A dynamic multi-attribute group decision-making method with R-numbers based on MEREC and CoCoSo method

Abstract

Dynamic multi-attribute group decision-making (DMAGDM) is a widespread practice in which evaluations are provided by multiple decision-makers at various times and early evaluations impact later evaluations. Additionally, attributes and alternatives can be added or removed over time. An R-numbers DMAGDM method is developed based on the advantages of R-numbers in capturing risks. This paper introduces the R-numbers Einstein weighted averaging (RNEWA) operator and R-numbers weighted Einstein geometric (RNEWG) operator, which are distinct from conventional algebraic operations, and examines their properties. Moreover, an expert weight determination model is constructed using the similarity measure of R-numbers. The attribute weight determination model in the R-numbers environment is also proposed with the method based on the criteria removal effects method (MEREC). A static rating calculation model, which utilizes the combination compromise solution (CoCoSo) method in the R-numbers environment, is built using the RNEWA operator and RNEWG operator. Furthermore, a new dynamic rating calculation model is proposed which does not require storage of all decision information over time. Finally, the applicability and effectiveness of the R-numbers DMAGDM method is demonstrated through a case study on supply chain risk assessment of manufacturing enterprises.

Introduction

Multi-attribute decision-making (MADM) is a significant branch in the field of decision-making, widely applied in various domains, such as industrial supplier selection [1, 2], enterprise performance evaluation [3, 4], risk assessment [5,6,7], modern agricultural development [8, 9], etc. And in many cases, decision-making is a dynamic process where early decisions will have a significant impact on subsequent outcomes. The process of obtaining the optimal result by multiple experts evaluating alternatives for multiple attributes at different times is known as dynamic multi-attribute group decision-making (DMAGDM). Decision-making also has a core set of procedures, and more and more scholars are conducting research to improve its efficiency, resulting in significant research achievements. The following topics are the main contents of these new decision models: (1) model for representing information of decision-makers' opinions, (2) model for aggregating decision-making information, (3) model for calculating the weights of decision-makers, (4) model for determining attribute weights, and (5) model for determining alternative priorities.

In the representation of decision-maker's opinions, to better describe the vagueness and uncertainty of decision information, researchers have proposed various models for fuzzy information representation. However, when the available data come from unreliable sources or involves future time, there is a certain degree of uncertainty and error in the existing fuzzy information representation models, resulting in a certain deviation between the available data and the definite values. Therefore, the concept of R-numbers [10] was proposed. R-numbers can capture and analyze errors and risks associated with fuzzy numbers in decision-making problems, particularly those involving future events. Additionally, the R-numbers method can simulate various risk scenarios by considering a range as the change of fuzzy data. Therefore, compared with the existing fuzzy set theory, R-numbers have more advantages in describing decision problems.

Information aggregation models play a crucial role in group decision-making environments, and the study of information aggregation operators has been active in recent years. To meet the needs of various information processing requirements, a variety of types of information fusion aggregation operators have been developed, including arithmetic averaging operator [11,12,13,14], ordered weighted averaging operator [11, 14, 15], hybrid weighted averaging operator [11, 14], weighted geometric operator [11,12,13, 16], ordered weighted geometric operator [11, 16], hybrid weighted geometric operator [11, 16], Bonferroni mean operator [17,18,19], Choquet integral operator [20, 21], Heronian mean operator [22, 23], etc. However, all of the above aggregation operators are based on a combination of algebraic products and algebraic sums. Algebraic operations include algebraic product and algebraic sum, but they are not the only operations that can be used to perform intersection and union operations. Various t-norms and t-conorms can also be used to establish the corresponding intersection and union operations, and the Einstein sum and Einstein product [24] are good choices, because they have the same smooth approximation as algebraic product and algebraic sum, respectively. Therefore, Einstein weighted average operator and Einstein weighted geometric operator are proposed in this paper to aggregate R-numbers.

In multi-attribute group decision-making (MAGDM), weight determination is an important concept. The core problem of this research is the determination of expert and attribute weights, which are unknown in this study. Objective weight determination methods are used for both types of weights. For expert weights, previous studies mainly rely on the concept of entropy. However, Chen and Yang [25] proposed a method to determine the objective weights by deriving expert weights from evaluation values. As different experts have different levels of knowledge and professional skills, the evaluation values given by experts may be unreasonable, and it is unreasonable to assign equal weights to experts. Most MAGDM studies adopt a subjective weight determination method based on expert experience [26,27,28]. However, assigning expert weights subjectively will result in significant deviation and massive workload. Therefore, we believe that for expert evaluation value, the smaller the deviation from common perception, the lower the expert weight, and the larger the deviation from common perception, the higher the expert weight. For attribute weights, previous studies mainly relied on the concept of maximum deviation. The attribute weights are usually determined by the deviation values of different alternatives under each attribute. The larger the deviation, the larger the attribute weight. Keshavarz-Ghorabaee [29] proposed a new method for determining objective attribute weights, referred to as the Method based on the Removal Effects of Criteria (MEREC). Attribute weights are calculated using the removal effect of each attribute on the overall performance of the alternatives. In this method, when the removal of an attribute has a greater impact on the overall performance of the scheme, the attribute weight is larger.

There are numerous models for determining the priority of alternatives. A recently proposed model for determining the priority of alternative options is the Combined Compromise Solution (CoCoSo) method introduced by Yazdani et al. [30]. The method employs several aggregation strategies to create a combination compromise algorithm, and it takes into account the weighted sum model (WSM) and the weighted product model (WPM) for increased flexibility in results. In the decision-making process, three equations are used to calculate the weights of alternatives. Finally, the alternatives are sorted and their priority is obtained using the aggregation multiplication rule. However, the calculation of WSM and WPM relies solely on traditional algebraic sum and algebraic product operations. In this study, we develop the CoCoSo method in the R-numbers environment, based on R-numbers Einstein weighted averaging (RNEWA) operator and R-numbers Einstein weighted geometric (RNEWG) operator, using Einstein sum and product operations.

However, the above decision-making research regarding the five subjects was conducted in a static environment, without considering changes over time, and the results are in a non-dynamic decision-making environment. The traditional DMAGDM method constructs a dynamic aggregation operator [31, 32] to aggregate multi-period information by collecting multi-period decision information to form the final decision result. However, this decision method requires not only the storage of all historical information but also certain requirements for the distribution of alternative programs over time. Furthermore, alternatives and attributes in the traditional DMAGDM do not change over time. Campanella and Ribeiro [33] proposed a new DMAGDM framework that addresses these shortcomings by incorporating a dynamic feedback mechanism into the information association aggregation process in each decision period. The framework includes three stages: (1) Calculating the static rating. Applying the classical static MADM model to obtain the static rating of each alternative; (2) Calculating dynamic rating. Based on historical information and appropriate aggregation processes, the dynamic rating of each alternative is obtained; (3) Updating historical information. Considering the dynamic rating values, alternatives are ranked and the information is stored in the historical collection for future use. In this framework, alternatives and attributes may change over time, and they may increase or decrease as time passes.

Despite extensive research and exploration carried out by researchers, there remains significant room for investigation in the following areas: (1) How to apply R-numbers to risk decision-making; (2) The new information fusion method of R-numbers; (3) How to determine the expert weight and attribute weight more reasonably in the R-numbers environment; (4) How to use CoCoSo method to calculate the static rating in the R-numbers environment; (5) How to establish a dynamic decision-making model that is more in line with the practical needs.

Therefore, based on the existing research and the characteristics of R-numbers, this paper discusses the DMAGDM framework in the R-numbers environment. The research objectives and contributions of this paper are as follows:

• (1) The Einstein operators of R-numbers are proposed. The Einstein sum and Einstein product of R-numbers are studied. Based on the Einstein sum and Einstein product of R-numbers, R-numbers Einstein weighted averaging operator and R-numbers weighted Einstein geometric operator are constructed, and their related properties are studied. This lays the foundation for determining attribute weights and researching the CoCoSo method in the R-numbers environment.

• (2) A model for determining expert weights in the R-numbers environment is proposed. The model is built on the idea of assigning lower weights to experts who deviate from common cognition and higher weights to experts who are close to common cognition.

• (3) An MEREC-based model for determining the attribute weights is proposed in the R-numbers environment. The model is based on the removal effect of attributes on the overall performance, where attributes having a significant impact on the overall performance are assigned higher weights and vice versa.

• (4) A model for determining static ratings is proposed in the R-numbers environment using the CoCoSo method. The traditional CoCoSo method, which is based on WSM and WPM, has some limitations. New WSM and WPM models based on RNEWA and RNEWG operators are proposed to address these limitations.

• (5) A dynamic decision-making model in the R-numbers environment is proposed. A new calculation formula is established for the feedback calculation technique of dynamic rating to satisfy the aggregate of static rating calculated using the CoCoSo method, thereby increasing flexibility.

The rest of this paper is structured as follows. In the next section, we review relevant literature research. In the following section, we review the basic concepts of triangular fuzzy number, Type-2 fuzzy sets, Type-2 triangular fuzzy number, R-numbers, and Einstein operation. In the next section, we describe the DMAGDM problem in the R-numbers environment and propose a solution framework based on R-numbers. In the next section, we propose the decision procedure under the proposed R-numbers DMAGDM framework. A transformation model is constructed to transform fuzzy numbers into R-numbers. Two aggregation operators based on R-numbers, RNEWA operator and RNEWG operator, are developed. An expert weight determination model based on R-numbers is proposed. An attribute weight determination model based on MEREC method in the R-numbers environment is constructed. A calculation model of static rating value of CoCoSo based on R-numbers is proposed, and a dynamic decision method based on R-numbers is given. In the following section, we verify the practicability and effectiveness of the method by analyzing the case of supply chain risk assessment of manufacturing enterprises. The superiority of this method is verified by comparative analysis. Finally, a conclusion is summarized the full text, and the shortcomings of this study and future research directions are pointed out in this section.

Literature review

An increasing number of scholars have been investigating DMAGDM in recent times. The following are some of the findings in regards to related topics.

R-numbers

There are few research results on R-numbers. Izadi et al. [34] proposed R-numbers cognitive map method for modeling problems in uncertainty and risky environment. Liu et al. [35] constructed risk-based decision framework based on R-numbers and best–worst method and its application to research and development project selection. Seiti et al. [36] developed the modified R-numbers for risk-based fuzzy information fusion and its application to failure modes, effects, and system resilience analysis (FMESRA). Zhao et al. [37] constructed a hybrid decision-making aided framework for multi-criteria decision-making with R-numbers and preference models.

Aggregation operators

In DMAGDM, each expert provides different evaluation information. A critical issue that requires discussion is how to aggregate this decision-making information, to obtain a comprehensive expression matrix and ultimately, arrive at the target decision outcome. In the process of information aggregation, a majority of studies adopt the arithmetic average operator [11, 13, 14] to aggregate the opinions of multiple experts. However, the arithmetic averaging operator compensates for differences, resulting in a bias in the decision outcome. Therefore, the geometric averaging operator [11, 13, 16] was proposed to handle some irrational and unexpected extreme information (i.e., excessively low or high values). Some studies considered the relationship between attributes and proposed Bonferroni operator [18, 19]. Others proposed Choquet integral operator [20, 21] to deal with abnormal information in evaluation information.

Weight determination methods

In DMAGDM, the determination of weights is crucial. These weights include expert weights and attribute weights. There are many methods for determining weights, mainly divided into three categories: subjective weighting method, objective weighting method, and subjective and objective weighting method. The subjective weighting method determines the weights based on the subjective judgment of the decision-maker, including Analytic Hierarchy Process (AHP) method [38], minimum square sum method [39], Simple Multi-Attribute Ranking Technique (SMART) [40], Level Based Weight Assessment (LBWA) method [41], etc. Objective weighting method uses the initial data to determine the weights based on the differences in the data, including Criteria Importance Through Inter-criteria Correlation (CRITIC) method [42], Simultaneous Evaluation of Criteria and Alternatives (SECA) method [43], Preference Selection Index (PSI) method [44], entropy method [45], etc. Subjective and objective weighting method considers both the subjective preference of decision-maker and the objective decision information, achieving a unification of subjectivity and objectivity, including the linear weighted single-objective optimization method [46], Frank–Wolfe method [47], etc.

MADM methods

In recent years, there have been numerous studies on MADM methods, which are employed to determine the priority of alternatives in a more convenient and rational manner. The core idea of Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method [48] is to select the optimal solution that is closest to the positive ideal solution and farthest from the negative ideal solution. The Weighted Aggregated Sum Product Assessment (WASPAS) method [49] is an MADM method based on the weighted sum model and the weighted product model, which can improve clustering accuracy and make the decision-making process more precise compared to a single method. The COmplex PRoportional ASsessment (COPRAS) method [50] considers the importance and validity of different options in the process of evaluating and ranking alternatives. The Evaluation based on the Distance from the Average Solution (EDAS) method [51] focuses on selecting the optimal solution that is closest to the average solution. The basic principle of Multi-Attributive Border Approximation area Comparison (MABAC) method [52] is to sort the alternatives and select the optimal solution according to the distance between the alternatives and the boundary approximation region.

Preliminaries

R-numbers

Definition 1

[38]. A fuzzy number $\tilde{A}$ on R can be a triangular fuzzy number (TFN) if its membership function $\mu_{{\tilde{A}}} \left( x \right):R \to \left[ {0,1} \right]$ is defined as

$$ \mu_{{\tilde{A}}} \left( x \right) = \left\{ \begin{gathered} {{\left( {x - a_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {x - a_{1} } \right)} {\left( {a_{2} - a_{1} } \right)}}} \right. \kern-0pt} {\left( {a_{2} - a_{1} } \right)}}, \quad a_{1} \le x < a_{2} \hfill \\ {{\left( {a_{3} - x} \right)} \mathord{\left/ {\vphantom {{\left( {a_{3} - x} \right)} {\left( {a_{3} - a_{2} } \right)}}} \right. \kern-0pt} {\left( {a_{3} - a_{2} } \right)}}, \quad a_{2} \le x \le a_{3} \hfill \\ 0,{\text{ otherwise}} \hfill \\ \end{gathered} \right.. $$

(1)

Here, $a_{1}$ and $a_{3}$ are the lower and upper bounds of the fuzzy number $\tilde{A}$, respectively, and $a_{2}$ is the modal value. The TFN can also be defied as $\tilde{A} = \left( {a_{1} ,a_{2} ,a_{3} } \right)$.

