A decision-making framework for China’s rare earth industry security evaluation by neutrosophic soft CoCoSo method

Abstract

The rare earth industry is a crucial strategic industry that is related to the national economy and national security. In the context of economic globalization, international competition is becoming increasingly fierce, and the rare earth industry is facing a more severe survival and development environment than ever before. Although China is the greatest world’s rare earth country in rare earth reserves, production, consumption and export volume, it is not a rare earth power. The rare earth industry has no right to speak in the international market. The comparative advantage is weakening and the security of rare earth industry appears. Therefore, studying the rare earth industry security has important theoretical and practical significance. When measuring the China’s rare earth industry security, the primary problem involves tremendous uncertainty. Neutrosophic soft set (NSS), depicted by the parameterized form of truth membership, falsity membership and indeterminacy membership, is a more serviceable pattern for capturing uncertainty. In this paper, five dimensions of rare earth industry security are identified and then prioritized against twelve different criteria relevant to structure, organization, layout, policy and ecological aspects of industry security. Then, the objective weight is computed by CRITIC (Criteria Importance Through Inter-criteria Correlation) method while the integrated weight is determined by concurrently revealing subjective weight and objective weight. Later, neutrosophic soft decision making method based CoCoSo (Combined Compromise Solution) is explored for settling the issue of low discrimination. Lastly, the feasibility and validity of the developed algorithm is verified by the issue of China’s rare earth industry security evaluation.

Keywords

Rare earth industry security neutrosophic soft set CoCoSo CRITIC

1 Introduction

Rare earth is a kind of rare metal possessing unique physical and chemical functions such as light, electricity, magnetism and catalysis [1 –4], so it has been successfully and widely used in petrochemical, metallurgy, machinery, light industry, agriculture, environmental protection and other fields of the national economy [5], and has become an indispensable resource for national production and development [6]. More importantly, the rare earth industry provides irreplaceable rare raw materials and key products for information technology, biotechnology, new material technology, aerospace, military and national defense industries [7], and is an important strategic industry that concerns national economic development and national security [8]. Entire countries in the world pay enormous importance to the security of the rare earth industry.

The world’s rare earth resources are abundant, but the distribution is extremely uneven. China’s rare earth resources are widely distributed, unique, and full of varieties. It is the country with the largest reserves of rare earth resources in the world. In addition, the Canada, Russia, Australia, United States, India and other countries are also rich in rare earth resources. China’s rare earth resources are primarily distributed in Guangdong, Inner Mongolia, Shandong, Sichuan, Jiangxi and Guizhou. After the founding of the China, its rare earth industry has made remarkable achievements and achieved the world’s largest reserves, production, consumption and export volume. However, in recent years, the security of rare earth industry has become increasingly prominent. Although China ranks among the four “first” in the world, it does not have the right to voice in the world’s rare earth market and precious rare earth resources are only sold at a low price. In 2016, China’s export of rare earth products showed an obvious increase in volume and decline in price. The export of rare earth products increased by 27.2% year on year, but the export amount decreased by 3.2% year on year. Due to the extensive production mode in the past, a series of problems occurred frequently, such as excessive consumption of total resources, waste of resources [9], extensive production, overcapacity and backward technological level [10].

In recent years, the international market environment of rare earth is constantly changing, and the international market pattern of rare earth is quietly changing [11]. In 2014, the WTO ruled that China’s measures to limit rare earth exports violated international trade regulations, and China was forced to cancel its rare earth export quotas, posing new challenges to the security of China’s rare earth industry and raising higher requirements for ensuring China’s rare earth industry security [12].

Considering the rare earth industry security could reduce the excessive consumption of total resources, waste of resources and extensive production, and promote the healthy and sustainable development [13]. On the other hand, ignoring the rare earth industry security can lead to some environmental and development problems [14] such as environmental pollution, that is motivated for evaluating and choosing the most reasonable annual rare earth industry security and adjust corresponding national strategy for reducing the loss. Therefore, selecting the optimal year of rare earth industry security under chosen conflicting criteria could help the rare earth decision makers (DMs) or experts in China for the long-term planning in their reserves, production, consumption and export volume. In this manner [15, 16], new decision making methods and their applications can be deemed as the important methods, which can effectively assess the year of rare earth industry security and rare earth industry security policies [17, 18].

