), \mu\_{F}^{\mathrm{L}}(\mathbf{x}) \right), \min \left( \mu\_{E}^{\mathrm{II}}(\mathbf{x}), \mu\_{F}^{\mathrm{II}}(\mathbf{x}) \right) \right], \left[ \max \left( \nu\_{E}^{\mathrm{L}}(\mathbf{x}), \nu\_{F}^{\mathrm{L}}(\mathbf{x}) \right), \max \left( \mu\_{E}^{\mathrm{II}}(\mathbf{x}), \mu\_{F}^{\mathrm{II}}(\mathbf{x}) \right) \right] \\ & \max \left( \nu\_{E}^{\mathrm{II}}(\mathbf{x}), \nu\_{F}^{\mathrm{II}}(\mathbf{x}) \right) \Big], \left[ \max \left( \nu\_{E}^{\mathrm{L}}(\mathbf{x}), \mu\_{F}^{\mathrm{L}}(\mathbf{x}) \right), \max \left( \mu\_{E}^{\mathrm{II}}(\mathbf{x}), \mu\_{F}^{\mathrm{II}}(\mathbf{x}) \right) \right] \Big| \, \left| \mathbf{x} \in \mathcal{U} \right]; \end{array}$$


#### *2.2. MGRSs*

As one of the most influential generalized rough set theories, the idea of MGRSs was initially established by Qian et al. [40–42] by means of parallel computational frameworks and risk-based information fusion strategies.

**Definition 4** ([40,41])**.** *Suppose R*1, *R*2, ... , *Rm are m crisp binary relations. For any X* ⊆ *U, the optimistic and pessimistic multigranulation approximations of X are provided as the following mathematical expressions:*

$$\begin{aligned} \sum\_{i=1}^{m} \mathcal{R}\_{i}^{\mathcal{O}} \left( X \right) &= \{ \left[ \mathbf{x} \right]\_{\mathcal{R}\_{1}} \subseteq X \lor \left[ \mathbf{x} \right]\_{\mathcal{R}\_{1}} \subseteq X \lor \dots \lor \left[ \mathbf{x} \right]\_{\mathcal{R}\_{m}} \left| \mathbf{x} \in \mathcal{U} \right. \}; \\ \overline{\sum\limits\_{i=1}^{m} \mathcal{R}\_{i}}^{\mathcal{O}} \left( X \right) &= \{ \sum\limits\_{i=1}^{m} \mathcal{R}\_{i} \; \left( X^{c} \right) \}^{c}; \\ \sum\limits\_{i=1}^{m} \mathcal{R}\_{i}^{\mathcal{P}} \left( X \right) &= \{ \left[ \mathbf{x} \right]\_{\mathcal{R}\_{1}} \subseteq X \land \left[ \mathbf{x} \right]\_{\mathcal{R}\_{1}} \subseteq X \land \dots \land \left[ \mathbf{x} \right]\_{\mathcal{R}\_{m}} \left| \mathbf{x} \in \mathcal{U} \right. \}; \\ \overline{\sum\limits\_{i=1}^{m} \mathcal{R}\_{i}}^{\mathcal{P}} \left( X \right) &= (\sum\limits\_{i=1}^{m} \mathcal{R}\_{i} \; \left( X^{c} \right))^{c}; \end{aligned} \right. \end{aligned}$$

*the pair* ( *m* ∑ *i*=1 *Ri O* (*X*), *m* ∑ *i*=1 *Ri O* (*X*)) *is named an optimistic MGRS with regard to X, whereas the pair* ( *m* ∑ *i*=1 *Ri P* (*X*), *m* ∑ *i*=1 *Ri P* (*X*)) *is named a pessimistic MGRS with regard to X.*

#### *2.3. PRSs*

Considering that the formulation of classical rough sets is fairly strict which may affect the application range of it, hence the concept of PRSs [43–45] was developed subsequently by means of the probabilistic measure theory.

**Definition 5** ([43])**.** *Suppose R is an equivalence relation over U, P is the probabilistic measure, then* (*U*, *R*, *P*) *is named a probabilistic approximation space. For any* 0 ≤ *β* < *α* ≤ 1 *and X* ⊆ *U, the lower and upper approximations of X are provided as the following mathematical expressions:*

$$\begin{aligned} \underline{R}\_{\alpha}(X) &= \{ P\left(X \,|\big[x\big]\_{R}\right) \ge \alpha \, |x \in \mathcal{U} \}; \\ \overline{R}\_{\beta}(X) &= \{ P\left(X \,|\big[x\big]\_{R}\right) > \beta \, |x \in \mathcal{U} \}. \end{aligned}$$

*the pair* (*R<sup>α</sup>* (*X*), *R<sup>β</sup>* (*X*)) *is named a PRS of X with regard to* (*U*, *R*, *P*)*.*

#### **3. IN MG-PRSs over Two Universes**

In what follows, prior to the introduction of new theoretical models, we shall revisit the formulation of IN relations within the context of two universes [28] at first.

**Definition 6** ([28])**.** *Suppose U and V are two an arbitrary universes of discourse. An IN relation over two universes R over U* × *V is provided as the following mathematical expression:*

$$\mathcal{R} = \{ \langle (\mathbf{x}, y), [\mu\_{\mathcal{R}}^{\mathcal{L}}(\mathbf{x}, y), \mu\_{\mathcal{R}}^{\mathcal{U}}(\mathbf{x}, y)], [\nu\_{\mathcal{R}}^{\mathcal{U}}(\mathbf{x}, y), \nu\_{\mathcal{R}}^{\mathcal{U}}(\mathbf{x}, y)], [\omega\_{\mathcal{R}}^{\mathcal{L}}(\mathbf{x}, y), \omega\_{\mathcal{R}}^{\mathcal{U}}(\mathbf{x}, y)] \rangle \, |(\mathbf{x}, y) \in \mathcal{U} \times V\},$$

*where <sup>μ</sup>*, *<sup>ν</sup>*, *<sup>ω</sup>* : *<sup>U</sup>*, *<sup>V</sup>* <sup>→</sup> int [0, 1]*. Similar with INSs, there also exists the restriction of* <sup>0</sup> <sup>≤</sup> *<sup>μ</sup><sup>U</sup> <sup>R</sup>* (*x*, *y*) + *νU <sup>R</sup>* (*x*, *<sup>y</sup>*) + *<sup>ω</sup><sup>U</sup> <sup>R</sup>* (*x*, *y*) ≤ 3*. Moreover, a set that includes all IN relations over U* × *V is further named INR* (*U* × *V*)*.*

#### *3.1. Optimistic IN MG-PRSs over Two Universes*

It is noted that the term "optimistic" is originated from the first paper of MGRSs [40]. Within the context of MGRSs, the notion of single and multiple IN inclusion degrees is scheduled to propose at first in the current section.

**Definition 7.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the single IN membership degree of x in E in terms of Ri is provided as the following mathematical expression:*

$$\eta\_{E}^{R\_{i}}\left(\mathbf{x}\right) = \frac{\sum\_{y \in V} R\_{i}\left(\mathbf{x}, y\right) E\left(y\right)}{\sum\_{y \in V} R\_{i}\left(\mathbf{x}, y\right)}.$$

*based on ηRi <sup>E</sup>* (*x*)*, the multiple IN membership degrees of x in E with regard to Ri are provided as the following mathematical expressions:*

$$\begin{aligned} \Psi\_E^{\stackrel{m}{i=1}R\_i}(\mathbf{x}) &= \min\_{i=1}^m \eta\_E^{R\_i}(\mathbf{x});\\ \Omega\_E^{\stackrel{m}{i=1}R\_i}(\mathbf{x}) &= \max\_{i=1}^m \eta\_E^{R\_i}(\mathbf{x}) \end{aligned}$$

Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *is named a minimal IN membership degree, whereas we call* Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *a maximal IN membership degree.*

In light of maximal IN membership degrees, optimistic multigranulation probabilistic models can be put forward conveniently.

**Definition 8.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are represented by α and β with α* > *β, then the lower and upper approximations of E in optimistic multigranulation probabilistic models are provided as the following mathematical expressions:*

$$\begin{aligned} \sum\_{i=1}^{\underline{m}} R\_i \quad (E) &= \{\Omega\_E^{i=1} \mid \mathbf{x} \rangle \ge \alpha \mid \mathbf{x} \in \mathcal{U} \};\\ \overbrace{\sum\limits\_{i=1}^{\underline{m}} R\_i}^{\underline{m}} \quad (E) &= \{\Omega\_E^{i=1} \mid \mathbf{x} \rangle > \beta \mid \mathbf{x} \in \mathcal{U} \}.\end{aligned}$$

*the pair* ( *m* ∑ *i*=1 *Ri* Ω,*α* (*E*), *m* ∑ *i*=1 *Ri* Ω,*β* (*E*)) *is named an optimistic IN MG-PRS over two universes of E.*

In what follows, some key properties of lower and upper approximations for optimistic multigranulation probabilistic models are explored in detail.

**Proposition 1.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E*, *F* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are denoted by α and β with α* > *β, then the lower and upper approximations for optimistic multigranulation probabilistic models own the following properties:*

$$\begin{array}{llll} \text{1.} & \underset{\begin{subarray}{c} \stackrel{\scriptstyle \! \! .}{=} \stackrel{\scriptstyle \! \! \! .}{=} \,\,\,\overline{R\_{i}} \quad (\mathcal{O}) = \mathcal{O}; \; \overline{\sum\limits\_{i=1}^{m} \Omega\_{i}} \Omega\_{i} \quad (\mathcal{V}) = \mathcal{U};\\ \text{2.} & a\_{1} \leq a\_{2} \Rightarrow \sum\limits\_{i=1}^{m} R\_{i} \quad (\mathcal{E}) \subseteq \stackrel{\scriptstyle \! \! \! \! .}{=} \,\,^{\Omega\_{1,a}}\_{i} \quad (\mathcal{E}), \,\beta\_{1} \leq \beta\_{2} \Rightarrow \sum\limits\_{i=1}^{m} R\_{i} \quad (\mathcal{E}) \subseteq \stackrel{\scriptstyle \! \! \! \! \/ }{}\_{i} \,\, \mathcal{R}\_{i} \,\, \/ \\ \text{3.} & \underset{\begin{subarray}{c} E \subseteq F \Rightarrow \sum\limits\_{i=1}^{m} R\_{i} \quad (E) \subseteq \stackrel{\scriptstyle \! \! \! \/ } \mathcal{R}\_{i} \quad (F) \subseteq \stackrel{\scriptstyle \! \! \! \! \/ } \mathcal{R}\_{i} \,\, \/ \end{subarray}}{\begin{subarray}{c} \Omega\_{a} \,\, \/ \}} \mathcal{R}\_{i} \,\, \mathcal{R}\_{i} \,\, \/ \$$

**Proof.**

1. *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α* (∅) = {Ω *m* ∑ *i*=1 *Ri* <sup>∅</sup> (*x*) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> { *<sup>m</sup>* max *i*=1 <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*)<sup>∅</sup> <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> <sup>∅</sup>, *m* ∑ *i*=1 *Ri* Ω,*β* (*V*) = {Ω *m* ∑ *i*=1 *Ri <sup>V</sup>* (*x*) <sup>&</sup>gt; *<sup>β</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> { *<sup>m</sup>* max *i*=1 <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*)*<sup>V</sup>* <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*) <sup>&</sup>gt; *<sup>β</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> *<sup>U</sup>*. Thus, *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α* (∅) = ∅ and *m* ∑ *i*=1 *Ri* Ω,*β* (*V*) = *U* can be obtained.

2. Since *<sup>α</sup>*<sup>1</sup> <sup>≤</sup> *<sup>α</sup>*2, we have *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α*<sup>2</sup> (*E*) = {Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) ≥ *α*<sup>2</sup> |*x* ∈ *U* }⊆{Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) ≥ *α*<sup>1</sup> |*x* ∈ *U* } = *m* ∑ *i*=1 *Ri* Ω,*α*<sup>1</sup> (*E*). Thus, *<sup>α</sup>*<sup>1</sup> <sup>≤</sup> *<sup>α</sup>*<sup>2</sup> <sup>⇒</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α*<sup>2</sup> (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α*<sup>1</sup> (*E*) can be concluded. In a similar manner, we can also prove *<sup>β</sup>*<sup>1</sup> <sup>≤</sup> *<sup>β</sup>*<sup>2</sup> <sup>⇒</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*β*<sup>2</sup> (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*β*<sup>1</sup> (*E*).