Definition 2

[53]. A Type-2 fuzzy set $\tilde{A}$ can be defined by $\mu_{{\tilde{A}}} :X \times U \to U$, in which $x$ is the primary variable of $\tilde{A}$ and $X$ is the universe of discourse of $x$. The three-dimensional membership function of $\tilde{A}\left( {\mu_{{\tilde{A}}} \left( {x,u} \right)} \right)$ can be described by the following equation:

$$ \tilde{A} = \left\{ {\left( {\left( {x,u} \right),\mu_{{\tilde{A}}} \left( {x,u} \right)} \right)|x \in X,u \in \left[ {0,1} \right]} \right\}, $$

(2)

where $u$ is the secondary variable and $u \in \left[ {0,1} \right]$. Besides $\mu_{{\tilde{A}}} \left( {x,u} \right)$ is recognized as secondary membership grade of $x$ and the primary membership of $x$ is defined as $J_{x} = \left\{ {\left( {x,u} \right)|u \in \left[ {0,1} \right],\mu_{{\tilde{A}}} \left( {x,u} \right) > 0} \right\}$.

Definition 3

[54]. The Type-2 triangular fuzzy number $\tilde{\tilde{A}}$ can be defined using a triangular fuzzy number with fuzzy elements as follows:

$$ \tilde{\tilde{A}} = \left( {\tilde{A}_{l} ,\tilde{A}_{m} ,\tilde{A}_{u} } \right), $$

(3)

where $\tilde{A}_{l}$ and $\tilde{A}_{u}$ are the fuzzy lower and upper bounds of $\tilde{\tilde{A}}$, respectively, and $\tilde{A}_{m}$ is the fuzzy modal value.

R-number is a newly established fuzzy number form of pessimistic-optimistic type-2 triangular fuzzy number by considering fuzzy positive and negative risk, fuzzy negative positive acceptable risk, and fuzzy risk perception.

Definition 4

[10]. The R-numbers for arbitrary fuzzy number $\tilde{B}$ with lower limit $l_{{\tilde{B}}}$ and upper limit $u_{{\tilde{B}}}$ in beneficial and non-beneficial modes are denoted by $R_{b}( {\tilde{B}} )$ and $R_{c} ( {\tilde{B}})$, respectively, and can be described as follows:

$$ R_{b} \left( {\tilde{B}} \right) = \left( {R_{1b} \left( {\tilde{B}} \right),R_{2b} \left( {\tilde{B}} \right),R_{3b} \left( {\tilde{B}} \right)} \right), $$

(4)

_where

$$ \left\{ \begin{gathered} R_{1b} \left( {\tilde{B}} \right) = \max \left( {\tilde{B} \otimes \left( {1{\mathbf{ \ominus }}\min \left( {\frac{{\tilde{r}^{ - } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ - } }},\tau } \right) \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ - } } \right)} \right),l_{{\tilde{B}}} } \right), \hfill \\ R_{2b} \left( {\tilde{B}} \right) = \tilde{B}, \hfill \\ R_{3b} \left( {\tilde{B}} \right) = \min \left( {\tilde{B} \otimes \left( {1 \oplus \frac{{\tilde{r}^{ + } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ + } }} \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ + } } \right)} \right),u_{{\tilde{B}}} } \right), \hfill \\ 0 < \tilde{r}^{ - } < 1, \quad \tilde{r}^{ + } > 0. \hfill \\ \end{gathered} \right. $$

(5)

$$ {\text{and}}\,\,R_{c} \left( {\tilde{B}} \right) = \left( {R_{1c} \left( {\tilde{B}} \right),R_{2c} \left( {\tilde{B}} \right),R_{3c} \left( {\tilde{B}} \right)} \right), $$

(6)

where

$$ \left\{ \begin{gathered} R_{1c} \left( {\tilde{B}} \right) = \max \left( {\tilde{B} \otimes \left( {1{\mathbf{ \ominus }}\min \left( {\frac{{\tilde{r}^{ + } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ + } }},\tau } \right) \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ + } } \right)} \right),l_{{\tilde{B}}} } \right), \hfill \\ R_{2c} \left( {\tilde{B}} \right) = \tilde{B}, \hfill \\ R_{3c} \left( {\tilde{B}} \right) = \min \left( {\tilde{B} \otimes \left( {1 \oplus \frac{{\tilde{r}^{ - } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ - } }} \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ - } } \right)} \right),u_{{\tilde{B}}} } \right), \hfill \\ \tilde{r}^{ - } > 1, \quad 0 < \tilde{r}^{ + } < 1, \hfill \\ \end{gathered} \right. $$

(7)

in which $\tau$ is a number infinitely close to one and $\tilde{r}^{ - }$ and $\tilde{r}^{ + }$ are fuzzy negative and positive risks and are defined as the fuzzy error and risk, which leads the fuzzy numbers becomes worse or better in the future. The fuzzy amount of positive and negative risks that can be tolerated by decision-makers are denoted by $\widetilde{AR}^{ - }$ and $\widetilde{AR}^{ + }$, respectively, and $\widetilde{RP}^{ - }$ and $\widetilde{RP}^{ + }$ are experts’ fuzzy risk perceptions related to negative and positive risks. In these relations, $\widetilde{AR}^{ - }$ and $\widetilde{AR}^{ + }$ values are always between zero and one, but the possible range of $\widetilde{RP}^{ - }$ and $\widetilde{RP}^{ + }$ is defined as follows:

$$ \left\{ \begin{gathered} {\text{Optimistic experts 0 < }}\widetilde{RP}^{ - } ,\widetilde{RP}^{ + } < 1; \hfill \\ {\text{Neutral experts }}\widetilde{RP}^{ - } ,\widetilde{RP}^{ + } = {0}; \hfill \\ {\text{Pessimistic experts }} - \infty { < }\widetilde{RP}^ -- ,\widetilde{RP}^{ + } < 0. \hfill \\ \end{gathered} \right. $$

In this article, we assume that $\tilde{B}$ is a triangular fuzzy number, then $R ( {\tilde{B}})$ is the Type-2 triangular fuzzy number. defined as

$R( {\tilde{B}} ) = \left( {\left( {\mu_{11} ,\mu_{12} ,\mu_{13} } \right),\left( {\mu_{21} ,\mu_{22} ,\mu_{23} } \right),\left( {\mu_{31} ,\mu_{32} ,\mu_{33} } \right)} \right)$.

Definition 5

[10]. Let us assume that two R-numbers $R ( {\tilde{B}} )$ and $R ( {\tilde{C}} )$ are Type-2 triangular fuzzy numbers, then the operations between them are defined as

$$ \begin{gathered} R( {\tilde{B}}) \oplus R\left( {\tilde{C}} \right) \hfill \\ \qquad = \left( \begin{gathered} \left( {\mu_{{11\tilde{B}}} + \mu_{{11\tilde{C}}} ,\mu_{{12\tilde{B}}} + \mu_{{12\tilde{C}}} ,\mu_{{13\tilde{B}}} + \mu_{{13\tilde{C}}} } \right), \hfill \\ \left( {\mu_{{21\tilde{B}}} + \mu_{{21\tilde{C}}} ,\mu_{{22\tilde{B}}} + \mu_{{22\tilde{C}}} ,\mu_{{23\tilde{B}}} + \mu_{{23\tilde{C}}} } \right), \hfill \\ \left( {\mu_{{31\tilde{B}}} + \mu_{{31\tilde{C}}} ,\mu_{{32\tilde{B}}} + \mu_{{32\tilde{C}}} ,\mu_{{33\tilde{B}}} + \mu_{{33\tilde{C}}} } \right) \hfill \\ \end{gathered} \right); \hfill \\ \end{gathered} $$

(8)

$$ R( {\tilde{B}} ) \otimes R\left( {\tilde{C}} \right) = \left( \begin{gathered} \left( {\mu_{{11\tilde{B}}} \mu_{{11\tilde{C}}} ,\mu_{{12\tilde{B}}} \mu_{{12\tilde{C}}} ,\mu_{{13\tilde{B}}} \mu_{{13\tilde{C}}} } \right), \hfill \\ \left( {\mu_{{21\tilde{B}}} \mu_{{21\tilde{C}}} ,\mu_{{22\tilde{B}}} \mu_{{22\tilde{C}}} ,\mu_{{23\tilde{B}}} \mu_{{23\tilde{C}}} } \right), \hfill \\ \left( {\mu_{{31\tilde{B}}} \mu_{{31\tilde{C}}} ,\mu_{{32\tilde{B}}} \mu_{{32\tilde{C}}} ,\mu_{{33\tilde{B}}} \mu_{{33\tilde{C}}} } \right) \hfill \\ \end{gathered} \right); $$

(9)

$$ \lambda R( {\tilde{B}} ) = \left( \begin{gathered} \left( {\lambda \mu_{{11\tilde{B}}} ,\lambda \mu_{{12\tilde{B}}} ,\lambda \mu_{{13\tilde{B}}} } \right), \hfill \\ \left( {\lambda \mu_{{21\tilde{B}}} ,\lambda \mu_{{22\tilde{B}}} ,\lambda \mu_{{23\tilde{B}}} } \right), \hfill \\ \left( {\lambda \mu_{{31\tilde{B}}} ,\lambda \mu_{{32\tilde{B}}} ,\lambda \mu_{{33\tilde{B}}} } \right) \hfill \\ \end{gathered} \right); $$

(10)

$$ \left( {R( {\tilde{B}} )} \right)^{\lambda } = \left( \begin{gathered} \left( {\left( {\mu_{{11\tilde{B}}} } \right)^{\lambda } ,\left( {\mu_{{12\tilde{B}}} } \right)^{\lambda } ,\left( {\mu_{{13\tilde{B}}} } \right)^{\lambda } } \right), \hfill \\ \left( {\left( {\mu_{{21\tilde{B}}} } \right)^{\lambda } ,\left( {\mu_{{22\tilde{B}}} } \right)^{\lambda } ,\left( {\mu_{{23\tilde{B}}} } \right)^{\lambda } } \right), \hfill \\ \left( {\left( {\mu_{{31\tilde{B}}} } \right)^{\lambda } ,\left( {\mu_{{32\tilde{B}}} } \right)^{\lambda } ,\left( {\mu_{{33\tilde{B}}} } \right)^{\lambda } } \right) \hfill \\ \end{gathered} \right). $$

(11)

Definition 6

[10]. The defuzzification operation of R-number $R ( {\tilde{B}})$ is defined by

$$ \begin{gathered} COA\left( {R( {\tilde{B}} )} \right) = \frac{1}{9}\quad \left( \begin{gathered} \mu_{11} + \mu_{12} + \mu_{13} + \mu_{21} + \hfill \\ \mu_{22} + \mu_{23} + \mu_{31} + \mu_{32} + \mu_{33} \hfill \\ \end{gathered} \right). \end{gathered} $$

(12)

The COA method is one of the simplest but most effective methods and has a wide range of research applications. The COA method is used to defuzzify R-numbers to reduce the computational complexity of R-numbers.

Definition 7

[10]. Let us assume that two R-numbers $R( {\tilde{B}} )$ and $R( {{C}} )$ are R-numbers, then the distance of them is defined as

$$ \begin{gathered} d\left( {R( {\tilde{B}} ),R\left( {\tilde{C}} \right)} \right) = \hfill \\ \quad \sqrt {\frac{1}{3}\left( \begin{gathered} \left( {\sqrt {\frac{1}{3}\left( {\left( {\mu_{{11\tilde{B}}} - \mu_{{11\tilde{C}}} } \right)^{2} + \left( {\mu_{{12\tilde{B}}} - \mu_{{12\tilde{B}}} } \right)^{2} + \left( {\mu_{{13\tilde{B}}} - \mu_{{13\tilde{C}}} } \right)^{2} } \right)} } \right) + \hfill \\ \left( {\sqrt {\frac{1}{3}\left( {\left( {\mu_{{21\tilde{B}}} - \mu_{{21\tilde{C}}} } \right)^{2} + \left( {\mu_{{22\tilde{B}}} - \mu_{{22\tilde{C}}} } \right)^{2} + \left( {\mu_{{23\tilde{B}}} - \mu_{{23\tilde{C}}} } \right)^{2} } \right)} } \right) + \hfill \\ \left( {\sqrt {\frac{1}{3}\left( {\left( {\mu_{{31\tilde{B}}} - \mu_{{31\tilde{C}}} } \right)^{2} + \left( {\mu_{{32\tilde{B}}} - \mu_{{32\tilde{C}}} } \right)^{2} + \left( {\mu_{{33\tilde{B}}} - \mu_{{33\tilde{C}}} } \right)^{2} } \right)} } \right) \hfill \\ \end{gathered} \right)} . \hfill \\ \end{gathered} $$

(13)

Einstein operations

Definition 8

[24]. Set theoretic operators are an important part of the development of fuzzy set theory. The concepts of t-norm and t-conform are contained by all types of specific operators. The Einstein product $\otimes_{\varepsilon }$ is a t-norm and Einstein sum $\oplus_{\varepsilon }$ is a t-conorm, where

$$ a \oplus_{\varepsilon } b = \frac{a + b}{{1 + a \cdot b}}, \quad \forall \left( {a,b} \right) \in \left[ {0,1} \right]^{2} . $$

(14)

$$ a \otimes_{\varepsilon } b = \frac{a \cdot b}{{1 + \left( {1 - a} \right) \cdot \left( {1 - b} \right)}}, \quad \forall \left( {a,b} \right) \in \left[ {0,1} \right]^{2} . $$

(15)

The proposed R-numbers DMAGDM framework

The decision-making process in real life is complex and dynamic. Therefore, the decision-making information is also dynamic and varies in different periods. In addition, alternatives and attributes may increase or decrease in different periods. This paper proposes an R-numbers-based DMAGDM to address these issues. The method consists of six stages. In the first stage, decision information is transformed into R-numbers. In the second stage, we study the aggregation operator of R-numbers, construct RNEWA operator and RNEWG operator, and study their related properties. The third stage focuses on determining expert weights. In the fourth stage, based on the RNEWA operator and expert weights obtained in the third stage, the MEREC method is constructed in the R-numbers environment to determine the attribute weights. In the fifth stage, the static ratings of alternatives in each period are calculated using the CoCoSo method based on the RNEWA operator and RNEWG operator. In the sixth stage, the dynamic ratings of alternatives are calculated according to the calculation method of dynamic ratings, and the alternatives are ranked according to the dynamic ratings to obtain the optimal solution. The DMAGDM framework, consisting of the above six stages, is shown in Fig. 1.

Fig. 1

R-numbers DMAGDM framework

Description of the R-numbers DMAGDM problem

The proposed method should take into account the following considerations:

• (1) Alternatives and attributes may change over time, with alternatives potentially being deemed unavailable and removed, or new alternatives being considered and added, and the same holds true for attributes.