Nevertheless, the growingly complex decision making environments and indecisive DM have trouble in displaying preference information with uncertain when dealing the above DM issues. Considering a suitable uncertain approach among the classical sets with its extensions is interesting for the decision making. Hence, one of the suitable uncertain model to deal with uncertainty is neutrosophic soft set (NSS), which was originally developed by Maji [19]. This model allows DMs to assign their preferences and judgments by some truth memberships, indeterminacy memberships and falsity memberships with parameterized form for a candidate alternative. There are some studies in the literature that have solved industry selection problems based on the NSS theory [20, 21].

However, the existing decision making methods [21–23 , 57] have low distinguishability in distinguishing the best alternative, which may be unreasonable for decision makers to choose best alternative(s). The CoCoSo (combined compromise solution) approach, initially introduced by Yazdani et al. [24], is an effective method to deal the uncertain information in a logical and commonsensible way. Hence, the motivation is to deal DM issues by presenting novel DM method out of deficiency. In this study, a new modified CoCoSo model based on the NSS is developed for ranking the years of rare earth industry security. In developed soft computing based-neutrosophic soft-CoCoSo (NS-CoCoSo) approach, a group of professional rare earth DMs is built to assess the years of rare earth industry security versus the chosen rare earth industry security criteria, in which their preference information are denoted by single-valued neutrosophic number. Hence, the DMs’ weights are offered for denoting the importance of each rare earth DM’s preference. Moreover, the CRITIC method is developed for achieving the rare earth industry security criteria’s weights to determine the relative importance of selected rare earth industry security criteria. Further, the integrated weight model is explored by unifying the objective weight (by CRITIC) and subjective weight.

To sum up, the main novelty of this paper is listed as follows:

A novel CoCoSo method is explored under neutrosophic soft environment.

Five dimensions of rare earth industry security are identified and then prioritized against 12 different criteria relevant to structure, organization, layout, policy and ecological aspects of industrial security.

The integrated weight model is based on CRITIC and the nonlinear weighted synthesis method that ponders both subjective evaluation information and objective evaluation information.

To promote our discussion, the rest of this article is organized as follows: In Section 2, the literature of NSS theory, and background of CoCoSo and CRITIC are reviewed. In Section 3, some basic concepts of NS and NSS are primitively reviewed. In Section 4, we develop a new neutrosophic soft DM method based on CRITIC with CoCoSo. In Section 5, a case study of China’s rare earth industry security is presented, and the sensitive analysis and comparison analysis are shown. In Section 6, we give some conclusions with potential directions.

2 Literature review

2.1 A review on the evolution of CoCoSo method

The inherent characteristics of decision making make it popular in real-life environment. The biggest controversy with decision making methods is that when the same set of data is employing in dealing the same problem, they may produce different ordering results and optimal alternatives.

Among the famous decision making methods, combined compromise solution (CoCoSo), originally developed by Yazdani [25], employs a comparability sequence by synthetically integrating exponentially weighted product and simple additive weighting. Because of its high degree of differentiation, it has been successfully applied to manufacturing technology assessment [24], personnel information assessment [26], sustainable supplier evaluation [27], solid waste disposal site selection [28], sustainability selection of OPEC countries [29]. Nevertheless, some diverse real issues with different decision preference information will appear during the process of decision making. To better deal diverse application environments, the CoCoSo method has been applied to different uncertain forms such as hesitant fuzzy linguistic term set [26] and interval neutrosophic set [28, 30].

However, it can be easily found that neutrosophic soft decision making method based on CoCoSo is not yet used in China’s rare earth industry security evaluation.

2.2 A review on the evolution of CRITIC method

Most applications related to decision making depend on the computation of the criteria weights, which are important for ranking and selection [31]. Generally speaking, the weight determining methods [32] are classified into subjective weight and objective weight, relying on whether the weight information directly or indirectly comes from the DMs and evaluation information, respectively. Generally, the evaluation of criteria weights depends on DMs’ opinions, that is, subjective weights. Since the subjective weight determining method represents the subjective evaluation of the DM, the ranking of the weight analysis result or the optimal alternative will be influenced by the experience and knowledge level of the DMs in the relevant field [33]. On the other hand, the objective weight determining method determines the weight based on the known preference information by solving the mathematical model. Such methods may be particularly useful in the absence of weighting information or inconsistent subjective weights obtained prior to make decision.