3. Since *<sup>E</sup>* <sup>⊆</sup> *<sup>F</sup>*, we have *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α* (*E*) = {Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) ≥ *α* |*x* ∈ *U* } = { *m* max *i*=1 <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*)*E*(*y*) <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* }⊆{<sup>Ω</sup> *m* ∑ *i*=1 *Ri <sup>F</sup>* (*x*) ≥ *α* |*x* ∈ *U* } = { *m* max <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*)*F*(*y*) <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> *<sup>m</sup>* ∑ *Ri* Ω,*α* (*F*). Thus, *<sup>E</sup>* <sup>⊆</sup> *<sup>F</sup>* <sup>⇒</sup> *<sup>m</sup>* ∑ *Ri* Ω,*α* (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *Ri* Ω,*α* (*F*)

*i*=1 *i*=1 *i*=1 *i*=1 can be deduced. Similarly, *<sup>E</sup>* <sup>⊆</sup> *<sup>F</sup>* <sup>⇒</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*β* (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*β* (*F*) can also be proved.

$$\begin{array}{rcl} \text{4.} & \sum\_{i=1}^{m} R\_{i} \quad (E \cup F) & = & \{ \max\_{i=1}^{m} \frac{\sum\_{\forall i \in V} R\_{i}(x,y)(E \cup F)(y)}{\sum\_{\forall i} R\_{i}(x,y)} \ge a \, | \, \mathbf{x} \in \mathcal{U} \} \quad \geq & \{ \max\_{i=1}^{m} \frac{\sum\_{\forall i \in V} R\_{i}(x,y)E(y)}{\sum\_{\forall i} R\_{i}(x,y)} \ge a \, | \, \mathbf{x} \in \mathcal{U} \}, \\ & \stackrel{\text{IH}}{=} & \Omega\_{i} \quad (E \cup F) & = & \{ \max\_{i=1}^{m} \frac{\sum\_{\forall i \in V} R\_{i}(x,y)(E \cup F)(y)}{\sum\_{\forall i} R\_{i}(x,y)} \ge a \, | \, \mathbf{x} \in \mathcal{U} \} \quad \geq & \{ \max\_{i=1}^{m} \frac{\sum\_{\forall i} R\_{i}(x,y)F(y)}{\sum\_{\forall i} R\_{i}(x,y)} \ge a \, | \, \mathbf{x} \in \mathcal{U} \}. \\ & & \stackrel{\text{IH}}{=} & \Omega\_{i} \quad (E \cup F) & \supseteq \stackrel{\text{IH}}{=} \stackrel{\text{R},a}{R\_{i}} \quad (E \cup F) \cup \stackrel{\text{IH}}{=} \stackrel{\text{R},a}{R} \quad (F) \text{ can be concluded. Similarly, we can also show that} \end{array}$$

$$\begin{array}{ccccc}\underset{i=1}{\overbrace{\begin{subarray}{c}\overset{i=1}{m}\Omega\\\in\end{subarray}}}\alpha\_{i}\theta & \overset{i=1}{\overbrace{\begin{subarray}{c}\overset{i=1}{m}\Omega\\\in\end{subarray}}}\alpha\_{i}\theta & \overset{i=1}{\overbrace{\begin{subarray}{c}\overset{i=1}{m}\Omega\\\in\end{subarray}}}\alpha\_{i}\theta\\\end{array}\text{(F)}\\\text{5.}\quad\text{According to the above conclusions,}\ \sum\_{i=1}^{m}\underset{R\_{i}}{R\_{i}}\quad(E\cap F)\;\subseteq\ \sum\_{i=1}^{m}R\_{i}\quad(E)\cap\sum\_{i=1}^{m}R\_{i}\quad(F)\text{ and}\\\end{array}$$

$$\begin{array}{cccc}\hline\\\text{1.} & \text{Actualing} & \text{to use above} & \text{conclusion}\_{\text{'}}\\\hline\\\sum\limits\_{i=1}^{m}\Omega\_{i}\delta & \overline{(E \cup F)} \supseteq \sum\limits\_{i=1}^{m}\Omega\_{i}\delta & \overline{(E) \cup \sum\limits\_{i=1}^{m}\Omega\_{i}} & \overline{(F) \cup \sum\limits\_{i=1}^{m}\Omega\_{i}} \\\hline\\\hline\end{array}$$

#### *3.2. Pessimistic IN MG-PRSs over Two Universes*

According to previous definitions, starting from minimal IN membership degrees, pessimistic multigranulation probabilistic models can be established in a similar way.

**Definition 9.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are represented by α and β with α* > *β, then the lower and upper approximations of E in pessimistic multigranulation probabilistic models are provided as the following mathematical expressions:*

$$\begin{aligned} \sum\_{i=1}^{m} R\_i \quad &(E) = \{ \Psi\_E^{\stackrel{\text{in}}{\sum}R\_i} \left( \mathbf{x} \right) \ge \alpha \left| \mathbf{x} \in \mathcal{U} \right\rangle; \\\overline{\frac{m}{\sum}} \Psi\_i \quad &(E) = \{ \Psi\_E^{\stackrel{\text{in}}{\sum}R\_i} \left( \mathbf{x} \right) > \beta \left| \mathbf{x} \in \mathcal{U} \right\rangle}, \end{aligned}$$

*the pair* ( *m* ∑ *i*=1 *Ri* Ψ,*α* (*E*), *m* ∑ *i*=1 *Ri* Ψ,*β* (*E*)) *is named a pessimistic IN MG-PRS over two universes of E.*

Next, some key properties of lower and upper approximations for pessimistic multigranulation probabilistic models are presented, and we can prove them according to above-mentioned proofs for Proposition 1.

**Proposition 2.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E*, *F* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are denoted by α and β with α* > *β, then the lower and upper approximations for pessimistic multigranulation probabilistic models own the following properties:*

*1. <sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (∅) = ∅*, m* ∑ *i*=1 *Ri* Ψ,*β* (*V*) = *U;*

*Mathematics* **2020**, *8*, 223

$$\begin{array}{llll} 2. & a\_1 \leq a\_2 \Rightarrow \sum\_{i=1}^{m} R\_i \quad (E) \subseteq \sum\_{i=1}^{m} R\_i \quad (E), \ \beta\_1 \leq \beta\_2 \Rightarrow \overbrace{\sum\_{i=1}^{m} R\_i}^{\textbf{W}, \mu\_1} \quad (E) \subseteq \sum\_{i=1}^{m} R\_i \quad (E) \subseteq \sum\_{i=1}^{m} R\_i \quad (E);\\ 3. & E \subseteq F \Rightarrow \underbrace{\sum\_{i=1}^{m} R\_i}\_{=1} \quad (E) \subseteq \underbrace{\sum\_{i=1}^{m} R\_i}\_{=1} \quad (F) \\_ \sum\_{i=1}^{m} R\_i \quad (E) \subseteq \sum\_{i=1}^{m} R\_i \quad (F);\\ & & \max\_{\textbf{w}, \textbf{a}} \quad \textbf{w} \leq \textbf{a} \end{array}$$

*4. <sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*<sup>E</sup>* <sup>∪</sup> *<sup>F</sup>*) <sup>⊇</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*E*) <sup>∪</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*F*)*, m* ∑ *i*=1 *Ri* Ψ,*β* (*<sup>E</sup>* <sup>∩</sup> *<sup>F</sup>*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*β* (*E*) <sup>∩</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*β* (*F*)*; 5. <sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*<sup>E</sup>* <sup>∩</sup> *<sup>F</sup>*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*E*) <sup>∩</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*F*)*, m* ∑ *i*=1 *Ri* Ψ,*β* (*<sup>E</sup>* <sup>∪</sup> *<sup>F</sup>*) <sup>⊇</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*β* (*E*) <sup>∪</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*β* (*F*)*.*

#### *3.3. Adjustable IN MG-PRSs over Two Universes*

Starting from two mathematical formulations of optimistic and pessimistic multigranulation probabilistic models, the former one utilizes maximal IN membership degrees Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) to establish related IN approximations, which indicates a risk-seeking information fusion tactic in MAGDM issues. On the contrary, the latter one utilizes minimal IN membership degrees Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) to establish related IN approximations, which indicates a risk-averse information fusion tactic in MAGDM issues. However, it is noted that both optimistic and pessimistic information fusion tactics are qualitative and static, they lack the ability of expressing risks of information fusion tactics from quantitative and dynamic standpoints. In order to quantitatively and dynamically describe the risk preference of information fusion tactics, adjustable IN membership degrees should be put forward by introducing the notion of risk coefficients, then adjustable IN MG-PRSs over two universes can be developed conveniently.

In what follows, adjustable IN membership degrees are defined by means of Ω ∑ *i*=1 *Ri <sup>E</sup>* (*x*) and Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*).

*m*

**Definition 10.** *Suppose Ri is an IN relation over two universes over U* × *V, λ (λ* ∈ [0, 1]*) is a risk coefficient. For any E* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the adjustable IN membership degrees of x in E with regard to Ri are provided as the following mathematical expressions:*

$$\Xi\_{E}^{\stackrel{\text{in}}{i=1}}(\mathbf{x}) = \lambda \Omega\_{E}^{\stackrel{\text{in}}{i=1}}(\mathbf{x}) \boxplus \left(1 - \lambda\right) \Psi\_{E}^{i=1}(\mathbf{x}) \dots$$

Next, adjustable multigranulation probabilistic models can be designed similarly.

**Definition 11.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are represented by α and β with α* > *β, then the lower and upper approximations of E in adjustable multigranulation probabilistic models are provided as the following mathematical expressions:*

$$\begin{aligned} \sum\_{i=1}^{m} R\_i \quad (E) &= \{ \Xi\_E^{i=1} \mid \mathbf{x} \rangle \ge \alpha \mid \mathbf{x} \in \mathcal{U} \};\\ \overline{\frac{m}{\sum\_{i=1}^{m} R\_i}} \quad (E) &= \{ \Xi\_E^{i=1} \mid \mathbf{x} \rangle > \beta \mid \mathbf{x} \in \mathcal{U} \}. \end{aligned}$$

*the pair* ( *m* ∑ *i*=1 *Ri* Ξ,*α* (*E*), *m* ∑ *i*=1 *Ri* Ξ,*β* (*E*)) *is named an adjustable IN MG-PRS over two universes of E.*

In what follows, some key properties of lower and upper approximations for adjustable multigranulation probabilistic models are presented, and we can also prove them according to above-mentioned proofs for Proposition 1.

**Proposition 3.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E*, *F* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are denoted by α and β with α* > *β, then the lower and upper approximations for adjustable multigranulation probabilistic models own the following properties:*

$$\mathbf{1}. \quad \underbrace{\stackrel{m}{\sum\limits\_{i=1}^{m}R\_{i}}}\_{\mathbf{i}=1} \mathbf{2}. \quad \left(\bigcirc \mathbf{) = \bigcirc \quad \overbrace{\sum\limits\_{i=1}^{m}R\_{i}}^{m} \left(V\right) = \mathbf{U}; \mathbf{1}\right)$$

$$\begin{array}{ccccccccc} 2. & a\_1 \leq a\_2 \Rightarrow \sum\_{i=1}^{m} R\_i & (E) \subseteq \sum\_{i=1}^{m} R\_i & (E), \beta\_1 \leq \beta\_2 \Rightarrow \sum\_{i=1}^{\overline{m}} R\_i & (E) \subseteq \sum\_{i=1}^{\overline{m}} R\_i & (E) \subseteq \sum\_{i=1}^{\overline{m}} R\_i & (E); \end{array}$$

$$\begin{array}{ccccc} \text{3.} & E \subseteq F \Rightarrow \sum\_{i=1}^{m} R\_{i} \, ^{\mathsf{E},a}(E) \subseteq \underbrace{\sum\_{i=1}^{m} R\_{i}}\_{\in} & (F) \, \overbrace{\sum\_{i=1}^{m} R\_{i}}\_{\in} & (E) \subseteq \overbrace{\sum\_{i=1}^{m} R\_{i}}\_{\in} & (F) \, \overbrace{\sum\_{i=1}^{m} R\_{i}}\_{\in} \end{array}$$

$$\begin{array}{ccccccccc} \mathsf{4.} & \stackrel{\mathsf{m}}{\operatorname{\bf{\tiny\$}}} & \stackrel{\mathsf{\mathsf{E}},\mathsf{a}}{\operatorname{\bf{\hbox}}} & (\operatorname{\bf{\hbox}}\cup F) \supseteq \operatorname{\bf{\hbox}}\stackrel{\mathsf{\hbox}}{\operatorname{\bf{\hbox}}} & \stackrel{\mathsf{\hbox},\mathsf{a}}{\operatorname{\bf{\hbox}}} & (\operatorname{\bf{\hbox}}) \cup \operatorname{\bf{\hbox}}\stackrel{\mathsf{\hbox}}{\operatorname{\bf{\hbox}}} & \stackrel{\mathsf{\hbox},\mathsf{a}}{\operatorname{\bf{\hbox}}} & (F), \stackrel{\mathsf{\hbox}}{\operatorname{\bf{\hbox}}} & \stackrel{\mathsf{\hbox},\mathsf{\hbox}}{\operatorname{\bf{\hbox}}} & (E\cap F) \subseteq \operatorname{\bf{\hbox}}\stackrel{\mathsf{\hbox}}{\operatorname{\bf{\hbox}}} & \stackrel{\mathsf{\hbox},\mathsf{\texttt{\hbox}}}{\operatorname{\bf{\hbox}}} & (E)\cap \operatorname{\bf{\hbox}}\stackrel{\mathsf{\hbox}}{\operatorname{\bf{\hbox}}} & \stackrel{\mathsf{\hbox},\mathsf{\texttt{\hbox}}}{\operatorname{\bf{\hbox}}} & \stackrel{\mathsf{\hbox},\mathsf{\texttt{\hbox}}}{\operatorname{\bf{\hbox}}} & \end{array}$$

$$\overline{\square}.\quad\underbrace{\begin{array}{c}\underline{m}\quad\Sigma\_{i}\underline{a}\\\underline{i}\equiv1\end{array}}\_{\mid i\equiv1\rangle}\quad(\underline{E\cap F})\subseteq\underbrace{\begin{array}{c}\underline{m}\quad\Sigma\_{i}\underline{a}\\\underline{i}\equiv1\end{array}}\_{\mid i\equiv1\rangle}\quad(\underline{E})\cap\underbrace{\begin{array}{c}\underline{m}\quad\Sigma\_{i}\underline{a}\\\underline{i}\equiv1\end{array}}\_{\mid i\equiv1\rangle}\quad(\underline{E\cup F})\supseteq\overbrace{\begin{array}{c}\overline{m}\quad\Sigma\_{i}\underline{b}\\\underline{i}\equiv1\end{array}}\quad(\underline{E})\cup\overbrace{\begin{array}{c}\overline{m}\quad\Sigma\_{i}\underline{b}\\\underline{i}\equiv1\end{array}}\_{\mid i\equiv1\rangle}\quad(\underline{F}).$$

#### *3.4. Relationships between Optimistic, Pessimistic, and Adjustable IN MG-PRSs over Two Universes*

In previous sections, three types of multigranulation probabilistic models with IN Information are investigated in detail. The following section aims to discuss relationships between optimistic, pessimistic, and adjustable multigranulation probabilistic models.