• (2) Experts provide evaluation information for alternatives under multiple attributes in the form of R-numbers in different periods.

• (3) The weights of experts and attributes are unknown.

• (4) The final decision of each period must consider the feedback of previous decisions.

The notations used in the proposed R-numbers DMAGDM method are as follows.

• (1) $P = \left\{ {1,2,3,...} \right\}$: the set of discrete periods, where $p \in P$ is the $pth$ period.

• (2) $E^{p} = \left\{ {e_{1}^{p} ,e_{2}^{p} ,...,e_{r}^{p} } \right\}$: the set of experts in the period $p$, where $e_{f}^{p} \left( {f = 1,2,...,r} \right)$ indicates the $fth$ expert in the period $p$.

• (3) $A^{p} = \left\{ {A_{1}^{p} ,A_{2}^{p} ,...,A_{m}^{p} } \right\}$: the set of alternatives in the period $p$, where $A_{i}^{p} \left( {i = 1,2,...,m} \right)$ is the $ith$ alternative, $m$ is the number of alternatives in the period $p$.

• (4) $A = A^{1} \cup A^{2} \cup ... \cup A^{p} = \left\{ {a_{1} ,a_{2} ,...,a_{g} } \right\}$: the set of alternatives in all periods, where $a_{g} \left( {g = 1,2,...,s} \right)$ is the $gth$ alternative, $s$ is the number of alternatives in all periods.

• (5) $C^{p} = \left\{ {C_{1}^{p} ,C_{2}^{p} ,...,C_{n}^{p} } \right\}$: the set of attributes in the period $p$, where $C_{j}^{p} \left( {j = 1,2,...,n} \right)$ is the $jth$ attribute, $n$ is the number of attributes in the period $p$.

• (6) $C = C^{1} \cup C^{2} \cup ... \cup C^{p} = \left\{ {c_{1} ,c_{2} ,...,c_{h} } \right\}$: the set of attributes in all periods, where $c_{h} \left( {h = 1,2,...,t} \right)$ is the $hth$ attribute, $t$ is the number of attributes in all periods.

• (7) ${\varvec{w}}^{{{\varvec{e}}_{{\varvec{f}}}^{{\varvec{p}}} }} = \left( {w_{ij}^{{e_{f}^{p} }} } \right)_{m \times n}$: the expert weight in the period $p$, where $w_{ij}^{{e_{f}^{p} }}$ denotes the expert weight about the $ith$ alternative of $jth$ attribute., $0 \le w_{ij}^{{e_{f}^{p} }} \le 1$, and $\sum\nolimits_{f = 1}^{r} {w_{ij}^{{e_{f}^{p} }} } = 1$.

• (8) $\user2{\varpi }^{{\mathbf{p}}} = \left( {\varpi_{1}^{p} ,\varpi_{2}^{p} ,...,\varpi_{n}^{p} } \right)^{T}$: the weight vector of attributes in the period $p$, where $\varpi_{j}^{p} \left( {j = 1,2,...,n} \right)$ is the weight of $jth$ attribute in the period $p$, $0 \le \varpi_{j} \le 1$, and $\sum\nolimits_{j = 1}^{n} {\varpi_{j} } = 1$.

• (9) $\tilde{\user2{X}}^{{{\varvec{e}}_{{\varvec{f}}}^{{\varvec{p}}} }} = \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right)_{m \times n}$: the initial decision matrix in the form of fuzzy numbers from $fth$ expert in the period $p$, where $\tilde{x}_{ij}^{{e_{f}^{p} }}$ denotes the evaluation information about the $ith$ alternative of $jth$ attribute.

• (10) ${\varvec{X}}^{{{\varvec{e}}_{{\varvec{f}}}^{{\varvec{p}}} }} = \left( {x_{ij}^{{e_{f}^{p} }} } \right)_{m \times n}$: R-numbers matrix converted from the initial decision matrix.

The framework of the R-numbers DMAGDM problem

The following section describes the framework of R-numbers DMAGDM, which consists of six phases, as shown below.

Stage 1. Convert initial decision matrix to R-numbers matrix.

This study investigates DMAGDM in the R-numbers environment, whereby the decision information is transformed into the form of R-numbers in this stage.

Stage 2. Propose the Einstein aggregation operators of R-numbers.

The objective of this stage in the MAGDM environment is to aggregate evaluation information from multiple experts and multiple attributes. As a result, this stage aims to propose an aggregation operator in the form of R-numbers to aggregate evaluations from multiple experts. The conventional aggregation operators are based on ordinary algebraic sum and product operations. To supplement the research on aggregation operators, the RNEWA and RNEWG operators, based on Einstein sum and Einstein product operations, are proposed and their relevant properties are studied.

Stage 3. Construct the expert weight determination model.

Due to the different characteristics of experts, they may have biases when evaluating alternatives. To reduce the impact of these biases on the evaluation, different weights should be given to different experts. The weight is larger when the expert's evaluation is close to the average value and smaller when it is far from the average value. Thus, the goal of this stage is to establish a model for determining expert weights.

Stage 4. Construct MEREC model for determining the attribute weights.

In some practical problems, experts may not be able to provide accurate information on the preferences of alternative solutions due to the influence of factors such as the objective environment, professional level, and time. Thus, in this stage, we adopt an objective attribute weight determination method-MEREC, which derives attribute weights from the evaluation information itself. To determine the attribute weights, the MEREC method employs the removal effect on alternatives. Attributes with higher performance effects are assigned larger weights, while attributes with lower performance effects are assigned smaller weights.

Stage 5. Construct the CoCoSo model to calculate the static rating of alternatives in each period.

In this stage, the CoCoSo method is used to obtain the static rating of alternatives for each period. The objective of this stage is to extend the calculation method of static ratings in the R-numbers environment based on the general steps of the CoCoSo method. In the R-numbers CoCoSo method, the RNEWA and RNEWG operators are used to replace the simple weighted sum and exponential weighted product models in the original CoCoSo method, making it more flexible.

Stage 6. Construct the model to calculate the dynamic ratings of alternatives.

The priority of alternatives is affected not only by the information of the current period, but also by the information of the previous period. Therefore, the objective of this stage is to obtain the dynamic rating of each alternative based on its static rating and the calculation method of dynamic rating. A new dynamic rating calculation equation is proposed, which can adjust the influence of past information more flexibly based on the actual situation.

The procedure of the proposed R-numbers DMAGDM method

According to the R-numbers DMAGDM framework presented in the previous section, the procedure of the proposed method is as follows.

The model for transforming initial matrix to R-numbers matrix

Step 1. Construct the initial matrix. The initial matrix obtained is as follows:

$$ \begin{gathered} \tilde{\user2{X}}^{{{\varvec{e}}_{{\varvec{f}}}^{{\varvec{p}}} }} = \left[ {\tilde{x}_{ij}^{{e_{f}^{p} }} \, } \right]_{m \times n} , \hfill \\ p = 1,2,3,...,f = 1,2,...,r, \hfill \\ i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

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Step 2. Transform initial matrix to R-numbers matrix.

In the DMAGDM framework proposed in this paper, the values of risk parameters are given by experts, namely pessimistic risk $\tilde{r}^{ - }$, optimistic risk $\tilde{r}^{ + }$, pessimistic acceptable risk $\widetilde{AR}^{ - }$, optimistic acceptable risk $\widetilde{AR}^{ + }$, pessimistic risk perception $\widetilde{RP}^{ - }$, and optimistic risk perception $\widetilde{RP}^{ + }$. According to the R-numbers definition, convert the initial decision matrix to the form of R-numbers

$$ \begin{gathered} {\varvec{X}}^{{{\varvec{e}}_{{\varvec{f}}}^{{\varvec{p}}} }} = \left[ {R\left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right)} \right]_{m \times n} = \left[ {x_{ij}^{{e_{f}^{p} }} } \right]_{m \times n} , \hfill \\ p = 1,2,3,...,f = 1,2,...,r, \hfill \\ i = 1,2,...,m,j = 1,2,...,n, \hfill \\ \end{gathered} $$

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where $x_{ij}^{{e_{f}^{p} }} \, = R\bigg(\tilde{x}_{ij}^{{e_{f}^{p} }}\bigg) = \bigg(\bigg(x_{ij11}^{{e_{f}^{p} }} ,x_{ij12}^{{e_{f}^{p} }} ,x_{ij13}^{{e_{f}^{p} }} \bigg), \bigg(x_{ij21}^{{e_{f}^{p} }} ,x_{ij22}^{{e_{f}^{p} }} ,x_{ij23}^{{e_{f}^{p} }}\bigg),\bigg(x_{ij31}^{{e_{f}^{p} }} ,x_{ij32}^{{e_{f}^{p} }} ,x_{ij33}^{{e_{f}^{p} }} \bigg) \bigg).$ The beneficial and non-beneficial modes are as follows, respectively:$R_{b} \Big(\tilde{x}_{ij}^{{e_{f}^{p} }}\Big) = \Big(R_{1b} \Big(\tilde{x}_{ij}^{{e_{f}^{p} }} \Big),R_{2b} \Big(\tilde{x}_{ij}^{{e_{f}^{p} }} \Big),R_{3b} \Big(\tilde{x}_{ij}^{{e_{f}^{p} }} \Big) \Big),$ where

$$ \left\{ \begin{gathered} R_{1b} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \max \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} \otimes \left( {1{\mathbf{ \ominus }}\min \left( {\frac{{\tilde{r}^{ - } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ - } }},\tau } \right) \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ - } } \right)} \right),l_{{\tilde{B}}} } \right), \hfill \\ R_{2b} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \tilde{x}_{ij}^{{e_{f}^{p} }} , \hfill \\ R_{3b} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \min \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} \otimes \left( {1 \oplus \frac{{\tilde{r}^{ + } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ + } }} \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ + } } \right)} \right),u_{{\tilde{B}}} } \right), \quad \hfill \\ 0 < \tilde{r}^{ - } < 1,\tilde{r}^{ + } > 0. \hfill \\ \end{gathered} \right., $$

and $R_{c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \left( {R_{1c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right),R_{2c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right),R_{3c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right)} \right),$ where

$$ \left\{ \begin{gathered} R_{1c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \max \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} \otimes \left( {1{\mathbf{ \ominus }}\min \left( {\frac{{\tilde{r}^{ + } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ + } }},\tau } \right) \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ + } } \right)} \right),l_{{\tilde{B}}} } \right), \hfill \\ R_{2c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \tilde{x}_{ij}^{{e_{f}^{p} }} , \hfill \\ R_{3c} \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} } \right) = \min \left( {\tilde{x}_{ij}^{{e_{f}^{p} }} \otimes \left( {1 \oplus \frac{{\tilde{r}^{ - } }}{{1{\mathbf{ \ominus }}\widetilde{RP}^{ - } }} \otimes \left( {1{\mathbf{ \ominus }}\widetilde{AR}^{ - } } \right)} \right),u_{{\tilde{B}}} } \right), \hfill \\ \tilde{r}^{ - } > 1,\quad 0 < \tilde{r}^{ + } < 1. \hfill \\ \end{gathered} \right. $$

The Einstein aggregation operators of R-numbers

Definition 9

Let us assume that $R ( {\tilde{B}} )$ and $R ( {\tilde{C}} )$ are two R-numbers, then the operations between them based on the Einstein t-norms and t-conorms are defined as

$$ \begin{gathered} R( {\tilde{B}} ) \oplus_{\varepsilon } R\left( {\tilde{C}} \right) \hfill \\ = \left( \begin{gathered} \left( {\frac{{\mu_{{11\tilde{B}}} + \mu_{{11\tilde{C}}} }}{{1 + \mu_{{11\tilde{B}}} \cdot \mu_{{11\tilde{C}}} }},\frac{{\mu_{{12\tilde{B}}} + \mu_{{12\tilde{C}}} }}{{1 + \mu_{{12\tilde{B}}} \cdot \mu_{{12\tilde{C}}} }},\frac{{\mu_{{13\tilde{B}}} + \mu_{{13\tilde{C}}} }}{{1 + \mu_{{13\tilde{B}}} \cdot \mu_{{13\tilde{C}}} }}} \right), \hfill \\ \left( {\frac{{\mu_{{21\tilde{B}}} + \mu_{{21\tilde{C}}} }}{{1 + \mu_{{21\tilde{B}}} \cdot \mu_{{21\tilde{C}}} }},\frac{{\mu_{{22\tilde{B}}} + \mu_{{22\tilde{C}}} }}{{1 + \mu_{{22\tilde{B}}} \cdot \mu_{{22\tilde{C}}} }},\frac{{\mu_{{23\tilde{B}}} + \mu_{{23\tilde{C}}} }}{{1 + \mu_{{23\tilde{B}}} \cdot \mu_{{23\tilde{C}}} }}} \right), \hfill \\ \left( {\frac{{\mu_{{31\tilde{B}}} + \mu_{{31\tilde{C}}} }}{{1 + \mu_{{31\tilde{B}}} \cdot \mu_{{31\tilde{C}}} }},\frac{{\mu_{{32\tilde{B}}} + \mu_{{32\tilde{C}}} }}{{1 + \mu_{{32\tilde{B}}} \cdot \mu_{{32\tilde{C}}} }},\frac{{\mu_{{33\tilde{B}}} + \mu_{{33\tilde{C}}} }}{{1 + \mu_{{33\tilde{B}}} \cdot \mu_{{33\tilde{C}}} }}} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

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$$ \begin{gathered} R( {\tilde{B}} ) \otimes_{\varepsilon } R\left( {\tilde{C}} \right) \hfill \\ = \left( \begin{gathered} \left( \begin{gathered} \frac{{\mu_{{11\tilde{B}}} \mu_{{11\tilde{C}}} }}{{1 + \left( {1 - \mu_{{11\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{11\tilde{C}}} } \right)}},\frac{{\mu_{{12\tilde{B}}} \mu_{{12\tilde{C}}} }}{{1 + \left( {1 - \mu_{{12\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{12\tilde{C}}} } \right)}}, \hfill \\ \frac{{\mu_{{13\tilde{B}}} \mu_{{13\tilde{C}}} }}{{1 + \left( {1 - \mu_{{13\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{13\tilde{C}}} } \right)}} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{\mu_{{21\tilde{B}}} \mu_{{21\tilde{C}}} }}{{1 + \left( {1 - \mu_{{21\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{21\tilde{C}}} } \right)}},\frac{{\mu_{{22\tilde{B}}} \mu_{{22\tilde{C}}} }}{{1 + \left( {1 - \mu_{{22\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{22\tilde{C}}} } \right)}}, \hfill \\ \frac{{\mu_{{23\tilde{B}}} \mu_{{23\tilde{C}}} }}{{1 + \left( {1 - \mu_{{23\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{23\tilde{C}}} } \right)}} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{\mu_{{31\tilde{B}}} \mu_{{31\tilde{C}}} }}{{1 + \left( {1 - \mu_{{31\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{31\tilde{C}}} } \right)}},\frac{{\mu_{{32\tilde{B}}} \mu_{{32\tilde{C}}} }}{{1 + \left( {1 - \mu_{{32\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{32\tilde{C}}} } \right)}}, \hfill \\ \frac{{\mu_{{33\tilde{B}}} \mu_{{33\tilde{C}}} }}{{1 + \left( {1 - \mu_{{33\tilde{B}}} } \right) \cdot \left( {1 - \mu_{{33\tilde{C}}} } \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