Among the famous objective weight determining methods, CRITIC (criteria importance through inter-criteria correlation), firstly developed by Diakoulaki et al. [34], integrates whole preference information included in the assessment criteria by the evaluation matrix. That is, the objective weights are obtained by quantifying the intrinsic information of each criterion from the DM. Its core process of calculating the objective weight involves the standard deviation of the criterion as well as the correlation between the criterion and other criteria. Up to now, it has been widely used in diverse application scenarios such as IPO performance evaluation [35], construction equipment selection [36], supply chain risk assessment [37], mini-grid business assessment [38] and attendance software evaluation [39].

2.3 A review on the evolution of NSS theory

The neutrosophic soft set (NSS) theory, combined with neutrosophic set [40, 41] and soft set [42], has been deemed as an effective and available tool by allowing the parameterized form of falsity membership, truth membership and indeterminacy membership for an alternative among the conflicting criteria under a set [43]. In this case, Maji [44] proposed a recognition strategy for dealing neutrosophic soft DM issue based on multiobserver input parameter method. Peng and Liu [22] developed three neutrosophic soft DM methods based on similarity measure, level soft set and EDAS. Moreover, the neutrosophic soft based choice value methods are employed in reference [45, 46] for dealing some complex decision making problems. Riaz and Hashmi [21] constructed the comparison table for obtaining the optimal alternative under neutrosophic soft environment. Dey et al. [23] investigated the neutrosophic soft DM based on information entropy and grey relational projection (GRP) method. Saeed et al. [47] extended the TOPSIS method under the neutrosophic soft environment to evaluate champion of FIFA 2018. Peng and Dai [48] presented an overview on the neutrosophic set by providing a systematic and clear perspective on the diverse concepts, tools and trends related to corresponding extensions. In this case, Guan et al. [49] and Jha et al. [50] prepared some forecasting models for dealing the stock trending analysis with neutrosophic soft information.

Considering the theory of NSS has certain reference value for solving group decision making (GDM) issues in the rare earth industry security field. Karaaslan [51] constructed a GDM method based on intersection of NSSs and compare matrix for dealing the candidates issue. Jana and Pal [52] employed some robust neutrosophic soft aggregation operators for integrating multiple DMs’ preference information and obtaining the optimal alternative. For aggregating an intrinsic stochastic randomness, Dong et al. [53] explored a stochastic GDM based on prospect theory under neutrosophic soft environment.

3 Preliminaries

In this section, some mathematical definitions, fundamental operations, and score function of neutrosophic set and neutrosophic soft set are offered to promote the understanding of the developed algorithm based CoCoSo.

Definition 1. [41] Let X be the universe of discourse, with a set of elements in X signified by x. The single-valued neutrosophic set (SVNS) A in X is portrayed by the truth-membership function T_A (x), the indeterminacy-membership function I_A (x), and the falsity-membership function F_N (x). Then an SVNS A can be defined as $A = {< x, T_{A} (x), I_{A} (x), F_{A} (x) > ∣ x \in X},$ (1) where T_A (x) , I_A (x) , F_A (x) ∈ [0, 1] for ∀x ∈ X. The sum of T_A (x) , I_A (x), and F_A (x) should meet the limiting condition 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3. For an SVNS A in X, the triplet (T_A (x) , I_A (x) , F_A (x)) is named as single-valued neutrosophic number (SVNN). For the sake of simplicity, we can employ x = (T_x, I_x, F_x) as an SVNN.

Definition 2. [41] Let x = (T_x, I_x, F_x) and y = (T_y, I_y, F_y) be two SVNNs, then operations can be denoted in the following.

(1) x^c = (F_x, 1 - I_x, T_x);

(2) x ⋃ y = (max {T_x, T_y} , min {I_x, I_y} , min {F_x, F_y});

(3) x ⋂ y = (min {T_x, T_y} , max {I_x, I_y} , max {F_x, F_y}).

To compare any two SVNNs, Peng and Dai [32] presented a score function for an SVNN x = (T_x, I_x, F_x) as follows: $s (x) = \frac{2}{3} + \frac{T_{x}}{3} - \frac{I_{x}}{3} - \frac{F_{x}}{3} .$ (2)

Definition 3. [19] A pair (ϝ , A) is called a neutrosophic soft set (NSS) over U, where ϝ is a mapping given by $ϝ : A \to \tilde{P} (U)$ . That is, the NSS is not a kind of set, but a parameterized family of subsets of the set U. For any parameter e ∈ A, ϝ (e) may be deemed as the set of e-approximate elements of the NSS (ϝ , A). Let ϝ (e) (x) be the membership value that object x holds parameter e, then ϝ (e) can be rewritten as an SVNS that ϝ (e) = {x/ϝ (e) (x) ∣ x ∈ U} = {x/(T_ϝ (e) (x) , I_ϝ (e) (x) , F_ϝ (e) (x)) ∣ x ∈ U}.