**Proposition 4.** *Suppose Ri is an IN relation over two universes over U* × *V. For any E*, *F* ∈ *IN* (*V*)*, x* ∈ *U, y* ∈ *V, the two IN thresholds are denoted by α and β with α* > *β; then, we have:*

$$\begin{array}{llll} 1. & \frac{\begin{array}{l} \Psi, \alpha\\ \sum\end{array} \quad \begin{array}{l} \Psi, \alpha\\ (E) \subseteq \sum\limits\_{i=1}^{m} R\_{i} \end{array} \quad (E) \subseteq \underbrace{\begin{array}{l} \square, \mathsf{a} \\ \sum\limits\_{i=1}^{m} R\_{i} \end{array}}\_{\mathsf{T}, \mathsf{a}} & (E) \subseteq \mathop{\begin{array}{l} \sum\limits\_{i=1}^{m} R\_{i} \end{array}}\_{\mathsf{E}, \mathsf{a}} & (E); \end{array} \\ 2. & \sum\limits\_{i=1}^{m} R\_{i} \quad (E) \subseteq \mathop{\begin{array}{l} \sum\limits\_{i=1}^{m} R\_{i} \end{array}}\_{\mathsf{T}, \mathsf{a}} & (E) \subseteq \mathop{\begin{array}{l} \sum\limits\_{i=1}^{m} R\_{i} \end{array}}\_{\mathsf{T}, \mathsf{a}} & (E). \end{array}$$

**Proof.**

1. *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α* (*E*) = {Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> { *<sup>m</sup>* max *<sup>i</sup>*=<sup>1</sup> *<sup>η</sup>Ri <sup>E</sup>* (*x*) ≥ *α* |*x* ∈ *U* } ≥ {(*<sup>λ</sup> <sup>m</sup>* max *<sup>i</sup>*=<sup>1</sup> *<sup>η</sup>Ri <sup>E</sup>* (*x*) (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*) *<sup>m</sup>* min *<sup>i</sup>*=<sup>1</sup> *<sup>η</sup>Ri <sup>E</sup>* (*x*)) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ξ,*α* (*E*) ≥ { *m* min *<sup>i</sup>*=<sup>1</sup> *<sup>η</sup>Ri <sup>E</sup>* (*x*) <sup>≥</sup> *<sup>α</sup>* <sup>|</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* } <sup>=</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*E*). Thus, *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ξ,*α* (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α* (*E*) can be obtained. 2. Likewise, *<sup>m</sup>* ∑ *i*=1 *Ri* Ψ,*α* (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ξ,*α* (*E*) <sup>⊆</sup> *<sup>m</sup>* ∑ *i*=1 *Ri* Ω,*α* (*E*) can also be proved.

#### **4. IN MAGDM Based on Multigranulation Probabilistic Models**

In the following section, we aim to sum up a viable and effective MAGDM approach by using the newly developed theoretical models. As pointed out in Section 1, multigranulation probabilistic models can manage the three challenges of typical MAGDM situations well—to be specific, the overall study context of IN MG-PRSs over two universes is IN information, which excels in depicting indeterminate

and incomplete information at the same time. In addition, the development of multigranulation probabilistic models provide decision makers with an effective strategy in MAGDM information fusion and analysis by taking superiorities of MGRSs and PRSs, and the proposed IN MG-PRSs over two universes are also equipped with the ability of describing risk preferences of information fusion quantitatively and dynamically. Hence, IN MG-PRSs over two universes play a significant role in solving MAGDM problems, and it is necessary to put forward corresponding MAGDM methods.

Next, for the sake of exploring MAGDM methods in a real-world scenario, we put the following discussions in the background of M&A TSs. We first let the universe *U* = {*x*1, *x*2,..., *xj*} be a set containing selectable M&A targets, whereas universe *V* = {*y*1, *y*2,..., *yk*} is a set containing assessment criteria. Then, we also let *E* ∈ *IN* (*V*) be a standard set containing several needs of corporate acquirers from the aspect of assessment criteria. Afterwards, *m* decision makers in a group provide several relations *Ri* ∈ *INR* (*U* × *V*) (*i* = 1, 2, . . . , *m*) between the above-mentioned universes. Finally, an information system (*U*, *V*, *Ri*, *E*) for M&A TSs can be established as the input for the following MAGDM algorithm based on multigranulation probabilistic models.

**Remark 1.** *In what follows, we intend to interpret the scheme of selecting the parametric value λ. According to m* ∑ *Ri*

*Definition 10, the adjustable IN membership degrees of x in E with regard to Ri are provided as* Ξ *i*=1 *<sup>E</sup>* (*x*) = *λ*Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) (1 − *λ*) Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*)*. It is not difficult to see* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) = Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *when λ* = 1*, whereas* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) = Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *when λ* = 0*. Hence, maximal IN membership degrees* Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *and minimal IN membership degrees* Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *are two extreme cases of adjustable IN membership degrees* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*)*. In light of the standpoint of risk decision-making with uncertainty from classical operational research [1],* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) = Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *can be seen as the "completely risk-seeking" strategy,* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) = Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *can be seen as the "completely risk-averse" strategy, and* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) = 0.5Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) (1 − 0.5) Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *can be seen as the "risk-neutral" strategy. Moreover,* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *can be seen as the "somewhat risk-seeking" strategy when λ* ∈ (0.5, 1)*, and* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *can be seen as the "somewhat risk-averse" strategy when λ* ∈ (0, 0.5)*. According to the above-stated theoretical explanations, the parametric value λ represents the risk preference of different decision makers in M&A TSs. In specific, the larger of the parametric value λ when all decision makers are more risk-seeking, whereas the smaller of the parametric value λ when all decision makers are more risk-averse. In general, the parametric value λ is determined by all decision-makers' risk preference or the empirical studies and inherent knowledge in advance. In practical MAGDM situations, suppose there are m decision makers in a group, each decision maker provides his or her risk preference λ<sup>i</sup> (λ<sup>i</sup>* ∈ [0, 1]*, i* = 1, 2, ... , *m), and then the final risk preference when computing* Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) *can be determined as λ* = *m* ∑ *i*=1 *λi <sup>m</sup> .*

In what follows, an algorithm for M&A TSs by virtue of adjustable IN MG-PRSs over two universes is established.

#### **5. An Illustrative Example**

For the sake of making an efficient comparative analysis with existing similar IN MAGDM approaches, we plan to utilize the case study that was previously investigated in [28]. In what follows, we first present the general context of M&A TSs and show basic steps of obtaining the optimal M&A target by means of the newly proposed algorithm developed in Section 4.

**Remark 2.** *Section 4 acts as a transition part which links between the theoretical models proposed in Section 3 and the application case presented in Section 5. To be specific, we first put forward two special theoretical models named optimistic and pessimistic IN MG-PRSs over two universes. Then, we further generalize these two special theoretical models to adjustable IN MG-PRSs over two universes. All the proposed three theoretical models are foundations for addressing MAGDM problems. Next, we propose a novel algorithm for M&A TSs in light of adjustable IN MG-PRSs over two universes in Section 4. Finally, in order to show the reasonability and effectiveness of the proposed algorithm, the following section plans to conduct several quantitative and qualitative analysis via an illustrative example.*

#### *5.1. MAGDM Procedures*

In this illustrative example, we use the case study that was previously investigated in [28]. The detailed case descriptions and datasets can be found at the webpage (https://www.mdpi.com/2073-8994/9/7/126/htm).

According to Algorithm 1, we aim to obtain the optimal M&A target by means of IN MG-PRSs over two universes. First, we calculate single IN membership degrees as follows.

**Algorithm 1** An algorithm for M&A TSs in light of adjustable IN MG-PRSs over two universes.

**Require:** An information system (*U*, *V*, *Ri*, *E*) for M&A TSs. **Ensure:** The optimal alternative.

Step 1. Calculate single IN membership degrees *ηRi <sup>E</sup>* (*x*) <sup>=</sup> <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*)*E*(*y*) <sup>∑</sup>*y*∈*<sup>V</sup> Ri*(*x*,*y*) ; *m*

Step 2. Calculate maximal IN membership degrees Ω ∑ *i*=1 *Ri <sup>E</sup>* (*x*) <sup>=</sup> *<sup>m</sup>* max *<sup>i</sup>*=<sup>1</sup> *<sup>η</sup>Ri <sup>E</sup>* (*x*) and minimal IN membership degrees Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) <sup>=</sup> *<sup>m</sup>* min *<sup>i</sup>*=<sup>1</sup> *<sup>η</sup>Ri <sup>E</sup>* (*x*); Step 3. Calculate the risk coefficient *λ* = *m* ∑ *i*=1 *λi <sup>m</sup>* ; Step 4. Calculate adjustable IN membership degrees Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) = *λ*Ω *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) (1 − *λ*) Ψ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*); Step 5. Determine score values of Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*) for all selectable M&A targets; Step 6. The best alternative is the one with the largest score value for Ξ *m* ∑ *i*=1 *Ri <sup>E</sup>* (*x*).

With regard to the relation presented in *R*1, we obtain

$$\eta\_{E}^{R\_{1}}\left(\mathbf{x}\_{1}\right) = \frac{\sum\_{y \in V} R\_{1}\left(\mathbf{x}\_{1}, y\right) \mathbb{E}\left(y\right)}{\sum\_{y \in V} R\_{1}\left(\mathbf{x}\_{1}, y\right)} = \left\langle \left[0.8477, 0.9207\right], \left[0.0378, 0.1345\right], \left[0.1771, 0.316\right] \right\rangle \dots$$

In a similar manner, we also obtain

$$\begin{aligned} \eta\_{E}^{R\_{1}}(\mathbf{x}\_{2}) &= \langle \left[ 0.7672, 0.894 \right], \left[ 0.0459, 0.1677 \right], \left[ 0.3197, 0.5164 \right] \rangle; \\ \eta\_{E}^{R\_{1}}(\mathbf{x}\_{3}) &= \langle \left[ 0.8212, 0.9314 \right], \left[ 0.05, 0.1497 \right], \left[ 0.2351, 0.3925 \right] \rangle; \\ \eta\_{E}^{R\_{1}}(\mathbf{x}\_{4}) &= \langle \left[ 0.8493, 0.9465 \right], \left[ 0.0232, 0.101 \right], \left[ 0.2147, 0.4122 \right] \rangle; \\ \eta\_{E}^{R\_{1}}(\mathbf{x}\_{5}) &= \langle \left[ 0.8774, 0.9643 \right], \left[ 0.0466, 0.1559 \right], \left[ 0.07, 0.1923 \right] \rangle. \end{aligned}$$

With regard to the relation presented in *R*2, we also obtain

$$\begin{aligned} \eta\_{E}^{R\_{2}}(\mathbf{x}\_{1}) &= \langle \left[ 0.8185, 0.9419 \right], \left[ 0.033, 0.1355 \right], \left[ 0.17, 0.3418 \right] \rangle : \\ \eta\_{E}^{R\_{2}}(\mathbf{x}\_{2}) &= \langle \left[ 0.7728, 0.9177 \right], \left[ 0.04, 0.1423 \right], \left[ 0.285, 0.4756 \right] \rangle : \\ \eta\_{E}^{R\_{2}}(\mathbf{x}\_{3}) &= \langle \left[ 0.8217, 0.9407 \right], \left[ 0.038, 0.1294 \right], \left[ 0.1799, 0.3357 \right] \rangle : \\ \eta\_{E}^{R\_{2}}(\mathbf{x}\_{4}) &= \langle \left[ 0.8327, 0.9548 \right], \left[ 0.022, 0.097 \right], \left[ 0.2434, 0.4106 \right] \rangle : \\ \eta\_{E}^{R\_{2}}(\mathbf{x}\_{5}) &= \langle \left[ 0.8591, 0.9645 \right], \left[ 0.031, 0.1583 \right], \left[ 0.096, 0.2316 \right] \rangle . \end{aligned}$$

With regard to the relation presented in *R*3, we also obtain

$$\begin{aligned} \eta\_{E}^{R\_{3}}(\mathbf{x}\_{1}) &= \langle [0.844, 0.9472], [0.03, 0.1422], [0.2104, 0.3615] \rangle; \\ \eta\_{E}^{R\_{3}}(\mathbf{x}\_{2}) &= \langle [0.799, 0.91], [0.0336, 0.1571], [0.3151, 0.5472] \rangle; \\ \eta\_{E}^{R\_{3}}(\mathbf{x}\_{3}) &= \langle [0.8388, 0.9368], [0.0424, 0.1466], [0.1803, 0.3484] \rangle; \\ \eta\_{E}^{R\_{3}}(\mathbf{x}\_{4}) &= \langle [0.8362, 0.9366], [0.0188, 0.1069], [0.206, 0.3834] \rangle; \\ \eta\_{E}^{R\_{3}}(\mathbf{x}\_{5}) &= \langle [0.8739, 0.9581], [0.029, 0.1351], [0.075, 0.2316] \rangle. \end{aligned}$$

Next, maximal and minimal IN membership degrees can be calculated in light of Definition 7. For maximal IN membership degrees, we have

$$\begin{aligned} \Omega\_{E}^{\frac{\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\Pi}}}}}}}}}}}}}} \right)} \text{\}} \text{\}} \\ \Omega\_{E}^{\frac{\text{\tiny\text{\tiny\text{\text$$