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$$ \begin{gathered} \lambda \cdot_{\varepsilon } R( {\tilde{B}} ) \hfill \\ = \left( \begin{gathered} \left( \begin{gathered} \frac{{\left( {1 + \mu_{{11\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{11\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{11\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{11\tilde{B}}} } \right)^{\lambda } }},\frac{{\left( {1 + \mu_{{12\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{12\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{12\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{12\tilde{B}}} } \right)^{\lambda } }}, \hfill \\ \frac{{\left( {1 + \mu_{{13\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{13\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{13\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{13\tilde{B}}} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{\left( {1 + \mu_{{21\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{21\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{21\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{21\tilde{B}}} } \right)^{\lambda } }},\frac{{\left( {1 + \mu_{{22\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{22\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{22\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{22\tilde{B}}} } \right)^{\lambda } }}, \hfill \\ \frac{{\left( {1 + \mu_{{23\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{23\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{23\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{23\tilde{B}}} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{\left( {1 + \mu_{{31\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{31\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{31\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{31\tilde{B}}} } \right)^{\lambda } }},\frac{{\left( {1 + \mu_{{32\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{32\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{32\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{32\tilde{B}}} } \right)^{\lambda } }}, \hfill \\ \frac{{\left( {1 + \mu_{{33\tilde{B}}} } \right)^{\lambda } - \left( {1 - \mu_{{33\tilde{B}}} } \right)^{\lambda } }}{{\left( {1 + \mu_{{33\tilde{B}}} } \right)^{\lambda } + \left( {1 - \mu_{{33\tilde{B}}} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right), \hfill \\ \lambda > 0, \hfill \\ \end{gathered} $$

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$$ \begin{gathered} R( {\tilde{B}} )^{{ \wedge_{\varepsilon } \lambda }} \hfill \\\qquad = \left( \begin{gathered} \left( \begin{gathered} \frac{{2\left( {\mu_{{11\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{11\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{11\tilde{B}}} } \right)^{\lambda } }},\frac{{2\left( {\mu_{{12\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{11\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{11\tilde{B}}} } \right)^{\lambda } }}, \hfill \\ \frac{{2\left( {\mu_{{13\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{13\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{13\tilde{B}}} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{2\left( {\mu_{{21\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{21\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{21\tilde{B}}} } \right)^{\lambda } }},\frac{{2\left( {\mu_{{22\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{22\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{22\tilde{B}}} } \right)^{\lambda } }}, \hfill \\ \frac{{2\left( {\mu_{{23\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{23\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{23\tilde{B}}} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{2\left( {\mu_{{31\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{31\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{31\tilde{B}}} } \right)^{\lambda } }},\frac{{2\left( {\mu_{{32\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{32\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{32\tilde{B}}} } \right)^{\lambda } }}, \hfill \\ \frac{{2\left( {\mu_{{33\tilde{B}}} } \right)^{\lambda } }}{{\left( {2 - \mu_{{33\tilde{B}}} } \right)^{\lambda } + \left( {\mu_{{33\tilde{B}}} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right), \hfill \\ \lambda > 0. \hfill \\ \end{gathered} $$

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Example 1

Let $R ( {\tilde{B}} ) = \left( {\left( {0.1,0.2,0.3} \right),\left( {0.2,0.3,0.4} \right)}\right.,$ $\left.{\left( {0.3,0.4,0.5} \right)} \right)$, $R ( {\tilde{C}}) = \left( {\left( {0.2,0.3,0.4} \right),\left( {0.3,0.4,0.5} \right)}\right.,$ $\left.{\left( {0.4,0.5,0.6} \right)} \right)$, be two R-numbers, and suppose $\lambda = 3$. Then using the operation rules in Definition 9, we can get

$$ \begin{gathered} R ( {\tilde{B}} ) \oplus_{\varepsilon } R\left( {\tilde{C}} \right) \hfill \\ = \left( \begin{gathered} \left( {\frac{0.1 + 0.2}{{1 + 0.1 \times 0.2}},\frac{0.2 + 0.3}{{1 + 0.2 \times 0.3}},\frac{0.3 + 0.4}{{1 + 0.3 \times 0.4}}} \right), \hfill \\ \left( {\frac{0.2 + 0.3}{{1 + 0.2 \times 0.3}},\frac{0.3 + 0.4}{{1 + 0.3 \times 0.4}},\frac{0.4 + 0.5}{{1 + 0.4 \times 0.5}}} \right), \hfill \\ \left( {\frac{0.3 + 0.4}{{1 + 0.3 \times 0.4}},\frac{0.4 + 0.5}{{1 + 0.4 \times 0.5}},\frac{0.5 + 0.6}{{1 + 0.5 \times 0.6}}} \right) \hfill \\ \end{gathered} \right) \hfill \\ = \left( \begin{gathered} \left( {0.2941,0.4717,0.6250} \right),\left( {0.4717,0.6250,0.7500} \right), \hfill \\ \left( {0.6250,0.7500,0.8462} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} R ( {\tilde{B}} ) \otimes_{\varepsilon } R\left( {\tilde{C}} \right) \hfill \\ = \left( \begin{gathered} \left( \begin{gathered} \frac{0.1 \times 0.2}{{1 + \left( {1 - 0.1} \right) \cdot \left( {1 - 0.2} \right)}},\frac{0.2 \times 0.3}{{1 + \left( {1 - 0.2} \right) \cdot \left( {1 - 0.3} \right)}}, \hfill \\ \frac{0.3 \times 0.4}{{1 + \left( {1 - 0.3} \right) \cdot \left( {1 - 0.4} \right)}} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{0.2 \times 0.3}{{1 + \left( {1 - 0.2} \right) \cdot \left( {1 - 0.3} \right)}},\frac{0.3 \times 0.4}{{1 + \left( {1 - 0.3} \right) \cdot \left( {1 - 0.4} \right)}}, \hfill \\ \frac{0.4 \times 0.5}{{1 + \left( {1 - 0.4} \right) \cdot \left( {1 - 0.5} \right)}} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{0.3 \times 0.4}{{1 + \left( {1 - 0.3} \right) \cdot \left( {1 - 0.4} \right)}},\frac{0.4 \times 0.5}{{1 + \left( {1 - 0.4} \right) \cdot \left( {1 - 0.5} \right)}}, \hfill \\ \frac{0.5 \times 0.6}{{1 + \left( {1 - 0.5} \right) \cdot \left( {1 - 0.6} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right) \hfill \\ = \left( \begin{gathered} \left( {0.0116,0.0385,0.0845} \right),\left( {0.0385,0.0845,0.1538} \right), \hfill \\ \left( {0.0845,0.1538,0.2500} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} 3 \cdot_{\varepsilon } R\left( {\tilde{B}} \right) \hfill \\ = \left( \begin{gathered} \left( \begin{gathered} \frac{{\left( {1 + 0.1} \right)^{3} - \left( {1 - 0.1} \right)^{3} }}{{\left( {1 + 0.1} \right)^{3} + \left( {1 - 0.1} \right)^{3} }},\frac{{\left( {1 + 0.2} \right)^{3} - \left( {1 - 0.2} \right)^{3} }}{{\left( {1 + 0.2} \right)^{3} + \left( {1 - 0.2} \right)^{3} }}, \hfill \\ \frac{{\left( {1 + 0.3} \right)^{3} - \left( {1 - 0.3} \right)^{3} }}{{\left( {1 + 0.3} \right)^{3} + \left( {1 - 0.3} \right)^{3} }} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{\left( {1 + 0.2} \right)^{3} - \left( {1 - 0.2} \right)^{3} }}{{\left( {1 + 0.2} \right)^{3} + \left( {1 - 0.2} \right)^{3} }},\frac{{\left( {1 + 0.3} \right)^{3} - \left( {1 - 0.3} \right)^{3} }}{{\left( {1 + 0.3} \right)^{3} + \left( {1 - 0.3} \right)^{3} }}, \hfill \\ \frac{{\left( {1 + 0.4} \right)^{3} - \left( {1 - 0.4} \right)^{3} }}{{\left( {1 + 0.4} \right)^{3} + \left( {1 - 0.4} \right)^{3} }} \hfill \\ \end{gathered} \right), \hfill \\ \left( \begin{gathered} \frac{{\left( {1 + 0.3} \right)^{\lambda } - \left( {1 - 0.3} \right)^{\lambda } }}{{\left( {1 + 0.3} \right)^{\lambda } + \left( {1 - 0.3} \right)^{\lambda } }},\frac{{\left( {1 + 0.4} \right)^{3} - \left( {1 - 0.4} \right)^{3} }}{{\left( {1 + 0.4} \right)^{3} + \left( {1 - 0.4} \right)^{3} }}, \hfill \\ \frac{{\left( {1 + 0.5} \right)^{3} - \left( {1 - 0.5} \right)^{3} }}{{\left( {1 + 0.5} \right)^{3} + \left( {1 - 0.5} \right)^{3} }} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right) \hfill \\ = \left( \begin{gathered} \left( {0.2922,0.5429,0.7299} \right),\left( {0.5429,0.7299,0.8541} \right), \hfill \\ \left( {0.7299,0.8541,0.9286} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} R\left( {\tilde{B}} \right)^{{ \wedge_{\varepsilon } 3}} \hfill \\ = \left( \begin{gathered} \left( {\frac{{2 \times 0.1^{3} }}{{\left( {2 - 0.1} \right)^{3} + 0.1^{3} }},\frac{{2 \times 0.2^{3} }}{{\left( {2 - 0.2} \right)^{3} + 0.2^{3} }},\frac{{2 \times 0.3^{3} }}{{\left( {2 - 0.3} \right)^{3} + 0.3^{3} }}} \right), \hfill \\ \left( {\frac{{2 \times 0.2^{3} }}{{\left( {2 - 0.2} \right)^{3} + 0.2^{3} }},\frac{{2 \times 0.3^{3} }}{{\left( {2 - 0.3} \right)^{3} + 0.3^{3} }},\frac{{2 \times 0.4^{3} }}{{\left( {2 - 0.4} \right)^{3} + 0.4^{3} }}} \right), \hfill \\ \left( {\frac{{2 \times 0.3^{3} }}{{\left( {2 - 0.3} \right)^{3} + 0.3^{3} }},\frac{{2 \times 0.4^{3} }}{{\left( {2 - 0.4} \right)^{3} + 0.4^{3} }},\frac{{2 \times 0.5^{3} }}{{\left( {2 - 0.5} \right)^{3} + 0.5^{3} }}} \right) \hfill \\ \end{gathered} \right) \hfill \\ = \left( \begin{gathered} \left( {0.0003,0.0027,0.0109} \right),\left( {0.0027,0.0109,0.0308} \right), \hfill \\ \left( {0.0109,0.0308,0.0714} \right) \hfill \\ \end{gathered} \right). \hfill \\ \end{gathered} $$

Definition 10

Let $R ( {\tilde{B}_{i} } )\,\,\,\left( {i = 1,2,...,m} \right)$ be a collection of R-numbers, ${\varvec{w}} = \left( {w_{1} ,w_{2} ,...,w_{m} } \right)^{T}$ be the weight vector of $R ( {\tilde{B}_{i} })$, satisfying $w_{i} \in \left[ {0,1} \right]$ and $\sum\nolimits_{i = 1}^{m} {w_{i} } = 1$. Then, the R-numbers Einstein weighted averaging (RNEWA) operator is defined as

$$ RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) = \mathop { \oplus_{\varepsilon } }\limits_{i = 1}^{m} w_{i} R\left( {\tilde{B}_{i} } \right). $$

(22)

Theorem 1

Let $R ( {\tilde{B}_{i} } )\,\,\,\left( {i = 1,2,...,m} \right)$ be a collection of R-numbers, ${\varvec{w}} = \left( {w_{1} ,w_{2} ,...,w_{m} } \right)^{T}$ be the weight vector of $R ( {\tilde{B}_{i} })$, satisfying $w_{i} \in \left[ {0,1} \right]$ and $\sum\nolimits_{i = 1}^{m} {w_{i} } = 1$. Then, the aggregated value by the RNEWA operator is still an R-number and

$$ \begin{gathered} RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) = \mathop { \oplus_{\varepsilon } }\limits_{i = 1}^{m} w_{i} R\left( {\tilde{B}_{i} } \right) \hfill \\\qquad = \left( \begin{gathered} \left( {\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{11i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{11i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{11i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{11i} } \right)^{{w_{i} }} } }},\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{12i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{12i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{12i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{12i} } \right)^{{w_{i} }} } }},\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{13i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{13i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{13i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{13i} } \right)^{{w_{i} }} } }}} \right), \hfill \\ \left( {\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{21i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{21i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{21i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{21i} } \right)^{{w_{i} }} } }},\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{22i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{22i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{22i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{22i} } \right)^{{w_{i} }} } }},\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{23i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{23i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{23i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{23i} } \right)^{{w_{i} }} } }}} \right), \hfill \\ \left( {\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{31i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{31i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{31i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{31i} } \right)^{{w_{i} }} } }},\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{32i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{32i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{32i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{32i} } \right)^{{w_{i} }} } }},\frac{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{33i} } \right)^{{w_{i} }} } - \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{33i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {1 + \mu_{33i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {1 - \mu_{33i} } \right)^{{w_{i} }} } }}} \right) \hfill \\ \end{gathered} \right). \hfill \\ \end{gathered} $$

(23)

Theorem 2

Let $R ( {\tilde{B}_{i} } )\,\,\,\left( {i = 1,2,...,m} \right)$ be a collection of R-numbers. And $R ( {\tilde{B}_{i} })\,\,\,\left( {i = 1,2,...,m} \right) = R( {\tilde{B}} ) =$ $\left( {\left( {\mu_{11} ,\mu_{12} ,\mu_{13} } \right),\left( {\mu_{21} ,\mu_{22} ,\mu_{23} } \right),\left( {\mu_{31} ,\mu_{32} ,\mu_{33} } \right)} \right)$, then

$$ RNEWA\left( {R( {\tilde{B}_{1} } ),R( {\tilde{B}_{2} } ),...,R( {\tilde{B}_{m} } )} \right) = R( {\tilde{B}} ) $$

(24)

This property is called Idempotency Property.