4 Novel neutrosophic soft DM method based on CoCoSo

4.1 The description of decision making issue

Assume that U = {x₁, x₂, ⋯ , x_m} and E = {ɛ₁, ɛ₂, ⋯ , ɛ_n} be a series of alternatives and parameters, respectively. The w = {w₁, w₂, ⋯ , w_n} is weight vector with limiting condition w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . The evaluation value of alternative x_i with respect to parameter ɛ_j is given by SVNN ϝ (ɛ_j) (x_i) = (T_ϝ (ɛ_j) (x_i) , I_ϝ (ɛ_j) (x_i) , F_ϝ (ɛ_j) (x_i)), which is shown in Table 1.

Table 1
Neutrosophic soft set (ϝ , E)

ɛ ₁ ɛ ₂ ⋯ ɛ _n

x ₁ ϝ (ɛ₁) (x₁) ϝ (ɛ₂) (x₁) ⋯ ϝ (ɛ_n) (x₁)

x ₂ ϝ (ɛ₁) (x₂) ϝ (ɛ₂) (x₂) ⋯ ϝ (ɛ_n) (x₂)

⋮ ⋮ ⋮ ⋱ ⋮

x _m ϝ (ɛ₁) (x_m) ϝ (ɛ₂) (x_m) ⋯ ϝ (ɛ_n) (x_m)

	ɛ ₁	ɛ ₂	⋯	ɛ _n
x ₁	ϝ (ɛ₁) (x₁)	ϝ (ɛ₂) (x₁)	⋯	ϝ (ɛ_n) (x₁)
x ₂	ϝ (ɛ₁) (x₂)	ϝ (ɛ₂) (x₂)	⋯	ϝ (ɛ_n) (x₂)
⋮	⋮	⋮	⋱	⋮
x _m	ϝ (ɛ₁) (x_m)	ϝ (ɛ₂) (x_m)	⋯	ϝ (ɛ_n) (x_m)

4.2 Method for determining integrated weight

4.2.1 Determine objective weight: CRITIC method

The CRITIC is a novel weight determining method to compute the objective weight in real decision making issue [34], which unifies the strength comparison of each criterion with the conflict between the criteria. The standard intensity comparison is served as the standard deviation, and the correlation coefficient is employed in calculating the conflict among them. To deal the evaluation information (SVNN) given by DMs, we introduce this method to neutrosophic soft environment.

Suppose that SVNN ϝ (ɛ_j) (x_i) represent the ith alternative with respect to jth parameter and $w_{j}^{o}$ represent the objective weight of jth parameter. C is a set of cost parameters and B is a set of benefit parameters.

In the following, the calculation procedure of objective weight based CRITIC under neutrosophic soft environment is shown as follows:

Step 1: Calculate score function ψ = (τ_ij) _m×n of each SVNN ϝ (ɛ_j) (x_i) as follows: $τ_{ij} = \frac{2}{3} + \frac{T_{ϝ} (ɛ_{j}) (x_{i})}{3} - \frac{I_{ϝ} (ɛ_{j}) (x_{i})}{3} - \frac{F_{ϝ} (ɛ_{j}) (x_{i})}{3} .$ (3)

Step 2: Switch the score matrix ϝ into normalized matrix $ψ^{'} = (τ_{ij}^{'})_{m \times n}$ as follows: $τ_{ij}^{'} = {\begin{matrix} \frac{τ_{ij} - τ_{j}^{-}}{τ_{j}^{+} - τ_{j}^{-}}, if j \in B, \\ \frac{τ_{j}^{+} - τ_{ij}}{τ_{j}^{+} - τ_{j}^{-}}, if j \in C, \end{matrix}$ (4) where $τ_{j}^{-} = \min_{i} τ_{ij}$ and $τ_{j}^{+} = \max_{i} τ_{ij}$

Step 3: Determine the criteria standard deviation as follows: $σ_{j} = \sqrt{\frac{\sum_{i = 1}^{m} (τ_{ij}^{'} - {\bar{τ}}_{j})^{2}}{m}},$ (5) where ${\bar{τ}}_{j} = \frac{\sum_{i = 1}^{m} τ_{ij}^{'}}{m}$ .