Similarly, for minimal IN membership degrees, we also have

$$\begin{aligned} \Psi\_{\overline{E}}^{\frac{3}{E}}(\mathbf{x}\_{1}) &= \langle [0.8185, 0.9207], [0.03, 0.1345], [0.17, 0.316] \rangle; \\ \Psi\_{\overline{E}}^{\frac{3}{E}}(\mathbf{x}\_{2}) &= \langle [0.7672, 0.894], [0.0336, 0.1423], [0.285, 0.4756] \rangle; \\ \Psi\_{\overline{E}}^{\frac{3}{E}}(\mathbf{x}\_{3}) &= \langle [0.8212, 0.9314], [0.038, 0.1294], [0.1799, 0.3357] \rangle; \\ \Psi\_{\overline{E}}^{\frac{3}{E}}(\mathbf{x}\_{4}) &= \langle [0.8327, 0.9366], [0.0188, 0.097], [0.206, 0.3834] \rangle; \\ \Psi\_{\overline{E}}^{\frac{3}{E}}(\mathbf{x}\_{5}) &= \langle [0.8591, 0.9581], [0.029, 0.1351], [0.07, 0.1923] \rangle. \end{aligned}$$

In order to make an efficient comparison with the MAGDM method proposed in [28], the risk coefficient *λ* = 0.6 is noted in [28]; then, we also take the risk coefficient *λ* = 0.6 in this case study. In what follows, adjustable IN membership degrees can be calculated. To be specific, for M&A target *x*1, we have

$$\stackrel{\stackrel{3}{\Sigma}}{\Xi^{i=1}\_{\Sigma}} \begin{pmatrix} \mathbf{x}\_{i} \\ \mathbf{x}\_{1} \end{pmatrix} = 0.6 \boldsymbol{\Omega}\_{\mathbf{E}}^{i=1} \begin{pmatrix} \mathbf{x}\_{i} \\ \mathbf{x}\_{1} \end{pmatrix} \boxplus \begin{pmatrix} \mathbf{1} - 0.6 \end{pmatrix} \overset{\stackrel{3}{\Sigma}}{\mathbf{F}^{i=1}\_{\Sigma}} \begin{pmatrix} \mathbf{x}\_{1} \\ \mathbf{x}\_{1} \end{pmatrix} = \left\langle \begin{bmatrix} 0.8366, 0.9379 \end{bmatrix}, \begin{bmatrix} 0.0345, 0.1391 \end{bmatrix}, \begin{bmatrix} 0.1932, 0.3426 \end{bmatrix} \right\rangle \begin{bmatrix} \mathbf{1} - \mathbf{1} \end{bmatrix} = \begin{bmatrix} \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{1} - \mathbf{1} \end{bmatrix}$$

In what follows, we also have

$$\begin{aligned} \Xi\_{E}^{\stackrel{3}{\to}}{E\_{E}^{i=1}}(\mathbf{x}\_{2}) &= \langle \left[ 0.7868, 0.9089 \right], \left[ 0.0405, 0.157 \right], \left[ 0.3054, 0.5173 \right] \rangle; \\ \Xi\_{E}^{\stackrel{3}{\to}}{E\_{E}^{i=1}}(\mathbf{x}\_{3}) &= \langle \left[ 0.832, 0.9371 \right], \left[ 0.0448, 0.1412 \right], \left[ 0.2112, 0.3687 \right] \rangle; \\ \Xi\_{E}^{\stackrel{3}{\to}}{E\_{E}}(\mathbf{x}\_{4}) &= \langle \left[ 0.8429, 0.9482 \right], \left[ 0.0213, 0.1028 \right], \left[ 0.2277, 0.4004 \right] \rangle; \\ \Xi\_{E}^{\stackrel{3}{\to}}{E\_{E}}(\mathbf{x}\_{5}) &= \langle \left[ 0.8705, 0.9621 \right], \left[ 0.0385, 0.1486 \right], \left[ 0.0846, 0.215 \right] \rangle. \end{aligned}$$

Finally, we calculate score values of Ξ 3 ∑ *i*=1 *Ri <sup>E</sup>* (*x*) for each *xi*, and the best *xi* is the one with the largest 3 ∑ *Ri*

score value for Ξ *i*=1 *<sup>E</sup>* (*x*); the final ranking result shows *x*<sup>5</sup> *x*<sup>1</sup> *x*<sup>4</sup> *x*<sup>3</sup> *x*2, i.e., the supreme alternative is *x*5.

#### *5.2. Sensitivity Analysis*

In the previous section, we obtain the optimal M&A target by using adjustable multigranulation probabilistic models with the risk coefficient *λ* = 0.6. The following sensitivity analysis aims to investigate the influence of the risk coefficient by changing the value of *λ*. To be specific, supposing the value of *λ* is taken as 0, 0.4, 0.5, 0.6, and 1, respectively, then we can obtain the final ranking orders in Table 1 below.


**Table 1.** Ranking orders of M&A targets with changing values of *λ*.

According to the final ranking orders in Table 1, it is easy to see that the best *xi* is insensitive to changing values of *λ*, that is, all results show the best *xi* is *x*5. Thus, the best alternative is reliable and stable. The only difference lies in the ranking order of *x*<sup>3</sup> and *x*<sup>4</sup> when *λ* = 0, i.e., *x*<sup>3</sup> is superior to *x*<sup>4</sup> when *λ* = 0, whereas *x*<sup>3</sup> is inferior to *x*<sup>4</sup> in other situations. The cause of this phenomenon is that the changing values of *λ* may affect the ranking order of *x*<sup>3</sup> and *x*<sup>4</sup> when the risk preferences is completely risk-averse.

#### *5.3. Comparative Analysis*

In what follows, we aim to compare with the MAGDM method proposed in [28] to present the merits of the proposed MAGDM algorithm. In [28], the authors put forward an algorithm for M&A TSs via IN MGRSs over two universes without the support of PRSs. The mathematical structures of optimistic and pessimistic IN MGRSs over two universes are presented below:

$$\underbrace{\sum\_{i=1}^{m}R\_{i}}\_{\begin{subarray}{c}\underline{i}=1\\\underline{i}=1\end{subarray}}\langle E\rangle = \{\langle\text{x},[\mu^{\mathrm{L}}\_{\begin{subarray}{c}\underline{m}&\boldsymbol{\mathcal{O}}\end{subarray}}\begin{subarray}{c}\left(\mathrm{x}\right),\mu^{\mathrm{L}}\_{\begin{subarray}{c}\underline{m}&\boldsymbol{\mathcal{O}}\\\underline{i}\in\boldsymbol{\mathcal{R}}\_{\begin{subarray}{c}\underline{\boldsymbol{m}}\end{subarray}}\begin{subarray}{c}\underline{\boldsymbol{\mathcal{R}}}\_{i}\end{subarray}\begin{subarray}{c}\boldsymbol{\mathcal{R}}\_{i}\end{subarray}\right\}\left\langle\nu^{\mathrm{L}}\_{\begin{subarray}{c}\underline{m}&\boldsymbol{\mathcal{O}}\\\underline{\boldsymbol{m}}\end{subarray}}\begin{subarray}{c}\left(\mathrm{x}\right),\nu^{\mathrm{L}}\_{\begin{subarray}{c}\underline{\boldsymbol{m}}\end{subarray}}\begin{subarray}{c}\left(\mathrm{x}\right),\nu^{\mathrm{L}}\_{\begin{subarray}{c}\underline{\boldsymbol{m}}\end{subarray}}\begin{subarray}{c}\boldsymbol{\mathcal{R}}\_{i}\end{subarray}\right\}\left\langle\mathrm{x}\right\rangle\end{subarray}\right\}\rangle$$

$$\begin{split} \xleftarrow{\sum\_{i=1}^{m}R\_{i}} \quad (E) &= \{ \langle \text{x}, [\mu \frac{\Box^{L}}{\sum\limits\_{i=1}^{m}R\_{i}}^{\text{U}}(\text{x}) \, , \mu \frac{\Box^{L}}{\sum\limits\_{i=1}^{m}R\_{i}}^{\text{U}}(\text{x}) \rangle, [\nu \frac{\Box^{L}}{\sum\limits\_{i=1}^{m}R\_{i}}^{\text{U}}(\text{x}) \, , \nu \frac{\Box^{L}}{\sum\limits\_{i=1}^{m}R\_{i}}^{\text{U}}(\text{x})], \\ & [\omega \frac{\Box^{L}}{\sum\limits\_{i=1}^{m}R\_{i}}^{\text{U}}(\text{x}) \, , \omega \frac{\Box^{L}}{\sum\limits\_{i=1}^{m}R\_{i}}^{\text{U}}(\text{x})] \} \mid \text{x} \in \mathsf{U} \}; \end{split}$$

$$\begin{array}{llll} & \stackrel{\scriptstyle m}{\sum}\_{i=1}^{m} R\_{i} & \left( E \right) = \{ \langle \text{x}, \left[ \mu\_{\stackrel{\scriptstyle \tiny \text{in}}{\text{m}}^{\text{L}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{P}} \left( \text{x} \right)} \left( \text{x} \right), \mu\_{\stackrel{\scriptstyle \text{m}}{\text{m}}^{\text{L}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{P}} \left( \text{x} \right) \right], \left[ \nu\_{\stackrel{\scriptstyle \text{m}}{\text{m}}^{\text{P}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{P}} \left( \text{x} \right) \right], \nu\_{\stackrel{\scriptstyle \text{m}}{\text{m}}^{\text{L}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{P}} \left( \text{E} \right)}} \left( \text{x} \right), \nu\_{\stackrel{\scriptstyle \text{m}}{\text{m}}^{\text{L}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{P}} \left( \text{E} \right)} \left( \text{x} \right) \rangle, \\ & \underbrace{ \left[ \omega\_{\stackrel{\scriptstyle \text{m}}{\text{m}}^{\text{L}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{P}} \left( \text{x} \right) \right], \omega \nu\_{\stackrel{\scriptstyle \text{m}}{\text{m}}^{\text{L}} \stackrel{\scriptstyle \text{P}}{\text{R}}^{\text{L}} \left( \text{E} \right)}}\_{\text{i=1}} \left( \text{x} \right) \rangle \rangle, \qquad \underbrace{ \left[ \begin{array}{c} \text{x} \right]}\_{} \text{$$

$$\begin{split} \underbrace{\sum\_{i=1}^{\overline{m}} R\_{i}}\_{i=1}^{P}(E) &= \{ \langle \mathbf{x}, [\mu^{\underline{L}}\_{\mathbf{x}}]\_{\mathbf{P}}^{\underline{L}}(\mathbf{x}), \mu^{\underline{IL}}\_{\overline{\mathbf{m}}\_{\mathbf{x}}}^{\mathbf{H}}(\mathbf{x}) \rangle, \mu^{\underline{IL}}\_{\overline{\mathbf{m}}\_{\mathbf{x}}}^{\mathbf{P}}(\mathbf{x}) \}, [\underbrace{\nu^{\underline{L}}\_{\mathbf{m}\_{\mathbf{x}}}}\_{i=1}^{P}(\mathbf{x}), \nu^{\underline{IL}}\_{\overline{\mathbf{m}}\_{\mathbf{x}}}^{\mathbf{P}}(\mathbf{x})] \}, \\ & [\underbrace{\omega^{\underline{L}}\_{\overline{\mathbf{m}}\_{\mathbf{m}}}}\_{i=1}^{P}(\mathbf{x}), \omega^{\underline{IL}}\_{\overline{\mathbf{m}}\_{\mathbf{x}}}^{\mathbf{H}}(\mathbf{x})] \rangle, \omega^{\underline{IL}}\_{i=1}^{\overline{M}}(\mathbf{x}) \}, \end{split}$$

where

*μL m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∨ *i*=1 <sup>∧</sup>*y*∈*V*{*ω<sup>L</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>μ</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *<sup>μ</sup><sup>U</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∨ *i*=1 <sup>∧</sup>*y*∈*V*{*ω<sup>U</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*y*)}, *νL m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∨</sup>*y*∈*V*{(<sup>1</sup> <sup>−</sup> *<sup>ν</sup><sup>U</sup> Ri* (*x*, *<sup>y</sup>*)) <sup>∧</sup> *<sup>ν</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *νU m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∨</sup>*y*∈*V*{(<sup>1</sup> <sup>−</sup> *<sup>ν</sup><sup>L</sup> Ri* (*x*, *<sup>y</sup>*)) <sup>∧</sup> *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*y*)}, *ω<sup>L</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∨</sup>*y*∈*V*{*μ<sup>L</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∧</sup> *<sup>ω</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *<sup>ω</sup><sup>U</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∨</sup>*y*∈*V*{*μ<sup>U</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∧</sup> *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*y*)}, *μL m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∨</sup>*y*∈*V*{*μ<sup>L</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∧</sup> *<sup>μ</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *<sup>μ</sup><sup>U</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∨</sup>*y*∈*V*{*μ<sup>U</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∧</sup> *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*y*)}, *νL m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∨ *i*=1 <sup>∧</sup>*y*∈*V*{*ν<sup>L</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>ν</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *<sup>ν</sup><sup>U</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∨ *i*=1 <sup>∧</sup>*y*∈*V*{*ν<sup>U</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>ν</sup><sup>U</sup> <sup>E</sup>* (*y*)}, *ω<sup>L</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∨ *i*=1 <sup>∧</sup>*y*∈*V*{*ω<sup>L</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>ω</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *<sup>ω</sup><sup>U</sup> m* ∑ *i*=1 *Ri O* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∨ *i*=1 <sup>∧</sup>*y*∈*V*{*ω<sup>U</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>ω</sup><sup>U</sup> <sup>E</sup>* (*y*)}. *μL m* ∑ *i*=1 *Ri P* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∧</sup>*y*∈*V*{*ω<sup>L</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>μ</sup><sup>L</sup> <sup>E</sup>* (*y*)}, *<sup>μ</sup><sup>U</sup> m* ∑ *i*=1 *Ri P* (*E*) (*x*) <sup>=</sup> *<sup>m</sup>* ∧ *i*=1 <sup>∧</sup>*y*∈*V*{*ω<sup>U</sup> Ri* (*x*, *<sup>y</sup>*) <sup>∨</sup> *<sup>μ</sup><sup>U</sup> <sup>E</sup>* (*y*)},