Proof

Since $R( {\tilde{B}_{i} })\,\,\,\left( {i = 1,2,...,m} \right) = R( {\tilde{B}} ) =\left( {\left( {\mu_{11} ,\mu_{12} ,}\right.}\right.$ $\left.{\left.{\mu_{13} } \right), \left( {\mu_{21} ,\mu_{22} ,\mu_{23} } \right),\left( {\mu_{31} ,\mu_{32} ,\mu_{33} } \right)} \right)$, then we have

$$ \begin{gathered} RNEWA\left( {R\left( {\tilde{B}} \right),R\left( {\tilde{B}} \right),,...,R\left( {\tilde{B}} \right)} \right) \hfill \\ = \left( \begin{gathered} \left( {\frac{{\left( {\left( {1 + \mu_{11} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{11} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{11} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{11} } \right)^{1/m} } \right)^{m} }},\frac{{\left( {\left( {1 + \mu_{12} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{12} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{12} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{12} } \right)^{1/m} } \right)^{m} }},\frac{{\left( {\left( {1 + \mu_{13} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{13} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{13} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{13} } \right)^{1/m} } \right)^{m} }}} \right), \hfill \\ \left( {\frac{{\left( {\left( {1 + \mu_{21} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{21} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{21} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{21} } \right)^{1/m} } \right)^{m} }},\frac{{\left( {\left( {1 + \mu_{22} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{22} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{22} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{22} } \right)^{1/m} } \right)^{m} }},\frac{{\left( {\left( {1 + \mu_{23} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{23} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{23} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{23} } \right)^{1/m} } \right)^{m} }}} \right), \hfill \\ \left( {\frac{{\left( {\left( {1 + \mu_{31} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{31} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{31} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{31} } \right)^{1/m} } \right)^{m} }},\frac{{\left( {\left( {1 + \mu_{32} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{32} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{32} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{32} } \right)^{1/m} } \right)^{m} }},\frac{{\left( {\left( {1 + \mu_{33} } \right)^{1/m} } \right)^{m} - \left( {\left( {1 - \mu_{33} } \right)^{1/m} } \right)^{m} }}{{\left( {\left( {1 + \mu_{33} } \right)^{1/m} } \right)^{m} + \left( {\left( {1 - \mu_{33} } \right)^{1/m} } \right)^{m} }}} \right) \hfill \\ \end{gathered} \right). \hfill \\ = \left( \begin{gathered} \left( {\frac{{\left( {1 + \mu_{11} } \right) - \left( {1 - \mu_{11} } \right)}}{{\left( {1 + \mu_{11} } \right) + \left( {1 - \mu_{11} } \right)}},\frac{{\left( {1 + \mu_{12} } \right) - \left( {1 - \mu_{12} } \right)}}{{\left( {1 + \mu_{12} } \right) + \left( {1 - \mu_{12} } \right)}},\frac{{\left( {1 + \mu_{13} } \right) - \left( {1 - \mu_{13} } \right)}}{{\left( {1 + \mu_{13} } \right) + \left( {1 - \mu_{13} } \right)}}} \right),\left( {\frac{{\left( {1 + \mu_{21} } \right) - \left( {1 - \mu_{21} } \right)}}{{\left( {1 + \mu_{21} } \right) + \left( {1 - \mu_{21} } \right)}},\frac{{\left( {1 + \mu_{22} } \right) - \left( {1 - \mu_{22} } \right)}}{{\left( {1 + \mu_{22} } \right) + \left( {1 - \mu_{22} } \right)}},\frac{{\left( {1 + \mu_{23} } \right) - \left( {1 - \mu_{23} } \right)}}{{\left( {1 + \mu_{23} } \right) + \left( {1 - \mu_{23} } \right)}}} \right), \hfill \\ \left( {\frac{{\left( {1 + \mu_{31} } \right) - \left( {1 - \mu_{31} } \right)}}{{\left( {1 + \mu_{31} } \right) + \left( {1 - \mu_{31} } \right)}},\frac{{\left( {1 + \mu_{32} } \right) - \left( {1 - \mu_{32} } \right)}}{{\left( {1 + \mu_{32} } \right) + \left( {1 - \mu_{32} } \right)}},\frac{{\left( {1 + \mu_{33} } \right) - \left( {1 - \mu_{33} } \right)}}{{\left( {1 + \mu_{33} } \right) + \left( {1 - \mu_{33} } \right)}}} \right) \hfill \\ \end{gathered} \right) \hfill \\ = \left( {\left( {\mu_{11} ,\mu_{12} ,\mu_{13} } \right),\left( {\mu_{21} ,\mu_{22} ,\mu_{23} } \right),\left( {\mu_{31} ,\mu_{32} ,\mu_{33} } \right)} \right) = R\left( {\tilde{B}} \right). \hfill \\ \end{gathered} $$

Therefore, $RNEWAA ( {R( {\tilde{B}_{1} } ),R ( {\tilde{B}_{2} } ),...,R ( {\tilde{B}_{m} } )} ) = R( {\tilde{B}})$.

Theorem 3

For $m$ R-numbers $R ( {\tilde{B}_{i} } )\,\,\,\left( {i = 1,2,...,m} \right)$, and $R {\tilde{B}_{i} })^{ - } = \left( \begin{gathered} \left( {\mathop {\min }\nolimits_{i} \left\{ {\mu_{11i} } \right\},\mathop {\min }\nolimits_{i} \left\{ {\mu_{12i} } \right\},\mathop {\min }\nolimits_{i} \left\{ {\mu_{13i} } \right\}} \right), \hfill \\ \left( {\mathop {\min }\nolimits_{i} \left\{ {\mu_{21i} } \right\},\mathop {\min }\nolimits_{i} \left\{ {\mu_{22i} } \right\},\mathop {\min }\nolimits_{i} \left\{ {\mu_{23i} } \right\}} \right), \hfill \\ \left( {\mathop {\min }\nolimits_{i} \left\{ {\mu_{31i} } \right\},\mathop {\min }\nolimits_{i} \left\{ {\mu_{32i} } \right\},\mathop {\min }\nolimits_{i} \left\{ {\mu_{33i} } \right\}} \right) \hfill \\ \end{gathered} \right)$,

$R ( {\tilde{B}_{i} } )^{ + } = \left( \begin{gathered} \left( {\mathop {\max }\nolimits_{i} \left\{ {\mu_{11i} } \right\},\mathop {\max }\nolimits_{i} \left\{ {\mu_{12i} } \right\},\mathop {\max }\nolimits_{i} \left\{ {\mu_{13i} } \right\}} \right), \hfill \\ \left( {\mathop {\max }\nolimits_{i} \left\{ {\mu_{21i} } \right\},\mathop {\max }\nolimits_{i} \left\{ {\mu_{22i} } \right\},\mathop {\max }\nolimits_{i} \left\{ {\mu_{23i} } \right\}} \right), \hfill \\ \left( {\mathop {\max }\nolimits_{i} \left\{ {\mu_{31i} } \right\},\mathop {\max }\nolimits_{i} \left\{ {\mu_{32i} } \right\},\mathop {\max }\nolimits_{i} \left\{ {\mu_{33i} } \right\}} \right) \hfill \\ \end{gathered} \right)$, we have

$$\begin{aligned}& R\left( \tilde{B}_{i}\right)^{ - } \le RNEWA\left( R\left( \tilde{B}_{1}\right),\right.\\ & \left. R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)\right) \le R\left( {\tilde{B}_{i} } \right)^{ + }.\end{aligned} $$

(25)

This property is called Boundedness Property.

Proof

Clearly, we can get $R ( {\tilde{B}_{i} } )^{ - } \le R ( {\tilde{B}_{i} } ) \le R ( {\tilde{B}_{i} } )^{ + }$. Thus, based on Theorems 1 and 2, we have

$$ \begin{gathered} RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) = R\left( {\tilde{B}_{i} } \right) \hfill \\ \qquad \ge RNEWA\left( {R\left( {\tilde{B}_{i} } \right)^{ - } ,R\left( {\tilde{B}_{i} } \right)^{ - } ,...,R\left( {\tilde{B}_{i} } \right)^{ - } } \right) = R\left( {\tilde{B}_{i} } \right)^{ - } , \hfill \\ RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) = R\left( {\tilde{B}_{i} } \right) \hfill \\ \qquad \le RNEWA\left( {R\left( {\tilde{B}_{i} } \right)^{ + } ,R\left( {\tilde{B}_{i} } \right)^{ + } ,...,R\left( {\tilde{B}_{i} } \right)^{ + } } \right) = R\left( {\tilde{B}_{i} } \right)^{ + } . \hfill \\ \end{gathered} $$

Theorem 4

Let $R\left( {\tilde{B}_{i} } \right)\,\,\left( {i = 1,2,...,m} \right)$ and $R ( {\tilde{B}_{i}^{\prime } } )$ $\left( {i = 1,2,...,m} \right)$ be two collection of R-numbers and ${\varvec{w}} = \left( {w_{1} ,w_{2} ,...,w_{m} } \right)^{T}$ be the weight vector of $R ( {\tilde{B}_{i} } )$ and $R ( {\tilde{B}_{i}^{\prime } } )$, satisfying $w_{i} \in \left[ {0,1} \right]$ and $\sum\nolimits_{i = 1}^{m} {w_{i} } = 1$. If $\mu_{11i} \le \mu^{\prime}_{11i} ,\mu_{12i} \le \mu^{\prime}_{12i} ,\mu_{21i} \le \mu^{\prime}_{21i} ,\mu_{13i} \le \mu^{\prime}_{13i} ,$ $\mu_{22i} \le \mu^{\prime}_{22i} ,$$\mu_{23i} \le \mu^{\prime}_{23i} ,$ $\mu_{31i} \le \mu^{\prime}_{31i} ,$ $\mu_{32i} \le \mu^{\prime}_{32i} ,$$\mu_{33i} \le \mu^{\prime}_{33i}$, then

$$ \begin{gathered} RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) \hfill \\ \le RNEWA\left( {R\left( {\tilde{B}^{\prime}_{1} } \right),R\left( {\tilde{B}^{\prime}_{2} } \right),...,R\left( {\tilde{B}^{\prime}_{m} } \right)} \right). \hfill \\ \end{gathered} $$

(26)

This property is called Monotonicity Property.

Proof

The above theorem obviously holds.

Theorem 5

Let $R( {\tilde{B}_{i} } )\,\,\, \left( {i = 1,2,...,m} \right) \hbox{ and }R ( {\tilde{B}_{i}^{\prime } } )$ $\left( {i = 1,2,...,m} \right)$ be two collection of R-numbers and ${\varvec{w}} = \left( {w_{1} ,w_{2} ,...,w_{m} } \right)^{T}$ be the weight vector of $R( {\tilde{B}_{i} } )$ and $R( {\tilde{B}_{i}^{\prime } } )$, satisfying $w_{i} \in \left[ {0,1} \right]$, $\sum\nolimits_{i = 1}^{m} {w_{i} } = 1$, and $R( {\tilde{B}^{\prime}_{i} } )\,\,\,\left( {i = 1,2,...,m} \right)$ be any permutation of $R( {\tilde{B}_{i} } )\,\,\,\left( {i = 1,2,...,m} \right)$. Then

$$ \begin{gathered} RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) \hfill \\ \qquad { = }RNEWA\left( {R\left( {\tilde{B}^{\prime}_{1} } \right),R\left( {\tilde{B}^{\prime}_{2} } \right),...,R\left( {\tilde{B}^{\prime}_{m} } \right)} \right). \hfill \\ \end{gathered} $$

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This property is called Commutativity Property.

Proof

Based on the Definition 10, we have

$$ \begin{gathered} RNEWA\left( {R\left( {\tilde{B}^{\prime}_{1} } \right),R\left( {\tilde{B}^{\prime}_{2} } \right),...,R\left( {\tilde{B}^{\prime}_{m} } \right)} \right) \hfill \\ \qquad = \mathop { \otimes_{\varepsilon } }\limits_{i = 1}^{m} R\left( {\tilde{B}^{\prime}_{i} } \right)^{{\omega_{i} }} = \mathop { \otimes_{\varepsilon } }\limits_{i = 1}^{m} R\left( {\tilde{B}_{i} } \right)^{{\omega_{i} }} \hfill \\ \qquad = RNEWA\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right). \hfill \\ \end{gathered} $$

Definition 11

Let $R( {\tilde{B}_{i} } )\left( {i = 1,2,...,m} \right)$ be a collection of R-numbers, ${\varvec{w}} = \left( {w_{1} ,w_{2} ,...,w_{m} } \right)^{T}$ be the weight vector of $R( {\tilde{B}_{i} } )$, satisfying $w_{i} \in \left[ {0,1} \right]$ and $\sum\nolimits_{i = 1}^{m} {w_{i} } = 1$. Then, the R-numbers Einstein weighted geometric (RNEWG) operator is defined as

$$ RNEWG\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) = \mathop { \otimes_{\varepsilon } }\limits_{i = 1}^{m} R\left( {\tilde{B}_{i} } \right)^{{\omega_{i} }} . $$

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Theorem 6

Let $R( {\tilde{B}_{i} } )\left( {i = 1,2,...,m} \right)$ be a collection of R-numbers, and ${\varvec{w}} = \left( {w_{1} ,w_{2} ,...,w_{m} } \right)^{T}$ be the weight vector of $R( {\tilde{B}_{i} } )$, satisfying $w_{i} \in \left[ {0,1} \right]$ and $\sum\nolimits_{i = 1}^{m} {w_{i} } = 1$. Then, the aggregated value by the RNEWG operator is still an R-number and

$$ \begin{gathered} RNEWG\left( {R\left( {\tilde{B}_{1} } \right),R\left( {\tilde{B}_{2} } \right),...,R\left( {\tilde{B}_{m} } \right)} \right) = \mathop { \otimes_{\varepsilon } }\limits_{i = 1}^{m} R\left( {\tilde{B}_{i} } \right)^{{\omega_{i} }} \hfill \\ \quad = \left( \begin{gathered} \left( {\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{11i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{n} {\left( {2 - \mu_{11i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{n} {\left( {\mu_{11i} } \right)^{{w_{i} }} } }},\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{12i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{n} {\left( {2 - \mu_{12i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{n} {\left( {\mu_{12i} } \right)^{{w_{i} }} } }},\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{13i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {2 - \mu_{13i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {\mu_{13i} } \right)^{{w_{i} }} } }}} \right), \hfill \\ \left( {\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{21i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{n} {\left( {2 - \mu_{21i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{n} {\left( {\mu_{21i} } \right)^{{w_{i} }} } }},\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{22i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{n} {\left( {2 - \mu_{22i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{n} {\left( {\mu_{22i} } \right)^{{w_{i} }} } }},\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{23i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{n} {\left( {2 - \mu_{23i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {\mu_{23i} } \right)^{{w_{i} }} } }}} \right), \hfill \\ \left( {\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{31i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {2 - \mu_{31i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {\mu_{31i} } \right)^{{w_{i} }} } }},\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{32i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {2 - \mu_{32i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {\mu_{32i} } \right)^{{w_{i} }} } }},\frac{{2\prod\nolimits_{i = 1}^{m} {\left( {\mu_{33i} } \right)^{{w_{i} }} } }}{{\prod\nolimits_{i = 1}^{m} {\left( {2 - \mu_{33i} } \right)^{{w_{i} }} } + \prod\nolimits_{i = 1}^{m} {\left( {\mu_{33i} } \right)^{{w_{i} }} } }}} \right) \hfill \\ \end{gathered} \right). \hfill \\ \end{gathered} $$

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Similar to RNEWA operator, RNEWG operator also has the properties of Idempotency, Boundedness, Monotonicity, and Commutativity.