Step 4: Compute the correlation among criteria pairs as follows: $ρ_{jk} = \frac{\sum_{i = 1}^{m} (τ_{ij}^{'} - {\bar{τ}}_{j}) (τ_{ik}^{'} - {\bar{τ}}_{k})}{\sqrt{\sum_{i = 1}^{m} (τ_{ij}^{'} - {\bar{τ}}_{j})^{2} \sum_{i = 1}^{m} (τ_{ik}^{'} - {\bar{τ}}_{k})^{2}}} .$ (6)

Step 5: Determine the quantity of information as follows: $c_{j} = σ_{j} \sum_{k = 1}^{n} (1 - ρ_{jk}) .$ (7)

Step 6: Compute the objective weight of each criterion as follows: $ω_{j} = \frac{c_{j}}{\sum_{j = 1}^{n} c_{j}} .$ (8)

4.2.2 Determine integrated weight: nonlinear weighted synthesis method

Assume that the subjective weight, directly given by DMs, is w = {w₁, w₂, ⋯, w_n} with limiting condition $\sum_{j = 1}^{n} w_{j} = 1, 0 \leq w_{j} \leq 1$ . The objective weight, determined by Equation (8), is ω = {ω₁, ω₂, ⋯ , ω_n}, where $\sum_{j = 1}^{n} ω_{j} = 1, 0 \leq ω_{j} \leq 1$ .

Therefore, the integrated weight ϖ = {ϖ₁, ϖ₂, ⋯, ϖ_n} can be denoted as follows: $ϖ_{j} = \frac{w_{j} * ω_{j}}{\sum_{j = 1}^{n} w_{j} * ω_{j}},$ (9) where $\sum_{j = 1}^{n} ϖ_{j} = 1, 0 \leq ϖ_{j} \leq 1$ .

4.3 The neutrosophic soft CoCoSo method

The CoCoSo, firstly explored by Yazdani et al. [25], is based on an integrated exponentially weighted product and simple additive weighting, which can be deemed as a compendium of compromise solutions.

In order to dispose of the real decision making issue, an NSS-CoCoSo approach is listed as follows.

Algorithm 1
NSS-CoCoSo

1: Establish the linguistic DM matrix G = (G_ij) _m×n (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) by means of Table 2.

2: Transform the linguistic decision matrix into the neutrosophic soft set (ϝ , A).

3: Calculate the score function ψ = (τ_ij) _m×n of each SVNN ϝ (ɛ_j) (x_i) by Equation 3.

4: Switch the matrix ϝ = (τ_ij) _m×n into a standard matrix $ψ^{'} = (τ_{ij}^{'})_{m \times n}$ by Equation 4.

5: Calculate integrated weight ϖ by Equation 9.

6: Determine the entire of the weighted comparability sequence as S_i.

$S_{i} = \sum_{j = 1}^{n} ϖ_{j} * τ_{ij}^{'}$ (10)

7: Determine the total of the power weight of comparability sequence as P_i.

$P_{i} = \sum_{j = 1}^{n} (τ_{ij}^{'})^{ϖ_{j}}$ (11)

8: Compute the relative weights of alternatives by the below three aggregation strategies.

$k_{ia} = \frac{P_{i} + S_{i}}{\sum_{i = 1}^{m} (P_{i} + S_{i})},$ (12)

$k_{ib} = \frac{S_{i}}{\min_{i} S_{i}} + \frac{P_{i}}{\min_{i} P_{i}},$ (13)

$k_{ic} = \frac{λ S_{i} + (1 - λ) P_{i}}{λ \max_{i} S_{i} + (1 - λ) \max_{i} P_{i}}, 0 \leq λ \leq 1 .$ (14)

9: Compute the assessment value k_i as follows:

$k_{i} = \sqrt[3]{k_{ia} k_{ib} k_{ic}} + \frac{k_{ia} + k_{ib} + k_{ic}}{3}$ (15)

10: Rank alternatives by assessment value k_i (i = 1, 2, ⋯ , m).

Remark 1. The developed algorithm not only inherits the idea of the original CoCoSo method, but also employs the neutrosophic soft set model in expanding its connotation. It has three advantages as follows: (1) The evaluation preference information is offered by the quantitative linguistic terms, which is more in line with human decision-making preference. (2) The objective weight determining method based on CRITIC can availably reveal the richness of the preference information embedded in each criterion. Moreover, the developed integrated weight determining method based on nonlinear weighted synthesis can simultaneously show subjective preference information and objective preference information. (3) The second aggregation strategy (Equation (13) in Algorithm 1) can greatly increase the ultimate decision value of k_i and better improve the discrimination degree than some existing neutrosophic soft decision making methods.