$$\begin{array}{lcl} & \sum\_{\begin{subarray}{c} \boldsymbol{x} \\ \sum\_{i} \in \mathcal{E}\_{i} \end{subarray}}^{p} \left( \mathbf{x} \right) = \underset{i=1}{\stackrel{\scriptstyle{\scriptstyle\prime}}{\operatorname{\prime}}}^{\mathcal{U}} \vee\_{\mathcal{V} \in \mathcal{V}} \{ \left( 1 - \nu\_{\mathcal{K}\_{i}}^{\mathcal{U}} (\mathbf{x}, y) \right) \wedge \nu\_{\mathcal{E}}^{\mathcal{U}} (\mathbf{y}) \}, \\ & \mu\_{\mathcal{U}}^{\mathcal{U}} \wedge\_{\mathcal{V} \in \mathcal{V}} \{ \left( 1 - \nu\_{\mathcal{K}\_{i}}^{\mathcal{U}} (\mathbf{x}, y) \right) \wedge \nu\_{\mathcal{E}}^{\mathcal{U}} (\mathbf{y}) \}, \\ & \mu\_{\mathcal{U} \in \mathcal{V}}^{\mathcal{U}} \wedge\_{\mathcal{E} \in \mathcal{V}} \{ \left( \nu\_{\mathcal{K}\_{i}}^{\mathcal{U}} (\mathbf{x}, y) \right) \wedge \nu\_{\mathcal{E}}^{\mathcal{U}} (\mathbf{y}) \}, \mu\_{\mathcal{U} \in \mathcal{V}}^{\mathcal{U}} \wedge\_{\mathcal{E} \in \mathcal{V}} \{ \left( \mathbf{x} \right) = \underset{i=1}{\stackrel{\scriptstyle \scriptstyle\!}{\mathcal{U}}}^{\mathcal{U}} \vee\_{\mathcal{V} \in \mathcal{V}} \{ \mu\_{\mathcal{K}\_{i}}^{\mathcal{U}} (\mathbf{x}, y) \land \nu\_{\mathcal{E}}^{\mathcal{U}} (\mathbf{y}) \}, \\ & \mu\_{\mathcal{E} \in \mathcal{E}\_{i} \atop \mathcal{E} \in \mathcal{E}\_{i} \end{array} \begin{cases} \nu\_{\mathcal{V}}^{\mathcal{U}} = \big( \$$

The pair ( *m* ∑ *i*=1 *Ri O* (*E*), *m* ∑ *i*=1 *Ri* (*E*)) is named an optimistic IN MGRS over two universes of *E*, whereas the pair ( *m* ∑ *i*=1 *Ri P* (*E*), *m* ∑ *i*=1 *Ri P* (*E*)) is named a pessimistic IN MGRS over two universes of *E*.

More concretely, the optimistic and pessimistic IN multigranulation rough approximations in terms of an information system (*U*, *V*, *Ri*, *E*)(*i* = 1, 2, 3) for M&A TSs are calculated at first, which are denoted by <sup>3</sup> ∑ *i*=1 *Ri O* (*E*), <sup>3</sup> ∑ *i*=1 *Ri O* (*E*), <sup>3</sup> ∑ *i*=1 *Ri P* (*E*) and <sup>3</sup> ∑ *i*=1 *Ri P* (*E*). Then, we further synthesize lower and upper versions of optimistic IN multigranulation rough approximations, which is denoted by <sup>3</sup> ∑ *i*=1 *Ri O* (*E*) <sup>3</sup> ∑ *i*=1 *Ri O* (*E*), whereas lower and upper versions of pessimistic counterparts

can also be synthesized, which is denoted by <sup>3</sup> ∑ *i*=1 *Ri P* (*E*) <sup>3</sup> ∑ *i*=1 *Ri P* (*E*). Let the risk coefficient *<sup>λ</sup>* <sup>=</sup> 0.6 when integrating <sup>3</sup> ∑ *i*=1 *Ri O* (*E*) <sup>3</sup> ∑ *i*=1 *Ri O* (*E*) with <sup>3</sup> ∑ *i*=1 *Ri P* (*E*) <sup>3</sup> ∑ *i*=1 *Ri P* (*E*), i.e., the synthesized set 0.6( 3 ∑ *i*=1 *Ri O* (*E*) <sup>3</sup> ∑ *i*=1 *Ri O* (*E*)) (1 − 0.6)( 3 ∑ *i*=1 *Ri P* (*E*) <sup>3</sup> ∑ *i*=1 *Ri P* (*E*)) should be further obtained. The above-stated sets are obtained as follows:

3 ∑ *i*=1 *Ri O* (*E*) <sup>3</sup> ∑ *i*=1 *Ri O* (*E*) = {*x*1,[0.82, 0.90] , [0.15, 0.24] , [0.24, 0.35], *x*2,[0.70, 0.80] , [0.15, 0.35] , [0.42, 0.56],*x*3,[0.85, 0.92] , [0.08, 0.18] , [0.30, 0.42], *x*4,[0.82, 0.90] , [0.10, 0.18] , [0.30, 0.42],*x*5,[0.76, 0.86] , [0.15, 0.24] , [0.14, 0.24]}; 3 ∑ *i*=1 *Ri P* (*E*) <sup>3</sup> ∑ *i*=1 *Ri P* (*E*) = {*x*1,[0.82, 0.90] , [0.10, 0.21] , [0.24, 0.35], *x*2,[0.70, 0.85] , [0.15, 0.28] , [0.40, 0.54],*x*3,[0.85, 0.92] , [0.08, 0.18] , [0.24, 0.35], *x*4,[0.79, 0.88] , [0.10, 0.21] , [0.28, 0.40],*x*5,[0.76, 0.86] , [0.15, 0.28] , [0.16, 0.27]}; 0.6( 3 ∑ *i*=1 *Ri O* (*E*) <sup>3</sup> ∑ *i*=1 *Ri O* (*E*)) (1 − 0.6)( 3 ∑ *i*=1 *Ri P* (*E*) <sup>3</sup> ∑ *i*=1 *Ri P* (*E*)) = {*x*1,[0.82, 0.90] , [0.13, 0.23] , [0.24, 0.35],*x*2,[0.70, 0.82] , [0.15, 0.32] , [0.41, 0.55], *x*3,[0.85, 0.92] , [0.08, 0.18] , [0.28, 0.39],*x*4,[0.81, 0.89] , [0.10, 0.19] , [0.29, 0.41], *x*5,[0.76, 0.86] , [0.15, 0.25] , [0.15, 0.25]}.

Finally, score values of 0.6( 3 ∑ *i*=1 *Ri O* (*E*) <sup>3</sup> ∑ *i*=1 *Ri O* (*E*)) (1 − 0.6)( 3 ∑ *i*=1 *Ri P* (*E*) <sup>3</sup> ∑ *i*=1 *Ri P* (*E*)) are calculated, and it is convenient to determine the ranking orders of five M&A targets via the above score values and obtain the optimal M&A target, which is *x*3, and *x*<sup>5</sup> is ranked second. The reason for the difference with the result obtained by using the proposed method lies in IN MGRSs over two universes lacking the ability of error tolerance; the MAGDM result is sensitive to outlier values from original information for M&A TSs.

#### *5.4. Discussion*

In order to address complicated MAGDM problems effectively, three key challenges are focused on at first. Then, under the guidance of multigranulation probabilistic models, we utilize the model of INSs, MGRSs and PRSs to handle the above-mentioned challenges. Moreover, compared with existing popular nonlinear modeling approaches, such as formal concept analysis [33,34,58,64–67], control systems [59,60] and sentiment analysis [61–63,68,69], which neither effectively handle indeterminate and incomplete information in complicated MAGDM problems, nor reasonably fuse and analyze multi-source information with incorrect and noisy data, it is necessary to combine INSs, MGRSs with PRSs to develop some meaningful hybrid models along with corresponding MAGDM approaches. In light of MAGDM procedures in the current section, we sum up the merits of the proposed MAGDM algorithm below:


Hence, the developed IN MG-PRSs over two universes perform outstandingly in MAGDM information representation, information fusion, and information analysis; they provide a beneficial tool for addressing complicated MAGDM problems.

#### **6. Conclusions**

This work mainly presents a general framework for dealing with complicated IN MAGDM problems by virtue of multigranulation probabilistic models. At first, three different types of multigranulation probabilistic models are put forward, that is, the optimistic version, the pessimistic version, and the adjustable version, and both definitions and key properties are discussed in detail. Then, relationships between optimistic, pessimistic, and adjustable multigranulation probabilistic models are further explored. Afterwards, corresponding IN MAGDM approaches are proposed in the background of M&A TSs. Finally, a practical example of M&A TSs is presented with several quantitative and qualitative analysis.

In the future, it is meaningful to generalize IN MG-PRSs over two universes to more extended neutrosophic contexts such as neutrosophic duplets, triplets and multisets. Furthermore, establishing efficient IN MAGDM approaches for problems with dynamic situations, high-dimensional attributes and large-scale of alternatives are also necessary. Another future interesting study option is to apply the presented theoretical models to other areas such as clustering, feature selections, compressed sensing, image processing, etc.

**Author Contributions:** Conceptualization, D.L.; writing—original draft preparation, C.Z.; validation, X.K. and Y.L.; writing—review and editing, S.B. and A.K.S.; supervision, D.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by the Key R&D program of Shanxi Province (International Cooperation, 201903D421041), National Natural Science Foundation of China (Nos. 61806116, 61672331, 61972238, 61802237, 61603278 and 61906110), Natural Science Foundation of Shanxi (Nos. 201801D221175, 201901D211176, 201901D211414), Training Program for Young Scientific Researchers of Higher Education Institutions in Shanxi, Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) (Nos. 201802014, 2019L0066, 2019L0500), and Cultivate Scientific Research Excellence Programs of Higher Education Institutions in Shanxi (CSREP) (2019SK036).

**Acknowledgments:** The authors are grateful to the editor and three anonymous reviewers for their helpful and valuable comments which helped to improve the quality of the research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **An Extended TOPSIS Method with Unknown Weight Information in Dynamic Neutrosophic Environment**

**Nguyen Tho Thong 1,2, Luong Thi Hong Lan 3,4, Shuo-Yan Chou 5,6, Le Hoang Son 1, Do Duc Dong <sup>2</sup> and Tran Thi Ngan 3,\***


Received: 9 February 2020; Accepted: 9 March 2020; Published: 11 March 2020

**Abstract:** Decision-making activities are prevalent in human life. Many methods have been developed to address real-world decision problems. In some practical situations, decision-makers prefer to provide their evaluations over a set of criteria and weights. However, in many real-world situations, problems include a lack of weight information for the times, criteria, and decision-makers (DMs). To remedy such discrepancies, an optimization model has been proposed to determine the weights of attributes, times, and DMs. A new concept related to the correlation measure and some distance measures for the dynamic interval-valued neutrosophic set (DIVNS) are defined in this paper. An extend Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method in the interval-valued neutrosophic set with unknown weight information in dynamic neutrosophic environments is developed. Finally, a practical example is discussed to illustrate the feasibility and effectiveness of the proposed method.

**Keywords:** dynamic neutrosophic environment; dynamic interval-valued neutrosophic set; unknown weight information

#### **1. Introduction**

Multiple criteria decision-making (MCDM) problems have gained more attention to researchers in recent years. The purpose of the MCDM process is to make the best ideal choice reaching the highest standard of achievement from a set of alternatives. Existing studies of MCDM attempt to handle various kinds of multi-criteria decision-making problems. The MCDM's evaluation is decided on the basis of alternative evaluations being withdrawn from to the weights of the criteria. They are completely unknown, based upon some diverse reasons, such as time pressure, partial knowledge, incomplete attribute information, and lack of decision-makers' information, so that the overall evaluation cannot be derived. Especially, in real-world situations of group decisions, the exact appreciation of weights is important for handling MCDM problems and for making a decision. For solving such problems, several studies have attempted to develop the methods to handle the MCDM problems using various kinds of information, such as fuzzy set [1], interval fuzzy set [2,3], intuitionistic fuzzy set [4,5], hesitant fuzzy set [6], neutrosophic set [7–10], interval neutrosophic set [11–15], or single neutrosophic set [16], etc. [17–19], and various methods (e.g., maximizing deviation method, entropy, optimization method) [20–22] in which the information of criteria weights are incompletely known.

Yue et al. [23] presented a TOPSIS model to calculate the weights of the DMs under a group decision environment with individual information described as interval numbers. Sajjad Ali Khan et al. [6] introduced a study based on the combination of the maximizing deviation method and the TOPSIS method for resolving MCDM problems where the valuation information is depicted as Pythagorean hesitant fuzzy numbers and information about attribute weight is incomplete. Broumi et al. [24] proposed an extended TOPSIS method for solving multiple attribute decision-making based on two new concepts of complex neutrosophic sets. Gupta et al. [4] also extended the TOPSIS method under intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets. They considered different variations of weights of attributes depending on their subjective impression, cognitive thinking, and their psychology. Wang and Mendel [25] presented an optimization model to solve the decision-making (DM) problems on the Interval Type-2 (IT2) fuzzy set. All the DMs' information is characterized by the IT2 fuzzy set and the attribute weights' information is completely unknown. Maghrabie et al. [26] proposed a new model that used the maximizing deviation method and grey systems theory to estimate the unknown criteria weights. Peng [27] proposed a novel model for achieving unknown attribute weights and handling an IoT (Internet of Things) industry decision-making issue based on interval neutrosophic sets. Tian et al. [28] combined single-valued neutrosophic sets with completely unknown criteria weights and qualitative flexible multiple criteria method for MCDM problems. In addition, for handling multi attribute decision-making problems with interval neutrosophic information, Hong et al. [29] discussed some distance measure based on the TODIM (an acronym in Portuguese for Interactive and Multicriteria Decision Making) method.