The model for determining expert weights

The model for determining expert weights is shown in Fig. 2

Fig. 2

The model for determining expert weights

Step 1. Based on the evaluation values from experts in the period $p$, we define the average value of these evaluation values as

$$ \begin{gathered} \overline{\user2{X}}^{{\varvec{p}}} = \left[ {\overline{x}_{ij}^{p} } \right], \quad p = 1,2,3,...,f = 1,2,3,...,r, \hfill \\ i = 1,2,...,m,j = 1,2,...,n, \hfill \\ \end{gathered} $$

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where

$$ \begin{aligned} \overline{x}_{ij}^{p}& = \frac{1}{r}\sum\limits_{f = 1}^{r} {x_{ij}^{{e_{f}^{p} }} } \hfill \\ & = \left( {\left( {\overline{x}_{ij11}^{p} ,\overline{x}_{ij12}^{p} ,\overline{x}_{ij13}^{p} } \right),\left( {\overline{x}_{ij21}^{p} ,\overline{x}_{ij22}^{p} ,\overline{x}_{ij23}^{p} } \right),}\right. \hfill\\ & \quad \times \left.{\left( {\overline{x}_{ij31}^{p} ,\overline{x}_{ij32}^{p} ,\overline{x}_{ij33}^{p} } \right)} \right). \hfill \\ \end{aligned} $$

Step 2. The degree of similarity between evaluation value $x_{ij}^{{e_{f}^{p} }}$ given by each expert and the average evaluation value $\overline{x}_{ij}^{p}$ is defined as

$$ \begin{gathered} s\left( {x_{ij}^{{e_{f}^{p} }} ,\overline{x}_{ij}^{p} } \right) = 1 - \frac{{d\left( {x_{ij}^{{e_{f}^{p} }} ,\overline{x}_{ij}^{p} } \right)}}{{\sum\nolimits_{f = 1}^{r} {d\left( {x_{ij}^{{e_{f}^{p} }} ,\overline{x}_{ij}^{p} } \right)} }}, \hfill \\ p = 1,2,3,...,f = 1,2,3,...,r, \hfill \\ i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

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where

$$\begin{gathered} d\left( {x_{ij}^{{e_{f}^{p} }} ,\overline{x}_{ij}^{p} } \right) = \sqrt {\frac{1}{3}\left( \begin{gathered} \left( {\sqrt {\frac{1}{3}\left( {\left( {x_{ij11}^{{e_{f}^{p} }} - \overline{x}_{ij11}^{p} } \right)^{2} + \left( {x_{ij12}^{{e_{f}^{p} }} - \overline{x}_{ij12}^{p} } \right)^{2} + \left( {x_{ij13}^{{e_{f}^{p} }} - \overline{x}_{ij13}^{p} } \right)^{2} } \right)} } \right) + \hfill \\ \left( {\sqrt {\frac{1}{3}\left( {\left( {x_{ij21}^{{e_{f}^{p} }} - \overline{x}_{ij21}^{p} } \right)^{2} + \left( {x_{ij22}^{{e_{f}^{p} }} - \overline{x}_{ij22}^{p} } \right)^{2} + \left( {x_{ij23}^{{e_{f}^{p} }} - \overline{x}_{ij23}^{p} } \right)^{2} } \right)} } \right) + \hfill \\ \left( {\sqrt {\frac{1}{3}\left( {\left( {x_{ij31}^{{e_{f}^{p} }} - \overline{x}_{ij31}^{p} } \right)^{2} + \left( {x_{ij32}^{{e_{f}^{p} }} - \overline{x}_{ij32}^{p} } \right)^{2} + \left( {x_{ij33}^{{e_{f}^{p} }} - \overline{x}_{ij33}^{p} } \right)^{2} } \right)} } \right) \hfill \\ \end{gathered} \right)} . \hfill \\ \end{gathered}$$

Step 3. The weight of expert is defined as

$$ \begin{gathered} w_{ij}^{{e_{f}^{p} }} = \frac{{S\left( {\overline{x}_{ij}^{p} ,x_{ij}^{{e_{f}^{p} }} } \right)}}{{\sum\nolimits_{f = 1}^{r} {S\left( {\overline{x}_{ij}^{p} ,x_{ij}^{{e_{f}^{p} }} } \right)} }}, \hfill \\ p = 1,2,3,...,f = 1,2,3,...,r, \hfill \\ i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

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The MEREC model for determining the attribute weights

In this section, MEREC is proposed to determine the attribute weights in each period in DMAGDM. The MEREC calculates the weight of criteria based on their removal effect on alternatives. The criteria with greater effect on performance are assigned greater weights. The attribute weight determination model is shown in Fig. 3.

Fig. 3

The model for determining attribute weights

Step 1. The RNEWA operator proposed in the section “The Einstein aggregation operators of R-numbers” and the expert weights calculated in the section “The model for determining expert weights” are used for aggregation of decision matrices in the form of R-numbers from different experts. The aggregation method is shown below

$$ \begin{gathered} x_{ij}^{p} = RNEWA\left( {x_{ij}^{{e_{1}^{p} }} ,x_{ij}^{{e_{2}^{p} }} ,...,x_{ij}^{{e_{r}^{p} }} } \right) = \mathop { \oplus_{\varepsilon } }\limits_{f = 1}^{r} w_{ij}^{{e_{f}^{p} }} x_{ij}^{{e_{f}^{p} }} \hfill \\ = \left( \begin{gathered} \left( {\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij11}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij11}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij11}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij11}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }},\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij12}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij12}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij12}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij12}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }},\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij13}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij13}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij13}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij13}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}} \right), \hfill \\ \left( {\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij21}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij21}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij21}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij21}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }},\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij22}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij22}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij22}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij22}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }},\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij23}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij23}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij23}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij23}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}} \right), \hfill \\ \left( {\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij31}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij31}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij31}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij31}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }},\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij32}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij32}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij32}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij32}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }},\frac{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij33}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } - \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij33}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}{{\prod\nolimits_{f = 1}^{r} {\left( {1 + x_{ij33}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } + \prod\nolimits_{f = 1}^{r} {\left( {1 - x_{ij33}^{{e_{f}^{p} }} } \right)^{{{w_{ij}{e_{f}^{p} }} }} } }}} \right) \hfill \\ \end{gathered} \right). \hfill \\ \end{gathered} $$

Step 2. The aggregation matrix in Step 1 is normalized using the following formula. This normalization method differs from the traditional normalization method as it switches the formulas of beneficial and non-beneficial criteria. Unlike many other studies, we convert all criteria to minimum type criteria

$$ \begin{gathered} x_{Nij}^{p} = \left( \begin{gathered} \frac{{a_{j}^{ - } }}{{\left( {x_{ij11}^{p} ,x_{ij12}^{p} ,x_{ij13}^{p} } \right)}},\frac{{a_{j}^{ - } }}{{\left( {x_{ij21}^{p} ,x_{ij22}^{p} ,x_{ij23}^{p} } \right)}}, \hfill \\ \frac{{a_{j}^{ - } }}{{\left( {x_{ij31}^{p} ,x_{ij32}^{p} ,x_{ij33}^{p} } \right)}} \hfill \\ \end{gathered} \right), \hfill \\ a_{j}^{ - } = \mathop {\min }\nolimits_{i} x_{ij11}^{p} ,{\text{for beneficial criteria}}{.} \hfill \\ \end{gathered} $$

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$$ \begin{gathered} x_{Nij}^{p} = \left( \begin{gathered} \frac{{\left( {x_{ij11}^{p} ,x_{ij12}^{p} ,x_{ij13}^{p} } \right)}}{{c_{j}^{ + } }},\frac{{\left( {x_{ij21}^{p} ,x_{ij22}^{p} ,x_{ij23}^{p} } \right)}}{{c_{j}^{ + } }}, \hfill \\ \frac{{\left( {x_{ij31}^{p} ,x_{ij32}^{p} ,x_{ij33}^{p} } \right)}}{{c_{j}^{ + } }} \hfill \\ \end{gathered} \right), \hfill \\ c_{j}^{ + } = \mathop {\max }\nolimits_{i} x_{ij33}^{p} ,{\text{for non - beneficial criteria}}{.} \hfill \\ \end{gathered} $$

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Step 3. Defuzzify the normalized matrix.

$$\begin{aligned}{}& COA\left( {x_{Nij}^{p} } \right) = \frac{1}{9}\\ & \quad \times \left( \begin{gathered} x_{Nij11}^{p} + x_{Nij12}^{p} + x_{Nij13}^{p} + x_{Nij21}^{p} + \hfill \\ x_{Nij22}^{p} + x_{Nij23}^{p} + x_{Nij31}^{p} + x_{Nij32}^{p} + x_{Nij33}^{p} \hfill \\ \end{gathered} \right).\end{aligned} $$

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Step 4. Calculate the overall performance of alternatives $S_{i}^{p}$. Based on the normalized value obtained in the step 2, we calculate the overall performance according to the following equation to ensure that smaller $COA\left( {x_{Nij}^{p} } \right)$ values result larger performance $S_{i}^{p}$ values

$$ \begin{gathered} S_{i}^{p} = \ln \left( {1 + \left( {\frac{1}{n}\sum\limits_{j} {\left| {\ln \left( {COA\left( {x_{Nij}^{p} } \right)} \right)} \right|} } \right)} \right), \hfill \\ p = 1,2,3,...,i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

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Step 5. Calculate the performance of the alternatives by removing each criteria

$$ \begin{gathered} S_{i}^{p\prime } = \ln \left( {1 + \left( {\frac{1}{n}\sum\limits_{k,k \ne j} {\left| {\ln \left( {COA\left( {x_{Nik}^{p} } \right)} \right)} \right|} } \right)} \right), \hfill \\ p = 1,2,3,...,i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

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where $S_{ij}^{p\prime }$ denotes the overall performance of $i{\text{th}}$ alternative concerning the removal of $j{\text{th}}$ criteria.

Step 6. Compute the summation of absolute deviations

$$ \begin{gathered} E_{j}^{p} = \sum\limits_{i} {\left| {S_{i}^{p\prime } - S_{i}^{p} } \right|} , \hfill \\ p = 1,2,3,...,i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

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Step 7. Determine the final attribute weight

$$ \begin{aligned} \varpi_{j}^{p} & = \frac{{E_{j}^{p} }}{{\sum\nolimits_{j} {E_{j}^{p} } }},p = 1,2,3,..., \\ i & = 1,2,...,m,j = 1,2,...,n.\end{aligned} $$

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The CoCoSo model to calculate static ratings for each period

The CoCoSo method is based on a comprehensive simple additive weighted and exponential weighted product model. Under the R-numbers environment, the CoCoSo method employs the RNEWA operator and RNEWG operator proposed in the section “The Einstein aggregation operators of R-numbers” for decision-making. The calculation model of static rating is illustrated in Fig. 4.

Fig. 4

The model for calculating static rating

Step 1: Normalize the expert aggregation matrix for each period in Step 1 of the section “The model for determining expert weights”

$$ \begin{gathered} x_{Nij}^{p} = \left( \begin{gathered} \left( {\frac{{x_{ij11}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{x_{ij12}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{x_{ij13}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }}} \right), \hfill \\ \left( {\frac{{x_{ij21}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{x_{ij22}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{x_{ij23}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }}} \right), \hfill \\ \left( {\frac{{x_{ij31}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{x_{ij32}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{x_{ij33}^{p} - a_{j}^{ - } }}{{c_{j}^{ + } - a_{j}^{ - } }}} \right) \hfill \\ \end{gathered} \right), \hfill \\ a_{j}^{ - } = \mathop {\min }\nolimits_{i} x_{ij11}^{p} ,c_{j}^{ + } = \mathop {\max }\nolimits_{i} x_{ij33}^{p} ,{\text{for beneficial criteria}}{.} \hfill \\ \end{gathered} $$

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$$ \begin{gathered} x_{Nij}^{p} = \left( \begin{gathered} \left( {\frac{{c_{j}^{ + } - x_{ij33}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{c_{j}^{ + } - x_{ij32}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{c_{j}^{ + } - x_{ij31}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }}} \right), \hfill \\ \left( {\frac{{c_{j}^{ + } - x_{ij23}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{c_{j}^{ + } - x_{ij22}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{c_{j}^{ + } - x_{ij21}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }}} \right), \hfill \\ \left( {\frac{{c_{j}^{ + } - x_{ij13}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{c_{j}^{ + } - x_{ij12}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }},\frac{{c_{j}^{ + } - x_{ij11}^{p} }}{{c_{j}^{ + } - a_{j}^{ - } }}} \right) \hfill \\ \end{gathered} \right), \hfill \\ a_{j}^{ - } \!=\! \mathop {\min }\nolimits_{i} x_{ij11}^{p} ,c_{j}^{ + } \!=\! \mathop {\max }\nolimits_{i} x_{ij33}^{p} ,{\text{for non-beneficial criteria}}{.} \hfill \\ \end{gathered} $$