5 A case study of China’s rare earth industry security

Rare earth was once hailed as an “industrial vitamin”, which shows the important role of the rare earth industry (REI) in the development of other industries. The REI security (REIS) refers to the state that the REI can resist and control risks, and its survival and development are not threatened under the open conditions [8]. As rare earth is a strategic scarce resource, the REIS is of great strategic significance to national economic security, steady economic development and even the overall national security, and plays a crucial role in military security, national defense security and social stability. In addition, the REIS should be considered at the strategic level. Given the key role and important position of the rare earth industry, it is necessary to have advanced strategic layout and arrangement to ensure the security of the industry.

5.1 Criteria for REIS

Proposing befitting evaluation criteria that responsibly measure rare earth industry security is significant and should be explicitly chosen on a case by case basis to determine [8]. According to the review on vast amounts of literature, the criteria for year of China’s REIS are identified. From the diverse of criteria that were considered during the preliminary and effectively screening, 12 criteria, covered all of structure, organization, layout, policy and ecological aspects [6], were ultimately taken into consideration. The chosen criteria have been shown in Table 3.

Table 2
Linguistic terms for evaluating the alternatives

Linguistic term Abbreviation SVNN

Extremely Low EL (0,1,1)

Very Low VL (0.1,0.8,0.9)

Low L (0.2,0.7,0.8)

Middle Low ML (0.3,0.6,0.7)

Below Middle BM (0.4,0.5,0.6)

Middle M (0.5,0.5,0.5)

Above Middle AM (0.6,0.4,0.5)

Middle High MH (0.7,0.3,0.4)

High H (0.8,0.2,0.3)

Very High VH (0.9,0.1,0.2)

Extremely High EH (1,0,0)

Linguistic term	Abbreviation	SVNN
Extremely Low	EL	(0,1,1)
Very Low	VL	(0.1,0.8,0.9)
Low	L	(0.2,0.7,0.8)
Middle Low	ML	(0.3,0.6,0.7)
Below Middle	BM	(0.4,0.5,0.6)
Middle	M	(0.5,0.5,0.5)
Above Middle	AM	(0.6,0.4,0.5)
Middle High	MH	(0.7,0.3,0.4)
High	H	(0.8,0.2,0.3)
Very High	VH	(0.9,0.1,0.2)
Extremely High	EH	(1,0,0)

Table 3

The criteria for China’s REIS

Criteria	Sub-criteria	Source
	Supply-demand ratio (ɛ₁)	[6, 12]
Industry structure security	The ratio of exports to total production (ɛ₂)	[6, 55]
	Degree of foreign capital control (ɛ₃)	[6, 55]
	Industrial concentration (ɛ₄)	[6, 12]
Industry organization security	Export price index (ɛ₅)	[6 , 55]
	Emerging sectors account for the proportion of rare earth consumption (ɛ₆)	[6]
	Proportion of rare earth reserves in the world (ɛ₇)	[6, 12]
Industry layout security	Working population (ɛ₈)	[6, 55]
	Rare earth patents (ɛ₉)	[6, 12]
Industry policy security	Effectiveness of industrial policy (ɛ₁₀)	[6]
	Environmental pollution status (ɛ₁₁)	[6, 55]
Industry ecological security	Environmental protection and governance (ɛ₁₂)	[6, 55]

5.2 Years of China’s REIS evaluation

How to rank years of China’s REIS and obtain an appropriate and secure year is a very important issue in future development. It can provide a model for the success or failure of China’s REI policy testing for the future China’s REIS.

In the process of evaluating the China’s REIS, it is necessary to give a reasonable evaluation system to ensure the availability of the assessment results. This section gives an evaluation criteria of years of China’s REIS as ɛ_j (j = 1, 2, ⋯ , 12), which is briefly described above.

The framework of neutrosophic soft decision making for years of China’s REIS evaluation is given in Fig. 1.

Fig. 1

The framework of neutrosophic soft decision making for years of China’s REIS evaluation.