According to above analyses, the motivations of this study are summarized as follows:


Therefore, we focus on the issue of multiple attribute group decision-making model based on an interval-valued neutrosophic fuzzy environment, and DMs' information is characterized by interval-valued neutrosophic fuzzy sets, and the information is completely and partially unknown. We study multiple attribute group decision-making methods with incompletely known weights of DMs, attributes, and time in the neutrosophic setting and the interval-valued neutrosophic setting.

In this paper, our aim is to propose a novel decision-making approach based on DIVNS for unknown weight information to effectively solve the above deficits. The main contributions of this paper can be summarized as follows:


To do that, the rest of this work is organized as follows. In Section 2, we review some basis concepts. In Section 3, we develop a TOPSIS approach to handle the MCDM problems under DIVNS in dynamic neutrosophic environments where all information of attributes, DMs, and time is completely and partially unknown. Section 4 presents the numerical results of applying our proposed method in

a practical problem to demonstrate the feasibility of this method. Some comparative analyses with existing algorithms are presented in Section 5. Finally, this paper ends with some conclusions of this study in Section 6.

#### **2. Preliminary**

In this section, we review some basic knowledge, such as dynamic interval-valued neutrosophic sets and MCDM.

#### *2.1. Dynamic Interval-Valued Neutrosophic Sets*

Neutrosophic sets are characterized by truth membership (T), indeterminacy membership (I), and falsity membership (F) with the conditions as 0 <= T <= 1; 0 <= I <= 1; 0 <= F <= 1. Moreover, three membership functions have to satisfy 0 <= T + I + F <= 3. Some other concepts were designed based on neutrosophic sets such as the neutrosophic probability and neutrosophic statistics, that refer to both randomness and indeterminacy with no such contraints of memberships [30]. Herein, we extend the neutrosophic set and logic to the dynamic interval-valued neutrosophic set where each element in the new neutrosophic set is expressed by the interval-valued neutrosophic number and time sequence.

**Definition 1** [31]. *Let U be a universe of discourse. A is an interval neutrosophic set expressed by:*

$$A = \left\{ \mathbf{x}, \left< \left[ T\_A^L(\mathbf{x}), T\_A^{\mathcal{U}}(\mathbf{x}) \right] \right> \left[ I\_A^L(\mathbf{x}), I\_A^{\mathcal{U}}(\mathbf{x}) \right] \left[ F\_A^L(\mathbf{x}), F\_A^{\mathcal{U}}(\mathbf{x}) \right] \right\} \middle| \mathbf{x} \in \mathcal{U} \right\} \tag{1}$$

*where TL <sup>A</sup>*(*x*), *<sup>T</sup><sup>U</sup> <sup>A</sup>* (*x*) ⊆ [0, 1]; *I L <sup>A</sup>*(*x*), *I U <sup>A</sup>* (*x*) ⊆ [0, 1]; *FL <sup>A</sup>*(*x*), *FU <sup>A</sup>* (*x*) ⊆ [0, 1] *represents truth, indeterminacy, and falsity membership functions of an element.*

Thong et al. [13] introduced the concept of a DIVNS, which is shown as follows.

**Definition 2** [13]. *Let U be a universe of discourse. A is a dynamic–valued neutrosophic set (DIVNS) expressed by,*

$$A = \left\{ \mathbf{x}, \left\langle \left[ T\_x^L(t), T\_x^{lI}(t) \right] \right\rangle \left[ I\_x^L(t), I\_x^{lI}(t) \right] \left[ F\_x^L(t), F\_x^{lI}(t) \right] \right\} \middle| \mathbf{x} \in \mathcal{U} \right\} \tag{2}$$

*where t* = {*t*1, *t*2, ... , *tk*}; *TL <sup>x</sup>* (*t*), *T<sup>U</sup> <sup>x</sup>* (*t*) ⊆ [0, 1]; *I L <sup>x</sup>* (*t*), *I U <sup>x</sup>* (*t*) ⊆ [0, 1]; *FL <sup>x</sup>* (*t*), *FU <sup>x</sup>* (*t*) ⊆ [0, 1] *and for convenience, we call<sup>n</sup>* = 6*T<sup>L</sup> <sup>x</sup>* (*t*), *T<sup>U</sup> <sup>x</sup>* (*t*) , *I L <sup>x</sup>* (*t*), *I U <sup>x</sup>* (*t*) , *FL <sup>x</sup>* (*t*), *FU <sup>x</sup>* (*t*) 7 *a dynamic interval–valued neutrosophic element (DIVNE).*

#### *2.2. MCDM Problems in a Dynamic Neutrosophic Environment*

Thong et al. [13] expressed MCDM problems in the dynamic neutrosophic environment as follows: Consider a MCDM problem containing *A* = {*A*1, *A*2, ... , *Av*} and *C* = {*C*1,*C*2, ... ,*Cn*} and *D* = {*D*1, *D*2, ... , *Dh*} are sets of alternatives, criteria, and decision-makers. For a decision-maker *Dq*(*q* = 1, 2, 3, ... , *h*) the evaluation characteristic of an alternatives *Am*(*m* = 1, 2, 3, ... , *v*) on a criteria *Cp*(*p* = 1, 2, 3, ... , *n*) in time sequence *t* = {*t*1, *t*2, ... , *tk*} is represented by the decision matrix where *d q mp*(*t*) = 8 *x q dmp* (*t*), *Tq dmp*, *t* , *I q dmp*, *t* , *Fq dmp*, *t* ; *<sup>t</sup>* <sup>=</sup> {*t*1, *<sup>t</sup>*2, ... , *tk*} taken by DIVNSs evaluated by decision-maker *Dq*.

#### **3. An Extended TOPSIS Method for Unknown Weight Information**

This section proposes the method to handle the MCDM problem that include a lack of the weight information for the times, criteria, and DMs in dynamic netrosophic environments.

#### *3.1. Correlation Coe*ffi*cient Measure for Dynamic Interval-Valued Neutrosophic Sets*

We propose a novel correlation coefficient measure for DIVNSs based on the idea in [32].

**Definition 3.** *Let <sup>Y</sup>*(*t*) = *x*(*t*), 6 *TY*(*x*, *tl*), *I <sup>Y</sup>*(*x*, *tl*), *FY*(*x*, *tl*) <sup>7</sup>, <sup>∀</sup>*tl* <sup>∈</sup> *<sup>t</sup>*, *<sup>x</sup>* <sup>∈</sup> *<sup>U</sup> and <sup>Z</sup>*(*t*) = *x*(*t*), 6 *TZ*(*x*, *tl*), *I <sup>Z</sup>*(*x*, *tl*), *FZ*(*x*, *tl*) <sup>7</sup>, <sup>∀</sup>*tl* <sup>∈</sup> *<sup>t</sup>*, *<sup>x</sup>* <sup>∈</sup> *<sup>U</sup> be two DIVNs in <sup>t</sup>* <sup>=</sup> {*t*1, *<sup>t</sup>*2, ... , *tk*} *and <sup>U</sup>* <sup>=</sup> (*x*1, *x*2, ... , *xn*). *A correlation coe*ffi*cient measure between A and B* is:

$$K(Y, Z) = \frac{\mathbb{C}(Y, Z)}{\max(T(Y), T(Z))} = \frac{\sum\_{i=1}^{n} \mathbb{C}(Y(\mathbf{x}\_i), Z(\mathbf{x}\_i))}{\max\left(\sum\_{i=1}^{n} T(Y(\mathbf{x}\_i)), \sum\_{i=1}^{n} T(Z(\mathbf{x}\_i))\right)} \tag{3}$$

*where C*(*Y*,*Z*) *is considered the correlation between two DIVNSs Y and Z; T*(*Y*) *and T*(*Z*) *refer to the information energies if the two DIVNSs, respectively. These components are provided by:*

*C*(*Y*,*Z*) = <sup>1</sup> *k k l*=1 *n i*=1 *C*(*Y*(*xi*, *tl*),*Z*(*xi*, *tl*)) = <sup>1</sup> *k k l*=1 *n i*=1 1 2 ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ inf*TY*(*xi*, *tl*) × inf*TZ*(*xi*, *tl*) + sup*TY*(*xi*, *tl*) × sup*TZ*(*xi*, *tl*) +inf*IY*(*xi*, *tl*) × inf*IZ*(*xi*, *tl*) + sup*IY*(*xi*, *tl*) × sup*IZ*(*xi*, *tl*) +inf*FY*(*xi*, *tl*) × inf*FZ*(*xi*, *tl*) + sup*FY*(*xi*, *tl*) × sup*FZ*(*xi*, *tl*) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ *T*(*Y*) = <sup>1</sup> *k k l*=1 *n i*=1 *T*(*Y*(*xi*, *tl*)) = <sup>1</sup> *k k l*=1 *n i*=1 1 2 ⎡ ⎢⎢⎢⎢⎣ (inf*TY*(*xi*, *tl*))<sup>2</sup> + (sup*TY*(*xi*, *tl*))<sup>2</sup> + (inf*IY*(*xi*, *tl*))<sup>2</sup> + (sup*IY*(*xi*, *tl*))<sup>2</sup> +(inf*FY*(*xi*, *tl*))<sup>2</sup> + (sup*FY*(*xi*, *tl*))<sup>2</sup> ⎤ ⎥⎥⎥⎥⎦ *T*(*Z*) = <sup>1</sup> *k k l*=1 *n i*=1 *T*(*Z*(*xi*, *tl*)) = <sup>1</sup> *k k l*=1 *n i*=1 1 2 ⎡ ⎢⎢⎢⎢⎣ (inf*TZ*(*xi*, *tl*))<sup>2</sup> + (sup*TZ*(*xi*, *tl*))<sup>2</sup> + (inf*IZ*(*xi*, *tl*))<sup>2</sup> + (sup*IZ*(*xi*, *tl*))<sup>2</sup> +(inf*FZ*(*xi*, *tl*))<sup>2</sup> + (sup*FZ*(*xi*, *tl*))<sup>2</sup> ⎤ ⎥⎥⎥⎥⎦

**Theorem 1.** *The correlation coe*ffi*cient K between Y and Z satisfies the follow properties:*

(i) 0 ≤ *K*(*Y*,*Z*) ≤ 1 (ii) *K*(*Y*,*Z*) = *K*(*Z*,*Y*) (iii) *K*(*Y*,*Z*) = 1 ⇔ *Y* = *Z*

**Proof.** (i) for any *i* = 1, 2, 3, ... , *n*, the values of [inf*TY*(*xi*, *tl*), sup*TY*(*xi*, *tl*)]; [inf*IY*(*xi*, *tl*), sup*IY*(*xi*, *tl*)]; [inf*FY*(*xi*, *tl*), sup*FY*(*xi*, *tl*)]; [inf*TZ*(*xi*, *tl*), sup*TZ*(*xi*, *tl*)]; [inf*IZ*(*xi*, *tl*), sup*IZ*(*xi*, *tl*)]; [inf*FZ*(*xi*, *tl*), sup*FZ*(*xi*, *tl*)] ⊆ [0, 1] exist for any *i* = 1, 2, 3, ... , *n*. Thus, it is hold that *C*(*Y*,*Z*) ≥ 0; *T*(*Y*) ≥ 0; *T*(*Z*) ≥ 0. Therefore

$$K(\mathcal{Y}, Z) = \frac{C(\mathcal{Y}, Z)}{\max(T(\mathcal{Y}), T(Z))} \ge 0$$

and according to the Cauchy–Schwarz inequality, it holds that:

$$K(\boldsymbol{Y}, \boldsymbol{Z}) = \frac{C(\boldsymbol{Y}, \boldsymbol{Z})}{\max(T(\boldsymbol{Y}), T(\boldsymbol{Z}))} \le 1$$

Therefore, 0 ≤ *K*(*Y*,*Z*) ≤ 1.

(ii) It is obvious that if *Y*(*t*) = *Z*(*t*), ∀*l* ∈ {1, 2, ... , *k*}. We have: inf*TY*(*xi*, *tl*) = inf*TZ*(*xi*, *tl*); sup*TY*(*xi*, *tl*) = sup*TZ*(*xi*, *tl*); inf*I <sup>Y</sup>*(*xi*, *tl*) = inf*I <sup>Z</sup>*(*xi*, *tl*); sup*I <sup>Y</sup>*(*xi*, *tl*) = sup*I <sup>Z</sup>*(*xi*, *tl*); inf*FY*(*xi*, *tl*) = inf*FZ*(*xi*, *tl*); sup*FY*(*xi*, *tl*) = sup*FZ*(*xi*, *tl*); Thus, *K*(*Y*,*Z*) = *K*(*Z*,*Y*). Theorem 1 is proved.

(iii) It is easily observed. -

#### *3.2. Distance Measures for Dynamic Interval-Valued Neutrosophic Sets*

In this section, we present the definitions of the Hamming and Euclidean distances between DIVNEs and distance of two dynamic interval-valued neutrosophic matrices.