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Step 2. Using the attribute weights calculated by MEREC method in the section “The MEREC model for determining the attribute weights”, simple additive weighted $S_{i}^{p}$ and exponential weighted product $P_{i}^{p}$ in the R-numbers environment are obtained based on RNEWA operator and RNEWG operator, respectively

$$ \begin{gathered} S_{i}^{p} = RNEWA\left( {x_{Ni1}^{p} ,x_{Ni2}^{p} ,...,x_{Ni3}^{p} } \right) = \mathop {\mathop \oplus \nolimits_{\varepsilon } }\limits_{j = 1}^{n} \varpi_{j}^{p} x_{Nij}^{p} \hfill \\ = \left( \begin{gathered} \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } }}} \right), \hfill \\ \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } }}} \right), \hfill \\ \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{r} {\left( {1 + x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {1 - x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } }}} \right) \hfill \\ \end{gathered} \right), \hfill \\ p = 1,2,3,...,i = 1,2,...,m,j = 1,2,...,n, \hfill \\ \end{gathered} $$

(42)

$$ \begin{gathered} P_{i}^{p} = RNEWG\left( {x_{Ni1}^{p} ,x_{Ni2}^{p} ,...,x_{Ni3}^{p} } \right) = \mathop {\mathop \otimes \nolimits_{\varepsilon } }\limits_{j = 1}^{n} \left( {x_{Nij}^{p} } \right)^{{\varpi_{j}^{p} }} \hfill \\ = \left( \begin{gathered} \left( {\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij11}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij12}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij13}^{p} } \right)^{{\varpi_{j}^{p} }} } }}} \right), \hfill \\ \left( {\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij21}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij22}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij23}^{p} } \right)^{{\varpi_{j}^{p} }} } }}} \right), \hfill \\ \left( {\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij31}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij32}^{p} } \right)^{{\varpi_{j}^{p} }} } }},\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } + \prod\nolimits_{j = 1}^{n} {\left( {x_{Nij33}^{p} } \right)^{{\varpi_{j}^{p} }} } }}} \right) \hfill \\ \end{gathered} \right), \hfill \\ p = 1,2,3,...,i = 1,2,...,m,j = 1,2,...,n. \hfill \\ \end{gathered} $$

(43)

Step 3. Defuzzify the $S_{i}^{p}$ and $P_{i}^{p}$

$$ COA\left( {S_{i}^{p} } \right) = \frac{1}{9}\left( \begin{gathered} S_{i11}^{p} + S_{i12}^{p} + S_{i13}^{p} + S_{i21}^{p} + \hfill \\ S_{i22}^{p} + S_{i23}^{p} + S_{i31}^{p} + S_{i32}^{p} + S_{i33}^{p} \hfill \\ \end{gathered} \right), $$

(44)

$$ COA\left( {P_{i}^{p} } \right) = \frac{1}{9}\left( \begin{gathered} P_{i11}^{p} + P_{i12}^{p} + P_{i13}^{p} + P_{i21}^{p} \hfill \\ + P_{i22}^{p} + P_{i23}^{p} + P_{i31}^{p} + P_{i32}^{p} + P_{i33}^{p} \hfill \\ \end{gathered} \right). $$

(45)

Step 4. The relative weights of the alternatives are obtained by the following aggregation method. The relative weights of other options are generated using three evaluation and scoring strategies, which can be obtained by the following formula:

$$ k_{ia}^{p} = \frac{{COA\left( {S_{i}^{p} } \right) + COA\left( {P_{i}^{p} } \right)}}{{\sum\nolimits_{i = 1}^{m} {\left( {COA\left( {S_{i}^{p} } \right) + COA\left( {P_{i}^{p} } \right)} \right)} }}, $$

(46)

$$ k_{ib}^{p} = \frac{{COA\left( {S_{i}^{p} } \right)}}{{\mathop {\min }\nolimits_{i} \left( {COA\left( {S_{i}^{p} } \right)} \right)}} + \frac{{COA\left( {P_{i}^{p} } \right)}}{{\mathop {\min }\nolimits_{i} \left( {COA\left( {P_{i}^{p} } \right)} \right)}}, $$

(47)

$$ \begin{gathered} k_{ic}^{p} = \frac{{\lambda COA\left( {S_{i}^{p} } \right) + \left( {1 - \lambda } \right)COA\left( {P_{i}^{p} } \right)}}{{\lambda \mathop {\max }\nolimits_{i} \left( {COA\left( {S_{i}^{p} } \right)} \right) + \left( {1 - \lambda } \right)\mathop {\max }\nolimits_{i} \left( {COA\left( {P_{i}^{p} } \right)} \right)}}, \hfill \\ 0 \le \lambda \le 1. \hfill \\ \end{gathered} $$

(48)

As can be seen from the above equation, Eq. (46) represents the arithmetic mean of the $COA\left( {S_{i}^{p} } \right)$ and $COA\left( {P_{i}^{p} } \right)$. Equation (47) represents the sum of relative scores of $COA\left( {S_{i}^{p} } \right)$ and $COA\left( {P_{i}^{p} } \right)$ compared to the optimal scores. Equation (48) expresses a balanced compromise of $COA\left( {S_{i}^{p} } \right)$ and $COA\left( {P_{i}^{p} } \right)$. In Eq. (48), $\lambda$ is determined by the expert and $\lambda$ is usually $0.5$. However, the flexibility and stability of the proposed CoCoSo method can depend on other values.

Step 5. Calculate the static rating $k_{i}^{p}$ of each alternative

$$ k_{i}^{p} = \left( {k_{ia}^{p} \cdot k_{ib}^{p} \cdot k_{ic}^{p} } \right)^{\frac{1}{3}} + \frac{1}{3}\left( {k_{ia}^{p} + k_{ib}^{p} + k_{ic}^{p} } \right). $$

(49)

The model to calculate dynamic ratings for each period

Information changes over time, where alternatives and attributes are variables, resulting in dynamic evolution. Hence, we consider the dynamic rating of each alternative. The evaluation of alternatives is not only based on the performance in the current stage but also on the impact of the historical stage. The evaluation scheme is as follows:

$$ E^{p} \left( {A_{i} } \right) = \left\{ \begin{gathered} R^{p} \left( {A_{i} } \right) \, A_{i} \in A^{p} \backslash H^{p - 1 \, } \, \hfill \\ \Phi \left( {E^{p - 1} \left( {A_{i} } \right),R^{p} \left( {A_{i} } \right)} \right) \, A_{i} \in A^{p} \cap H^{p - 1 \, } \hfill \\ E^{p - 1} \left( {A_{i} } \right) \, A_{i} \in H^{p - 1 \, } \backslash A^{p} \hfill \\ \end{gathered} \right., $$

(50)

where $R^{p} \left( {A_{i} } \right) = k_{i}^{p}$, and $\Phi$ is an associative aggregation function for aggregating performance values for the current and historical periods.

• (1) If the alternative $A_{i}$ only belongs to the current set of alternatives $A^{p}$ rather than the historical set of alternatives $H^{p - 1}$, then its evaluation value $E^{p} \left( {A_{i} } \right)$ is equal to the static rating $R^{p} \left( {A_{i} } \right)$ calculated in the previous stage.

• (2) If the alternative $A_{i}$ belongs to both the current set of alternatives $A^{p}$ and the historical set of alternatives $H^{p - 1}$, then its evaluation value is equal to $\Phi \left( {E^{p - 1} \left( {A_{i} } \right),R^{p - 1} \left( {A_{i} } \right)} \right)$.

• (3) If the alternative $A_{i}$ does not belong to the current set of alternatives $A^{p}$, but belongs to the historical set of alternatives $H^{p - 1}$, its evaluation value $E^{p} \left( {A_{i} } \right)$ is equal to the last calculated iteration value $E^{p - 1} \left( {A_{i} } \right)$.

Inspired by the literature [33], considering the characteristics of the static rating of the alternative calculated by CoCoSo method, we define $\Phi$ as

$$ \begin{gathered} \Phi \left( {E^{{_{p - 1} }} \left( {A_{i} } \right),R^{{_{p} }} \left( {A_{i} } \right)} \right) \hfill \\ \qquad = \left( {1 - \lambda } \right)E^{{_{p - 1} }} \left( {A_{i} } \right) + \lambda R^{{_{p} }} \left( {A_{i} } \right),0 < \lambda < 1. \hfill \\ \end{gathered} $$

(51)

where the value of $\lambda$ is determined according to the decision-maker's balance between the importance of the static rating in the current period and the historical period. In this paper, we believe that decision-makers should pay attention to the static rating of alternatives in the current period, so $\lambda$ is given a weight of 0.7, while the static rating in the historical period is given a weight of 0.3. This decision-making method not only considers the performance of alternatives in the historical period, but also meets the reality of decision-making in the current period.

In the proposed dynamic framework, the current-period static ratings of alternatives are obtained based on the alternatives and attributes. The dynamic ratings of alternatives can then be obtained by combining the current-period static ratings with stored historical information. The dynamic ratings of each period are obtained through this iterative method until the stopping criterion is met. The dynamic decision-making framework is shown in Fig. 5.

Fig. 5

Dynamic framework

Numerical example

This section illustrates the operability and effectiveness of the R-numbers DMAGDM method proposed in this paper through case studies of internal and external risk assessments of supply chains of several manufacturing enterprises in China. Internal and external risks are crucial to the sustainable development of enterprises. In reality, the risks faced by suppliers are unpredictable over time. Due to their own development, businesses may also launch markets or register new enterprises in this environment. Therefore, selecting the best supplier under such dynamic changes is evidently a DMAGDM problem. R-numbers have significant advantages in capturing risks and can improve decision-making reliability. Therefore, this paper employs R-numbers DMAGDM method to evaluate enterprise suppliers.

Suppose manufacturing enterprise A has multiple suppliers. Supplier enterprises and risk assessment indicators change over time. Three experts $E = \left\{ {1,2,3} \right\}$ were selected to participate in each stage of the decision-making process. For the sake of simplicity, we will consider only three periods $P = \left\{ {1,2,3} \right\}$ of decision-making. Table 1 is used to evaluate alternatives with different attributes in different periods, the pessimistic risk $\tilde{r}^{ - }$, optimistic risk $\tilde{r}^{ + }$, pessimistic acceptable risk $\widetilde{AR}^{ - }$, and optimistic acceptable risk $\widetilde{AR}^{ + }$ in the expert evaluation. Due to the risk perception in expert assessment, suitable $\widetilde{RP}$ were considered according to Table 2, where positive risk perception and negative risk perception were considered to be the same, i.e., $\widetilde{RP}^{ + } = \widetilde{RP}^{ - }$.

Table 1 Linguistic terms for evaluating alternatives, optimistic risk, pessimistic risk, optimistic acceptable risk, and pessimistic acceptable risk

Table 2 Linguistic terms for evaluating risk perception

In the first period, three supplier enterprises were evaluated $A^{1} = \left\{ {A_{1}^{1} ,A_{2}^{1} ,A_{3}^{1} } \right\}$ for four risk indicators $C^{1} = \left\{ {C_{1}^{1} ,C_{2}^{1} ,C_{3}^{1} ,C_{4}^{1} } \right\}$. The evaluation information obtained is shown in Table 21 (see Appendix 1).

In the second period, four supplier enterprises $A^{2} = \left\{ {A_{1}^{2} ,A_{2}^{2} ,A_{3}^{2} ,A_{4}^{2} } \right\}$ were evaluated for three risk indicators $C^{2} = \left\{ {C_{1}^{2} ,C_{2}^{2} ,C_{3}^{2} } \right\}$, and the evaluation information obtained is shown in Table 21 (see Appendix 1).

In the third period, four supplier enterprises $A^{3} = \left\{ {A_{2}^{3} ,A_{3}^{3} ,A_{4}^{3} ,A_{5}^{3} } \right\}$ were evaluated for three risk indicators $C^{3} = \left\{ {C_{2}^{3} ,C_{3}^{3} ,C_{4}^{3} } \right\}$, and the evaluation information obtained is shown in Table 21 (see Appendix 1).

Table 21 (see Appendix 1) also present three experts' language assessment information about pessimistic risk $\tilde{r}^{ - }$, optimistic risk $\tilde{r}^{ + }$, pessimistic acceptable risk $\widetilde{AR}^{ - }$, optimistic acceptable risk $\widetilde{AR}^{ + }$, pessimistic risk perception $\widetilde{RP}^{ - }$, optimistic risk perception $\widetilde{RP}^{ + }$ in each period, respectively. Since this case is an assessment of risk, the pessimistic acceptable risk and the optimistic acceptable risk are identified as 0. The given risk parameter linguistic terms are converted into the form of triangular fuzzy numbers using Tables 1, 2, and the conversion result is shown in Table 22 (see Appendix 2).

Stage 1. Convert the initial matrix to R-numbers matrix.

According to the definition of R-numbers, convert the data in Table 22 (see Appendix 2) into the form of R-numbers, as shown in Table 23 (see Appendix 3).

Stage 2. Determine the expert weights.

Step 1. The average evaluation matrix of experts is calculated using Eq. (30), and the results are shown in Table 3

Table 3 The average evaluation matrix of experts

Step 2. Calculate the degree of similarity between evaluation value $x_{ij}^{{e_{f}^{1} }}$ given by each expert and the average evaluation value $\overline{x}_{ij}^{1}$ using Eq. (31), and the results are shown in Table 4.

Table 4 The average evaluation matrix of experts

Step 3. Calculate the expert weights using Eq. (32), and results are shown in Table 5.

Table 5 The expert weights

Stage 3. Determine the attribute weights.

Step 1. The RNEWA operator proposed in the section “The Einstein aggregation operators of R-numbers” and the expert weights calculated in Stage 2 are used for aggregation of decision matrices in the form of R-numbers from different experts. The aggregation results are shown in Table 6.

Table 6 The aggregation matrix

Step 2. Normalize the aggregation matrix, and the results are shown in Table 7.

Table 7 Normalized matrix

Step 3. Defuzzify the normalized matrix, and the results are shown in Table 8.

Table 8 Defuzzification matrix

Step 4. Calculate the overall performance of alternatives $S_{i}^{1}$, and the results are shown in Table 9.

Table 9 Performance of alternatives

Step 5. Calculate the performance of the alternatives by removing each criteria, and the results are shown in Table 9.

Step 6. Calculate the summation of absolute deviations, and the results are shown in Table 9.

Step 7. Determine the final attribute weights, and the results are shown in Table 9.

Stage 4. Calculate static ratings for each period using the CoCoSo model.

Step 1. Normalize the expert aggregation matrix in Step 1 of Stage 3, and the results are shown in Table 10.

Table 10 Normalized matrix

Step 2. Using the attribute weights calculated by MEREC method in Stage 3, and RNEWA operator and RNEWG operator in the section “The Einstein aggregation operators of R-numbers”, the simple additive weighted $S_{i}^{1}$ and exponential weighted product $P_{i}^{1}$ are obtained in Table 11.