Example 1. Suppose that there are five years of China’s REIS U = {x₁ (2014), x₂ (2015), x₃ (2016), x₄ (2017), x₅ (2018)} to be considered for further evaluation. The parameter set E = {ɛ₁, ɛ₂, ɛ₃, ɛ₄, ɛ₅, ɛ₆, ɛ₇, ɛ₈, ɛ₉, ɛ₁₀, ɛ₁₁, ɛ₁₂} is employed in assessing the years of China’s REIS by domain expert. The ɛ₁, ɛ₃ and ɛ₁₁ are cost parameters. Others are benefit parameters. The weight information is assigned as w = (0.02, 0.05, 0.02, 0.11, 0.11, 0.13, 0.05, 0.05, 0.11, 0.13, 0.12, 0.1). The assessments for years of China’s REIS arise from the rare earth experts and the linguistic matrix is shown in Table 4.

Table 4

The linguistic matrix in Example 1

	ɛ ₁	ɛ ₂	ɛ ₃	ɛ ₄	ɛ ₅	ɛ ₆	ɛ ₇	ɛ ₈	ɛ ₉	ɛ ₁₀	ɛ ₁₁	ɛ ₁₂
x ₁	ML	VH	BM	M	MH	MH	AM	BM	H	MH	VH	BM
x ₂	ML	H	BM	M	MH	H	AM	BM	H	MH	VH	BM
x ₃	ML	H	BM	BM	H	MH	AM	BM	H	MH	VH	ML
x ₄	ML	H	BM	AM	MH	H	AM	BM	H	MH	VH	BM
x ₅	L	H	AM	AM	H	H	MH	M	VH	H	H	M

In the following, the developed method (λ = 0.5) is employed in choosing the optimal year of China’s REIS under neutrosophic soft information.

Step 1. Obtain the neutrosophic soft set (ϝ , A) employed by Table 2, as shown in Table 4.

Step 2. Switch the linguistic matrix into the neutrosophic soft set, which is presented in Table 5.

Step 3. Calculate the score function ϝ = (τ_ij) _5×12 of each SVNN F (ɛ_j) (x_i) by Eq. (3).

$ϝ = (τ_{ij})_{5 \times 12} = (\begin{matrix} 0.3333 & 0.8667 & 0.4333 & 0.5000 & 0.6667 & 0.6667 & 0.5667 & 0.4333 & 0.7667 & 0.6667 & 0.8667 & 0.4333 \\ 0.3333 & 0.7667 & 0.4333 & 0.5000 & 0.6667 & 0.7667 & 0.5667 & 0.4333 & 0.7667 & 0.6667 & 0.8667 & 0.4333 \\ 0.3333 & 0.7667 & 0.4333 & 0.4333 & 0.7667 & 0.6667 & 0.5667 & 0.4333 & 0.7667 & 0.6667 & 0.8667 & 0.3333 \\ 0.3333 & 0.7667 & 0.4333 & 0.5667 & 0.6667 & 0.7667 & 0.5667 & 0.4333 & 0.7667 & 0.6667 & 0.8667 & 0.4333 \\ 0.2333 & 0.7667 & 0.5667 & 0.5667 & 0.7667 & 0.7667 & 0.6667 & 0.5000 & 0.8667 & 0.7667 & 0.7667 & 0.5000 \end{matrix}) .$

Step 4. Switch the matrix ϝ = (τ_ij) _5×12 into a standard matrix $ϝ^{'} = (τ_{ij}^{'})_{5 \times 12}$ by Eq. (4).

$ϝ^{'} = (τ_{ij})_{5 \times 12} = (\begin{matrix} 0 & 1 & 1 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.6 \\ 0 & 0 & 1 & 0.5 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0.6 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0.6 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}) .$

Step 5. Calculate integrated weight ϖ by Eq. (9) as follows:

ϖ₁ = 0.0139, ϖ₂ = 0.0928, ϖ₃ = 0.0529, ϖ₄ = 0.1009,

ϖ₅ = 0.1666, ϖ₆ = 0.1842, ϖ₇ = 0.0348, ϖ₈ = 0.0348,

ϖ₉ = 0.0767, ϖ₁₀ = 0.0906, ϖ₁₁ = 0.0836, ϖ₁₂ = 0.0682.

Step 6. Determine the total of the weighted comparability sequence for every year of China’s REIS as S_i:

S₁ = 0.2371, S₂ = 0.3284, S₃ = 0.2194, S₄ = 0.3788, S₅ = 0.8543 .

Step 7. Calculate entire power weight of comparability sequences for every year of China’s REIS as P_i:

P₁ = 3.8982, P₂ = 3.8982, P₃ = 2, P₄ = 3.9658, P₅ = 10 .