**Definition 4.** *Let n*<sup>1</sup> *and n*<sup>2</sup> *be two DIVNEs. The dynamic interval-valued neutrosophic distance between n*<sup>1</sup> *and n*<sup>2</sup> is determined as follows:

*(i) The Hamming distance:*

$$d\_1(n\_1, n\_2) = \frac{1}{6 \times k} \sum\_{l=1}^{k} \left( \begin{array}{c} \left| T\_{n\_1}^{\mathcal{L}}(t\_l) - T\_{n\_2}^{\mathcal{L}}(t\_l) \right| + \left| T\_{n\_1}^{\mathcal{U}}(t\_l) - T\_{n\_2}^{\mathcal{U}}(t\_l) \right| + \left| I\_{n\_1}^{\mathcal{L}}(t\_l) - I\_{n\_2}^{\mathcal{L}}(t\_l) \right| \\ + \left| I\_{n\_1}^{\mathcal{U}}(t\_l) - I\_{n\_2}^{\mathcal{U}}(t\_l) \right| + \left| F\_{n\_1}^{\mathcal{L}}(t\_l) - F\_{n\_2}^{\mathcal{L}}(t\_l) \right| + \left| F\_{n\_1}^{\mathcal{U}}(t\_l) - F\_{n\_2}^{\mathcal{U}}(t\_l) \right| \end{array} \right) \tag{4}$$

*(ii) The Euclidean distance:*

$$d\_{2}(n\_{1},n\_{2}) = \sqrt{\frac{1}{6 \times k} \sum\_{l=1}^{k} \left( \begin{array}{l} \left(T\_{n\_{1}}^{L}(t\_{l}) - T\_{n\_{2}}^{L}(t\_{l})\right)^{2} + \left(T\_{n\_{1}}^{L}(t\_{l}) - T\_{n\_{2}}^{L}(t\_{l})\right)^{2} + \left(I\_{n\_{1}}^{L}(t\_{l}) - I\_{n\_{2}}^{L}(t\_{l})\right)^{2} \\ + \left(I\_{n\_{1}}^{L}(t\_{l}) - I\_{n\_{2}}^{L}(t\_{l})\right)^{2} + \left(F\_{n\_{1}}^{L}(t\_{l}) - F\_{n\_{2}}^{L}(t\_{l})\right)^{2} + \left(F\_{n\_{1}}^{L}(t\_{l}) - F\_{n\_{2}}^{L}(t\_{l})\right)^{2} \end{array}} \tag{5}$$

*(iii) The geometry distance:*

$$d\_3(n\_1, n\_2) = \left(\frac{1}{6\times k} \sum\_{l=1}^k \left( \begin{array}{c} \left(T\_{n\_1}^{\mathcal{L}}(t\_l) - T\_{n\_2}^{\mathcal{L}}(t\_l)\right)^{\alpha} + \left(T\_{n\_1}^{\mathcal{U}}(t\_l) - T\_{n\_2}^{\mathcal{U}}(t\_l)\right)^{\alpha} + \left(I\_{n\_1}^{\mathcal{L}}(t\_l) - I\_{n\_2}^{\mathcal{L}}(t\_l)\right)^{\alpha} \\ + \left(I\_{n\_1}^{\mathcal{L}}(t\_l) - I\_{n\_2}^{\mathcal{L}}(t\_l)\right)^{\alpha} + \left(F\_{n\_1}^{\mathcal{L}}(t\_l) - F\_{n\_2}^{\mathcal{L}}(t\_l)\right)^{\alpha} + \left(F\_{n\_1}^{\mathcal{U}}(t\_l) - F\_{n\_2}^{\mathcal{U}}(t\_l)\right)^{\alpha} \end{array} \right) \right)^{\frac{1}{\alpha}} \tag{6}$$

*where* α > 0 *and*


Therefore, the distance in Equation (6) is a generalization of distances in Equation (5) and Equation (4).

**Definition 5.** *Given two dynamic interval-valued neutrosophic matrices <sup>A</sup>*<sup>1</sup> <sup>=</sup> [α(*tl*)]*h*×*<sup>n</sup> and <sup>A</sup>*<sup>2</sup> <sup>=</sup> [β(*tl*)]*h*×*n, the elements of both A*<sup>1</sup> *and A*<sup>2</sup> *are described by DIVNS. After that the distance between A*<sup>1</sup> *and A*<sup>2</sup> is defined by:

$$d(A\_1, A\_2) = \frac{1}{\text{Im}} \sum\_{p}^{n} \sum\_{q}^{h} d\left(\alpha\_{qp}, \beta\_{qp}\right) \tag{7}$$

*where d* <sup>α</sup>*qp*, <sup>β</sup>*qp is the distance between two DIVNEs.*

#### *3.3. Unknown Weight Information in Dynamic Neutrosophic Environment*

#### 3.3.1. Determining the Weight of Time

It is common knowledge that the weights of time periods have an important role in MCDM problems practical application. In the followings, we present how to determine the weights of time periods in dynamic neutrosophic environments.

**Definition 6.** *Given a basic unit-interval monotonic (BUM) function g* : [0, 1] → [0, 1], *the time weight can be determined as follows:*

$$\lambda(t\_l) = \mathbf{g}\left(\frac{\mathcal{R}\_l}{TV}\right) - \mathbf{g}\left(\frac{\mathcal{R}\_{l-1}}{TV}\right) \tag{8}$$

*Mathematics* **2020**, *8*, 401

*where Rl* = *l j*=1 *Vj*; *TV* = *k i*=1 *Vi*; *Vi* = 1 + *T*(*MDi*)*; T*(*MDi*) *denotes the support of i th largest argument by all the other arguments:*

$$T(MD\_i) = \sum\_{\substack{j=1 \\ j \neq i}}^k \Omega \upmu[MD\_{i\prime}MD\_j],$$

$$\begin{split} \sup\_{\mathbf{h}} \left\{ \text{MD}\_{\text{i}}, \text{MD}\_{\text{j}} \right\} &= 1 - d \Big\{ \text{MD}\_{\text{i}}, \text{MD}\_{\text{j}} \Big\} \\ \leq 1 - \frac{1}{\ln} \sum\_{q}^{h} \sum\_{p=1}^{n} \sqrt{\frac{1}{\delta} \Bigg\{ \left( T^{\mathcal{L}} \Big( \mathbf{x}\_{pq}^{i} \Big) - T^{\mathcal{L}} \Big( \mathbf{x}\_{pq}^{j} \Big) \Big)^{2} + \left( T^{\mathcal{U}} \Big( \mathbf{x}\_{pq}^{i} \Big) - T^{\mathcal{U}} \Big( \mathbf{x}\_{pq}^{j} \Big) \Big)^{2} + \left( I^{\mathcal{L}} \Big( \mathbf{x}\_{pq}^{i} \Big) - I^{\mathcal{L}} \Big( \mathbf{x}\_{pq}^{j} \Big) \right)^{2} + \left( I^{\mathcal{L}} \Big( \mathbf{x}\_{pq}^{i} \Big) - I^{\mathcal{L}} \Big( \mathbf{x}\_{pq}^{j} \Big) \Big)^{2}}{\left( L^{\mathcal{U}} \Big( \mathbf{x}\_{pq}^{i} \Big) - L^{\mathcal{U}} \Big( \mathbf{x}\_{pq}^{j} \Big) \Big)^{2} + \left( I^{\mathcal{U}} \Big( \mathbf{x}\_{pq}^{i} \Big) - I^{\mathcal{U}} \Big( \mathbf{x}\_{pq}^{j} \Big) \right)^{2}} \right\} \end{split} \tag{9}$$

3.3.2. Determining the Weights of Decision-Makers

The weights of DMs play a critical role in MCDM problems. In this section, we present how to determine the weights of DMs in dynamic neutrosophic environment.

**Definition 7.** *Let <sup>D</sup>*<sup>1</sup> <sup>=</sup> [α(*tl*)]*v*×*<sup>n</sup> and <sup>D</sup>*<sup>2</sup> <sup>=</sup> [β(*tl*)]*v*×*<sup>n</sup> be two dynamic interval-valued neutrosophic matrices, in which the elements of both D*<sup>1</sup> *and D*<sup>2</sup> *are expressed by DIVNS. Then the correlation coe*ffi*cient between D*<sup>1</sup> *and D*<sup>2</sup> *is defined by:*

$$\mathbb{C}(D\_1, D\_2) = \frac{1}{n\upsilon} \sum\_{p}^{n} \sum\_{m}^{\upsilon} \mathbb{K}(\alpha\_{mp}, \beta\_{mp}) \tag{10}$$

*where K* <sup>α</sup>*mp*, <sup>β</sup>*mp is correlation coe*ffi*cient measure between two DIVNEs*.

**Theorem 2.** *For two Dynamic interval-valued neutrosophic matrices <sup>D</sup>*<sup>1</sup> <sup>=</sup> [α(*tl*)]*v*×*<sup>n</sup> and <sup>D</sup>*<sup>2</sup> <sup>=</sup> [β(*tl*)]*v*×*<sup>n</sup> where the elements of both D*<sup>1</sup> *and D*<sup>2</sup> *are expressed by DIVNSs, C*(*D*1, *D*2) *satisfies the three conditions:*


**Proof.** (i) According to Theorem 1, we have 0 ≤ *K* <sup>α</sup>*mp*, <sup>β</sup>*mp* <sup>≤</sup> 1; *<sup>m</sup>* <sup>=</sup> 1, 2, 3, ... , *<sup>v</sup>*; *p* = 1, 2, 3, ... , *n*, Thus,

$$0 \le \frac{1}{nv} \sum\_{p}^{n} \sum\_{m}^{v} K(\alpha\_{mp}, \beta\_{mp}) \le 1$$

(ii) According to Definition 3 and Theorem 1 it is easily observed.

(iii) According to Theorem 1 we obtain *C*(*D*1, *D*2) = 1 ⇔ *D*<sup>1</sup> = *D*<sup>2</sup>

Thus, Theorem 2 is proved. -

**Definition 8.** *For the decision-maker Dq*, *the weights of decision-makers can be defined as follows*:

$$
\omega\_q = \frac{\delta\_q}{\sum\_{q=1}^h \delta\_q} \tag{11}
$$

*where* δ*<sup>q</sup> has the form:*

$$\delta\_{\mathfrak{q}} = \sum\_{\substack{q'=1;\\ q'\neq q}}^h \mathbb{C}\{D\_{q'}D\_{q'}\}\tag{12}$$

*C Dq*, *Dq is the correlation coe*ffi*cient between two decision-makers q* and *q* .

#### 3.3.3. Determining the Weights of the Criteria

In real life applications, the attribute information may be completely unknown. Thus, we need to develop an integrated programming model for MCDM problems under the dynamic neutrosophic environment.

**Definition 9.** *Let Cp be the pth criterion and Am be the mth alternative, the deviation value between Am and all the other alternatives in dynamic neutrosophic environment can be calculated as:*

$$O\_{mp}(w) = \sum\_{\substack{k=1 \\ k \neq i}}^{v} d(n\_{mp}, n\_{kp}) w\_p \\ k \neq i$$

*where wp is weight of the pth criterion. d nmp*, *nkp is the distance between two DIVNEs.*

**Definition 10.** *The deviation among all the alternatives to the others can be computed by the global deviation function as follows:*

$$O\_p(w) = \sum\_{m=1}^{\upsilon} O\_{mp}(w) = \sum\_{m=1}^{\upsilon} \sum\_{k=1}^{\upsilon} d\binom{n}{m p} w\_p$$
 
$$k \neq m \tag{14}$$
 
$$\text{s.t. } \sum\_{p=1}^{n} w\_p = 1 ; w\_{p \ge 0};$$

By using the deviation degree between evaluations [33], the criteria weights can be calculated. Then, we construct optimization decision making model with the purpose of maximizing the decision space in the following:

$$\text{maxO}(w) = \sum\_{p=1}^{n} O\_p(w) = \sum\_{p=1}^{n} \sum\_{m=1}^{\upsilon} \sum\_{\substack{k=1 \\ k \neq m}}^{\upsilon} d \binom{n\_{mp}, n\_{kp}}{} w\_p^\* \to \max \tag{15}$$

where *d nmp*, *nkp* is the distance between two elements. The optimization model can be solved based on the Lagrange function. Let ξ be the Lagrange multiplier. We have:

$$L\{w\_{p^\star}^\star,\xi\} = O(w) - \frac{1}{2}\xi\left(\sum\_{p=1}^n \left(w\_p^\star\right)^2 - 1\right)$$

$$L\{w\_{p^\star}^\star,\xi\} = \sum\_{p=1}^n \sum\_{m=1}^\upsilon \sum\_{\substack{k=1\\k\neq m}}^\upsilon d\left(n\_{mp^\star}n\_{kp}\right)w\_p^\star - \frac{1}{2}\xi\left(\sum\_{p=1}^n \left(w\_p^\star\right)^2 - 1\right)$$

*Mathematics* **2020**, *8*, 401

$$\begin{cases} \begin{array}{ll} \frac{\partial L}{\partial \mathbf{w}\_p} = \sum\_{m=1}^{\upsilon} & \sum\_{k=1}^{\upsilon} d \left( n\_{\mathrm{mp}}, n\_{\mathrm{kp}} \right) - \xi; w\_p^\* = 0 & \sum\_{m=1}^{\upsilon} & d \left( n\_{\mathrm{mp}}, n\_{\mathrm{kp}} \right) \\ & k = 1; \end{cases} \\\ k \neq m \end{cases} \Longrightarrow w\_p^\* = \frac{\begin{array}{ll} \sum\_{k=1}^{\upsilon} & \sum\_{k=1}^{\upsilon} d \left( n\_{\mathrm{mp}}, n\_{\mathrm{kp}} \right) \\ \frac{\partial L}{\partial \zeta} = \frac{1}{2} \left( \sum\_{p=1}^{n} \left( w\_p^\* \right)^2 - 1 \right) = 0 \end{array}}{\xi}$$

Since *<sup>n</sup> m*=*p w*∗ *p* 2 = 1, the value of ξ can be calculated as follows:

$$\sum\_{p=1}^{n} \left| \frac{\sum\_{\substack{m=1\\m\neq m}}^{\upsilon} k = 1;}{k \neq m} \right|^2 = 1, \text{ thus, } \xi = \left[ \sum\_{p=1}^{n} \left| \sum\_{\substack{m=1\\m\neq m}}^{\upsilon} \sum\_{\substack{k=1\\k\neq m}}^{\upsilon} d(n\_{mp}, n\_{kp}) \right|^2 \right]$$

From the above equations, a formula to calculate the criteria weights can be obtained as follows:

$$w\_p^\* = \frac{\sum\_{m=1}^{\upsilon} \sum\_{k=1}^{\upsilon} \quad d(n\_{mp}, n\_{kp})}{k \neq m}$$

$$\left\{ \sum\_{p=1}^n \left[ \sum\_{m=1}^{\upsilon} \sum\_{\substack{k=1 \\ k = 1 \\ k \neq m}}^{\upsilon} d(n\_{mp}, n\_{kp}) \right]^2 \right\}$$

#### *3.4. TOPSIS Method with Unknown Weight Information in Dynamic Neutrosophic Environments*

In this section, we develop a MCDM approach based on the TOPSIS model with unknown weight information in dynamic neutrosophic environments. The scheme of the proposed MCDM technique is given in Figure 1. The detailed method is constructed as follows:

**Step 1**. Construct the dynamic interval-valued neutrosophic decision matrix as MCDM problems expressed in Section 2.2.