Table 11 Results of period $p = 1$

Step 3. Defuzzify the $S_{i}^{1}$ and $P_{i}^{1}$, and the results are shown in Table 11.

Step 4. Calculate the relative weights of the alternatives, and the results are shown in Table 11.

Step 5. Calculate the static rating $k_{i}^{1}$ of each alternative, and the results are shown in Table 11.

Stage 5 Calculate dynamic ratings of alternatives.

Since in period $p = 1$, there is no historical period for the alternatives, the dynamic ratings of the alternatives are equal to the static ratings of the alternatives. Therefore, we can get $E^{1} \left( {A_{1} } \right) = 3.3932,E^{1} \left( {A_{2} } \right) = 1.4135,E^{1} \left( {A_{3} } \right) = 2.5196$. And the optimal solution for period $p = 1$ is $A_{1}$.

Since some details have been explained in period $p = 1$, we only focus on the generation of static and dynamic ratings of alternatives in this period.

Stage 1. Convert the initial matrix to R-numbers matrix.

According to the definition of R-numbers, convert the data in Table 22 (see Appendix 2) into the form of R-numbers, as shown in Table 23 (see Appendix 3).

Stage 2. Determine the expert weights.

The expert weights are shown in Table 12.

Table 12 The results of expert weights

Stage 3. Determine the attribute weights.

The attribute weights calculated by MEREC method are $\varpi_{1}^{2} = 0.3116,\varpi_{2}^{2} = 0.2710,\varpi_{3}^{2} = 0.4174$ in period $p = 2$.

Stage 4. Calculate static ratings using the CoCoSo model in period $p = 2$.

Step 1. Normalize the expert aggregation matrix in Stage 3, and the results are shown in Table 13.

Table 13 Normalized matrix

Step 2. Using the attribute weights calculated by MEREC method in Stage 3, and RNEWA operator and RNEWG operator in the section “The Einstein aggregation operators of R-numbers”, the simple additive weighted $S_{i}^{2}$ and exponential weighted product $P_{i}^{2}$ are obtained in Table 14.

Table 14 Results of period $p = 2$

Step 3. Defuzzify the $S_{i}^{2}$ and $P_{i}^{2}$, and the results are shown in Table 14.

Step 4. The relative weights of the alternatives are obtained, and the results are shown in Table 14.

Step 5. Calculate the static rating $k_{i}^{2}$ of each alternative, and the results are shown in Table 14.

Stage 5. Calculate dynamic ratings of alternatives.

The alternative $A_{4}$ is added in period $p = 2$. And based on the dynamic ratings of alternatives $\left\{ {A_{1} ,A_{2} ,A_{3} } \right\}$ in period $p = 1$, dynamic ratings of alternatives can be calculated as $E^{3} \left( {A_{1} } \right) = 2.9031,$$E^{2} \left( {A_{2} } \right) = 1.5494,$ $E^{2} \left( {A_{3} } \right) = 2.3541,$ $E^{2} \left( {A_{4} } \right) = 2.0247$ in period $p = 2$. Therefore, the optimal solution for period $p = 2$ is $A_{1}$.

Stage 1. Convert the initial matrix to R-numbers matrix.

According to the definition of R-numbers, convert the data in Table 22 (see Appendix 2) into the form of R-numbers, as shown in Table 23 (see Appendix 3).

Stage 2. Determine the expert weights.

The expert weights are shown in Table 15.

Table 15 The results of experts

Stage 3. Determine the attribute weights.

The attribute weights calculated by MEREC method are $\varpi_{2}^{3} = 0.3127,\varpi_{3}^{3} = 0.3280,\varpi_{4}^{3} = 0.3593$ in period $p = 3$.

Stage 4. Calculate static ratings using the CoCoSo model.

Step 1. Normalize the expert aggregation matrix in Stage 3, and the results are shown in Table 16

Table 16 Normalized matrix

Step 2. Using the attribute weights calculated by MEREC method in Stage 3, and RNEWA operator and RNEWG operator in the section “The Einstein aggregation operators of R-numbers”, the simple additive weighted $S_{i}^{3}$ and exponential weighted product $P_{i}^{3}$ are obtained in Table 17.

Table 17 Results of period $p = 3$

Step 3. Defuzzify the $S_{i}^{3}$ and $P_{i}^{3}$, and the results are shown in Table 17.

Step 4. The relative weights of the alternatives are obtained, and the results are shown in Table 17.

Step 5. Calculate the static rating $k_{i}^{3}$ of each alternative, and the results are shown in Table 17.

Stage 5. Calculate dynamic ratings of alternatives

The alternative $A_{5}$ is added in period $p = 3$, and alternative $A_{1}$ is deleted. Based on the dynamic ratings of alternatives $\left\{ {A_{2} ,A_{3} ,A_{4} } \right\}$ in period $p = 2$, dynamic ratings of alternatives can be calculated as $E^{3} \left( {A_{2} } \right) = 1.9438,$$E^{3} \left( {A_{3} } \right) = 2.2538,$$E^{3} \left( {A_{4} } \right) = 1.9542,$ $E^{3} \left( {A_{5} } \right) = 1.472$ in period $p = 3$. Therefore, the optimal solution for period $p = 3$ is $A_{3}$.

Comparative analysis

To demonstrate the rationality and feasibility of the proposed method, this section makes a comparative analysis of the proposed method, including the comparison with the information fusion method and the comparison with the static MAGDM model.

Comparison with information fusion methods based on existing aggregation operators

Only Liu et al. [35] studied the information fusion method based on aggregation operator in the R-numbers environment in the existing literature. This section compares and analyzes this method with the operators proposed in this paper to prove the effectiveness of the proposed method.

First, we use the R-numbers weighted averaging operator defined by Liu et al. to replace the RNEWA operator in Step 1 of the section “The MEREC model for determining the attribute weights” for expert information aggregation. Second, we use R-numbers weighted averaging operator instead of RNEWA operator to calculate $S_{i}^{p}$ and R-numbers geometric weighted averaging operator instead of RNEWG operator to calculate $P_{i}^{p}$ in Step 2 of the section “The CoCoSo model to calculate static ratings for each period”. The final calculation results of the dynamic ratings of the alternatives for the three periods are shown in Table 18. It can be seen from Table 9 that the optimal solutions calculated by the two methods are the same in the three periods, namely $A_{1} ,A_{1} ,A_{3}$. Therefore, the R-numbers DMAGDM method proposed in this paper is stable and reliable.

Table 18 Results of sensitivity analysis

Comparison with static MAGDM

To illustrate the role of DMAGDM in decision-making, we compare static MAGDM with DMAGDM. The sorting results of alternatives in three periods obtained by static MAGDM and DMAGDM are shown in Table 19.

Table 19 Results of static MAGDM and DMAGDM

As shown in Table 19, the static ranking results in period $p = 1$ are the same as the dynamic ranking results, both of which are $A_{1}^{{}} \succ A_{3} \succ A_{2}^{{}}$.

In period $p = 2$, static ranking result is different from dynamic ranking result. The result of static ranking is $A_{4}^{{}} \succ A_{3} \succ A_{2}^{{}} \succ A_{1}^{{}}$, while the result of dynamic ranking is $A_{1}^{{}} \succ A_{3} \succ A_{4}^{{}} \succ A_{2}^{{}}$. The reason for this change is that the alternative performance in period $p = 1$ is considered in period $p = 2$. The ranking results of alternatives $\left\{ {A_{1} ,A_{2} ,A_{3} } \right\}$ change from $A_{3} \succ A_{2}^{{}} \succ A_{1}^{{}}$ calculated by static method to $A_{1}^{{}} \succ A_{3} \succ A_{2}^{{}}$ calculated by dynamic method, because it integrates the performance of alternatives in period $p = 1$, and the dynamic ranking result in period $p = 1$ is $A_{1}^{{}} \succ A_{3} \succ A_{2}^{{}}$, This result also shows the necessity of dynamic decision-making. For alternative $A_{4}$, the dynamic rating is equal to the static rating. The change of dynamic ranking compared with static ranking is due to the change of dynamic rating of other alternatives $\left\{ {A_{1} ,A_{2} ,A_{3} } \right\}$, rather than its own influence.

In period $p = 3$, static ranking result is also different from dynamic ranking result. The result of static ranking is $A_{2}^{{}} \succ A_{3}^{{}} \succ A_{4}^{{}} \succ A_{5}^{{}}$, while the result of dynamic ranking is $A_{3}^{{}} \succ A_{4}^{{}} \succ A_{2}^{{}} \succ A_{5}^{{}}$. For alternative $\left\{ {A_{2} ,A_{3} } \right\}$, static ranking is $A_{2}^{{}} \succ A_{3}^{{}}$, while dynamic ranking is $A_{3}^{{}} \succ A_{2}^{{}}$. In period $p = 1$, the static and dynamic gap value of $A_{3}^{{}} \succ A_{2}^{{}}$ is 1.1061. In period $p = 2$, the static and dynamic gap values of $A_{3}^{{}} \succ A_{2}^{{}}$ are 0.0957 and 0.8030, respectively. In period $p = 3$, the static gap value of $A_{2}^{{}} \succ A_{3}^{{}}$ is 0.8403 and the dynamic gap value of $A_{3}^{{}} \succ A_{2}^{{}}$ is 0.3100. Over time, the static gap value of $A_{3}^{{}} \succ A_{2}^{{}}$ narrowed until it became $A_{2}^{{}} \succ A_{3}^{{}}$ in period $p = 3$. While in periods $p = 2$ and $p = 3$, with the consideration of the performance of $A_{3}^{{}} \succ A_{2}^{{}}$ in the historical period, $A_{3}^{{}} \succ A_{2}^{{}}$ still exists in periods $p = 2$ and $p = 3$, but this advantage gradually decreases. To sum up, the $A_{3}^{{}} \succ A_{2}^{{}}$ obtained under dynamic decision-making is practical in period $p = 3$.

As shown in Fig. 6, the static ranking results of alternatives vary greatly in different periods. As shown in Fig. 7, the ranking results of alternatives in the three periods are relatively stable and in line with reality. In real life, the ranking results of periods with small difference will not change greatly.

Fig. 6

Static ranking

Fig. 7

Dynamic ranking

Sensitivity analysis

It is noted that the dynamic decision model we proposed has an important parameter $\lambda$. Obviously, the parameter $\lambda$ has a significant impact on the final decision result. This section aims to investigate the impact of parameters on decision results.

We assign different values to the parameter $\lambda$ in the dynamic model, and the decision results are shown in Table 20. From Table 20, we can see that the values of the parameter $\lambda$ are different, and the calculated results of dynamic rating are different, but the order is basically the same. In period $p = 1$, the ranking result is not affected by the parameters, and the ranking result is $A_{1}^{{}} \succ A_{3} \succ A_{2}^{{}}$. Because the calculation results in period $p = 1$ are not affected by the historical results, but only determined by the current static rating calculation results. In period $p = 2$, no matter how parameter $\lambda$ changes, the best alternative is $A_{1}$. In period $p = 3$, when $\lambda = 0.3$ and $\lambda = 0.5$, the ranking result is $A_{2}^{{}} \succ A_{3}^{{}} \succ A_{4}^{{}} \succ A_{5}^{{}}$, while when parameter $\lambda = 0.7$, the ranking result is $A_{3}^{{}} \succ A_{4}^{{}} \succ A_{2}^{{}} \succ A_{5}^{{}}$. The reason for the change in the ranking result of alternative $A_{2}$ is that the static rating of $A_{2}$ in period $p = 3$ is a maximum value, so when $\lambda = 0.7$, the weight of the static rating is 0.3, and the weight is greatly reduced, which leads to a large change in the ranking result of $A_{2}$.

Table 20 Results of sensitivity analysis

Conclusion

This paper proposes a new DMAGDM framework which processes the R-numbers information representation model. In DMAGDM, the alternatives and attributes vary over time, addressing the requirement of time variable and R-numbers information representation model. To obtain comprehensive evaluation results, the model must solve the following problems: determining expert and attribute weights, as well as determining Einstein aggregation operator of R-numbers information. The key of the model is to determine the static ratings using the CoCoSo method and finally to obtain the priority of the alternatives using the new dynamic rating calculation model.

The R-numbers DMAGDM model proposed in this paper has important significance in enterprise project risk assessment. On the one hand, it can accurately capture the risk indicators in the environment. On the other hand, it can describe the dynamic changes of alternatives and attributes, and obtain the optimal solution using the proposed dynamic decision model. In the process of project selection and decision-making, many projects can show good profitability and profitability in the investment evaluation stage. However, once the development starts, various uncertainties emerge, resulting in insufficient profitability or project failure. The fundamental reason is a lack of clear prediction and understanding of project risks. The key to influencing the development of enterprise project is to carry out a reasonable prediction and evaluation of project risk at the beginning stage. The R-numbers DMAGDM model can help enterprise managers make accurate decisions in a changing internal and external risk environment, avoiding failures caused by insufficient risk prediction capabilities. The model can also be extended to new application fields, such as energy exploitation, medical system, financial investment, and so on for risk decision-making.

The contribution of this paper lies in: (1) defining new information fusion methods for R-numbers, namely RNEWA and RNEWG operators, and studying their related properties; (2) proposing a model for determining expert weights based on R-number similarity; (3) presenting a model for determining attribute weights based on MEREC method in the R-numbers environment; (4) constructing a static rating calculation model based on the CoCoSo method; (5) constructing a novel dynamic decision-making model based on R-numbers.

This study also has some limitations. For example, the risk evaluation value of R-numbers depends on the expertise of the experts, so it requires high professionalism of the experts. Additionally, this research only provides a case study of supply chain risk research in the manufacturing industry, and more application cases need to be expanded.

Based on the R-numbers DMAGDM model proposed in this paper, the following issues are worth further research: (1) The information representation model in the R-numbers DMAGDM model is R-numbers. The proposed model's application for other information representation models should be investigated; (2) In this paper, the objective weight determination method is adopted to determine the expert and attribute weights. More reasonable weight determination methods are worth further study; (3) The dynamic priority determination model of alternatives should be investigated further; (4) Attention should be paid to dynamic decision models in large-scale group decision environments; (5) The R-numbers DMAGDM method can be extended to new applications domains, such as energy exploration, medical systems, and financial investment, etc.

Appendices

Appendix 1: Linguistic term matrix

See Table 21.

Table 21 Linguistic terms matrix

Appendix 2: Triangular fuzzy number matrix

See Table 22.

Table 22 Triangular fuzzy number matrix

Appendix 3. The results of converting to R-numbers

See Table 23.

Table 23 The results of converting to R-numbers