**Step 2**. Using Equation (8) to determine the time weights λ = (λ1, λ2, ... , λ*k*) of *k* time sequence:

$$\text{g}(\mathbf{x}) = \frac{e^{\alpha \mathbf{x}} - 1}{e^{\alpha} - 1} \tag{17}$$

**Step 3**. Using Equations (10)–(12) to determine the DMs' weights ω = (ω1, ω2, ... , ω*h*) of *h* decision-makers.

**Step 4**. I the criteria weight information is completely unknown, we determine the criteria weights *w* = (*w*1, *w*2, ... , *wn*) *<sup>T</sup>* of *n* criteria by using Equation (16), otherwise go to Step 5.

**Step 5**. Suppose *<sup>W</sup>* <sup>=</sup> [ψ(*tl*)]*p*×*q*; *<sup>p</sup>* <sup>=</sup> 1, 2, 3, ... , *<sup>n</sup>*; *<sup>q</sup>* <sup>=</sup> 1, 2, 3, ... , *<sup>h</sup>*; *<sup>l</sup>* <sup>=</sup> 1, 2, 3, ... , *<sup>k</sup>* be dynamic interval-valued neutrosophic matrix of important criteria weights. ψ*pq*(*tl*) is the weight of decision-maker *qth* to criterion *pth* in time sequence *tl*. The criteria weights *w* = (*w*1, *w*2, ... , *wn*) *T* can be calculated by:

*Mathematics* **2020**, *8*, 401

$$w\_p = \left\langle \begin{bmatrix} \left\langle 1 - \left\{ 1 - \left( 1 - \sum\_{q=1}^h T\_{pq}^L(\psi\_{l\_l}) \right)^{\frac{1}{h}} \right\}^{\frac{1}{h}} \\ \left\langle \sum\_{q=1}^h I\_{pq}^L(\psi\_{l\_l}) \right\rangle^{\frac{1}{h\kappa}}, \left\langle \sum\_{q=1}^h I\_{pq}^{\text{II}}(\psi\_{l\_l}) \right\rangle^{\frac{1}{h\kappa}} \end{bmatrix}, \left\{ \left\langle \sum\_{q=1}^h I\_{pq}^{\text{II}}(\psi\_{l\_l}) \right\rangle^{\frac{1}{h\kappa}}, \left\langle \sum\_{q=1}^h I\_{pq}^{\text{II}}(\psi\_{l\_l}) \right\rangle^{\frac{1}{h\kappa}} \right\} \end{bmatrix} \right\rangle \right\rangle \tag{18}$$

**Step 6**. The aggregate ratings of alternative *m* and criteria *p* can be estimated as:

$$\mathbf{x}\_{mp} = \frac{1}{\hbar k} \otimes \left\{ \begin{bmatrix} \left( 1 - \left( 1 - \left( 1 - \sum\_{q=1}^{h} T\_{pmq}^{\mathrm{L}}(\mathbf{x}\_{i}) \right)^{\frac{1}{\hbar}} \right)^{\frac{1}{\hbar}} \\\\ \left( \sum\_{q=1}^{h} I\_{pmq}^{\mathrm{L}}(\mathbf{x}\_{i}) \right)^{\frac{1}{\hbar \mathrm{L}}}, \left( \sum\_{q=1}^{h} I\_{pmq}^{\mathrm{L}}(\mathbf{x}\_{i}) \right)^{\frac{1}{\hbar \mathrm{L}}} \end{bmatrix}, \overline{F\_{mp}(\mathbf{x})} = \left[ \left( \sum\_{q=1}^{h} F\_{pmq}^{\mathrm{L}}(\mathbf{x}\_{i}) \right)^{\frac{1}{\hbar \mathrm{L}}}, \left( \sum\_{q=1}^{h} F\_{pmq}^{\mathrm{L}}(\mathbf{x}\_{i}) \right)^{\frac{1}{\hbar \mathrm{L}}} \right] \end{bmatrix} \right\}. \tag{19}$$

**Step 7**: Average weighted ratings of alternatives can be calculated as follows:

**Case 1**: If the information about the criteria weights is known, the criteria weights is a collection of DIVNEs and the average weighted ratings of alternatives in *tl*, calculated by:

$$\mathbf{G}\_{m} = \frac{1}{p} \sum\_{p=1}^{n} \left\{ \left< \begin{bmatrix} I\_{mp}^{L}(\mathbf{x}) \times I\_{p}^{L}(\mathbf{w}), I\_{mp}^{LI}(\mathbf{x}) \times I\_{p}^{LI}(\mathbf{w}) \\ \left[ I\_{mp}^{L}(\mathbf{x}) + I\_{p}^{L}(\mathbf{w}) - I\_{mp}^{L}(\mathbf{x}) \times I\_{p}^{L}(\mathbf{w}), I\_{mp}^{IL}(\mathbf{x}) + I\_{p}^{IL}(\mathbf{w}) - I\_{mp}^{IL}(\mathbf{x}) \times I\_{p}^{IL}(\mathbf{w}) \right]\_{\mathbf{r}} \\ \left[ I\_{mp}^{L}(\mathbf{x}) + I\_{p}^{L}(\mathbf{w}) - I\_{mp}^{L}(\mathbf{x}) \times I\_{p}^{L}(\mathbf{w}), I\_{mp}^{IL}(\mathbf{x}) + I\_{p}^{IL}(\mathbf{w}) - I\_{mp}^{IL}(\mathbf{x}) \times I\_{p}^{L}(\mathbf{w}) \right] \\ \mathbf{m} = 1,2,3,...,w; p = 1,2,3,...,n; \end{pmatrix} \right\} \tag{20}$$

**Case 2**: If the information about the criteria weights is unknown, the criteria weights is a collection of DIVNEs and average weighted ratings of alternatives in *tl*, calculated by:

$$G\_{m} = \frac{1}{p} \sum\_{p=1}^{n} \left\langle \left\langle \begin{bmatrix} 1 - \left(1 - T\_{mp}^{l}(\mathbf{x})\right)^{w\_{p}}, 1 - \left(1 - T\_{mp}^{lI}(\mathbf{x})\right)^{w\_{p}}\\ \left[\left(I\_{mp}^{l}(\mathbf{x})\right)^{w\_{p}}, \left(I\_{mp}^{lI}(\mathbf{x})\right)^{w\_{p}}\right], \left[\left(F\_{mp}^{l}(\mathbf{x})\right)^{w\_{p}}, \left(F\_{mp}^{lI}(\mathbf{x})\right)^{w\_{p}}\right] \end{bmatrix} \right\rangle \right. \tag{21}$$

**Step 8**: Determine the interval neutrosophic positive ideal solution (PIS, *A*+) and the interval neutrosophic negative ideal solution (NIS, *A*−):

$$A^{+} = \{ \mathbf{x}, ([1, 1], [0, 0], [0, 0]) \} \tag{22}$$

$$A^- = \{ \mathbf{x}, ([0,0], [1,1], [1,1]) \} \tag{23}$$

**Step 9**: Compute the distance of alternatives.

The distances of each alternative in time sequence *tl*, are calculated:

$$d\_m^+ = \sqrt{\left(G\_m - A^+\right)^2} \tag{24}$$

$$d\_m^- = \sqrt{\left(G\_m - A^-\right)^2} \tag{25}$$

where *d*<sup>+</sup> *<sup>m</sup>* and *d*<sup>−</sup> *<sup>m</sup>* represent the shortest and farthest distances of alternative *Am*.

**Step 10**: Determine the relative closeness coefficient.

The closeness coefficient values are calculated below:

$$\text{CC}\_{m} = \frac{d\_{m}^{-}}{d\_{m}^{+} + d\_{m}^{-}} \tag{26}$$

**Step 11**: Rank the alternatives based on the relative closeness coefficients.

**Figure 1.** TOPSIS method with unknown weight information.

#### **4. Experiments**

This section applies the proposed method with dataset in [17] to evaluate lecturers' performances from ULIS, Vietnam National University, Hanoi, Vietnam. The hierarchical structure of the constructed multi-criteria decision-making problem is depicted in Figure 2 for the dataset.

**Figure 2.** Evaluation lecturer's performance problem.

According to the language labels in Table 1 below, the rating of lectures through criteria sets are done by decision-makers.

**Table 1.** Language variables.


**Step 1**: Dynamic interval-valued neutrosophic decision matrix shown in Table 2.

**Step 2**: Bases on Equation (8) and BUM function in Equation (17), we receive the weights of the time periods:

$$\lambda\_1 = 0.280; \lambda\_2 = 0.330; \lambda\_3 = 0.390$$

**Step 3**: Using Equations (10)–(12) to calculate weights of the DMs, we receive the weights of the DMs as follows:

$$\alpha\_1 = 0.330; \alpha\_2 = 0.337; \alpha\_3 = 0.333$$

**Step 4**: Based on the basic of maximizing deviation method and Equation (16), we receive the weights of the criteria as follows:

$$w\_1 = 0.160; w\_2 = 0.165; w\_3 = 0.171; w\_4 = 0.166; w\_5 = 0.175; w\_6 = 0.163$$

**Step 5**: Average weighted ratings are shown in Table 3.

#### *Mathematics* **2020**, *8*, 401


**Table 2.** Dynamic interval-valued neutrosophic decision matrix.

**Table 3.** Average weighted ratings of lectures.


**Step 6**: Compute the distance of each lecture from (PIS, *A*+) and (NIS, *A*−). The results are shown in Table 4 below.


**Table 4.** The distance of each lecture.

**Step 7**: Calculate the closeness coefficient for lectures. Table 5 shows the values of the closeness coefficient.


**Table 5.** The closeness coefficient of lectures.

**Step 8**: Rank the lectures based on the values of the closeness coefficients.

Table 5 shows the ranking order is *A*<sup>2</sup> *A*<sup>3</sup> *A*<sup>4</sup> *A*<sup>1</sup> *A*<sup>5</sup> and *A*<sup>2</sup> is the best lecture.

#### **5. Comparison with the Related Methods**

In this section, we compare the proposed method with those in Thong et al. [17] and Peng [29] to demonstrate the advantages for unknown weight information in dynamic neutrosophic environments. Data used to prove the performance of the method are in [17]. Table 6 shows that the rankings of lectures by Thong et al. [17] as *A*<sup>2</sup> *A*<sup>3</sup> *A*<sup>4</sup> *A*<sup>1</sup> *A*<sup>5</sup> and Peng [29] as *A*<sup>2</sup> *A*<sup>3</sup> *A*<sup>1</sup> *A*<sup>4</sup> *A*5. Thus, *A*<sup>2</sup> is still the best option. These results are the same as our proposed method. However, the proposed method can be solved with unknown weight information in a dynamic neutrosophic environment. Moreover, it is more generalized and flexible than Thong et al. [17]'s method with unknown weight information in a dynamic neutrosophic environment.



#### **6. Conclusions**

In this paper, we proposed a novel approach to solve MCDM problems in dynamic neutrosophic environments where all the information supplied by the DMs is described as interval-valued neutrosophic sets and the information about the weight of attributes, DMs, and time may be incompletely known. A new concept related to the correlation measure and some distance measures for dynamic interval-valued neutrosophic sets are defined. Then, we have proposed an extended TOPSIS method to solve MCDM problems, are is expressed with the interval-valued neutrosophic setting in dynamic neutrosophic environments. Finally, the effectiveness of the proposed method has been demonstrated with the purpose of evaluating lecturers' performance in ULIS, Vietnam National University, Hanoi, Vietnam. We considered in this situation that all the weight information about the criteria, DMs, and time is expressed with various conditions is unknown.

Since the proposed method has not demonstrated its practicality and effectiveness with more real applications and the weight information about the criteria and DMs that change over time is not mentioned in our method, in the future, we will conduct further studies to handle unknown weight information in which the criteria and DMs vary with time periods and with more real decision-making data.

**Author Contributions:** Data curation, L.T.H.L.; methodology, L.H.S., D.D.D. and T.T.N.; validation, N.T.T. and D.D.D.; writing—original draft, N.T.T.; writing—review & editing, S.-Y.C., L.H.S., T.T.N. and D.D.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the "Center for Cyber-physical System Innovation" from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan.

**Acknowledgments:** The first author thanks to the support from the Domestic Master/Ph.D. Scholarship Programme of the Vingroup Innovation Foundation. The authors would like to express their greatest thanks the Center for IoT Innovation (CITI), National Taiwan University of Science and Technology for their support.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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