Step 8. Three aggregation strategies are employed in generating relative weights of other options, which are obtained by Eqs. (1)-(1) and presented as follows:

k_1a = 0.1604, k_2a = 0.1639, k_3a = 0.0861, k_4a = 0.1685, k_5a = 0.4210 .

k_1b = 3.0294, k_2b = 3.4455, k_3b = 2.0000, k_4b = 3.7092, k_5b = 8.8930 .

k_1c = 0.3810, k_2c = 0.3894, k_3c = 0.2045, k_4c = 0.4003, k_5c = 1.0000 .

Step 9. Calculate assessment value k_i by Eq. (1) as follows:

k₁ = 1.7602, k₂ = 1.9366, k₃ = 1.0913, k₄ = 2.0561, k₅ = 4.9908 .

Step 10. Rank five years of China’s REIS by the decreasing values of assessment value as follows: x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃.

Consequently, we can conclude that the x₅ (2018) is optimal year of China’s REIS for selection.

5.2.1 The analysis of weight information

In the following, a comparison among the original weight, the objective weight (Dey et al. [23]), two existing integrated weights (Peng and Liu [22] and Dong et al. [53]) and the developed integrated weight is presented for illustrating the weight information problem.

According to Fig. 2, the developed integrated weight determining method can concurrently reflect the objective weight information and subjective weight information to some extent.

Fig. 2

The comparison of the weight information.

(1) For objective weight (Dey et al. [23]), the weight values have significant difference compared with the original weight given by DM. In other words, a powerful generation gap has existed in the original weight and objective weight, which may result in different orders or optimal alternatives during the procedure of decision making. Consequently, to reduce uncertainty and increase certainty, it is necessary to consider both of them. Ulteriorly, the integrated weight is a good choice for efficaciously alleviating conflicts. Moreover, it is easily found that the weight value of w₈ is equal to 0. Obviously, it is extremely unreasonable. The main reason is that the indeterminacy memberships of all years of China’s REIS are 0.5.

(2) For integrated weight (Peng and Liu [22]), it is almost the same as the original weight. In other words, it fails to reflect the objective information transmitted by DMs.

(3) For integrated weight (Dong et al. [53]), although it can reflect both the objective information and subjective information to some extent, it also encountered the situation of Peng and Liu [22] that the weight value of w₈ is equal to 0.

5.2.2 Effect of the parameter λ in the developed method

Based on the presented model, it can be easily found that the impact of the ultimate ranking is determined by the parameter λ that stems from CoCoSo approach.

According to Fig. 3, no matter how the value of the parameter λ changes, we can see that the decision values of year of China’s REIS x₅ are still 4.9908. The main reason is that the effect of the parameterλ is limited by the Eq. (1), which S₅ and P₅ are the maximal of S_i and P_i in corresponding year of China’s REIS x₅, respectively. That is, the homologous values of Eq. (1) is 1. In addition, we can conclude that the decision values of years of China’s REIS (x₁, x₂) decrease with the increasing of the parameter λ and years of China’s REIS (x₃, x₄) increase with the increasing of the parameter λ. The explanation of the increasing and decreasing trend of years of China’s rare earth industry security (x₂, x₃, x₄, x₅) can be given in below. We can find that the main trend comes from the k_ic in Eq. (1) and the relation P_i > S_i in Example 1. Hence, we can proceed to have a deeper discussion in theoretical level.

Fig. 3

The changing trend of diverse λ in Algorithm 1.

We can easily achieved the expressions as follows:

$k_{1 c} = \frac{λ S_{1} + (1 - λ) P_{1}}{λ \max_{i} S_{i} + (1 - λ) \max_{i} P_{i}} = \frac{P_{1} + λ (S_{1} - P_{1})}{0.8543 λ + 10 (1 - λ)} = \frac{3.8982 - 3.6611 λ}{10 - 9.1457 λ}$ ,

$k_{2 c} = \frac{λ S_{2} + (1 - λ) P_{2}}{λ \max_{i} S_{i} + (1 - λ) \max_{i} P_{i}} = \frac{P_{2} + λ (S_{2} - P_{2})}{0.8543 λ + 10 (1 - λ)} = \frac{3.8982 - 3.5698 λ}{10 - 9.1457 λ}$ ,

$k_{3 c} = \frac{λ S_{3} + (1 - λ) P_{3}}{λ \max_{i} S_{i} + (1 - λ) \max_{i} P_{i}} = \frac{P_{3} + λ (S_{3} - P_{3})}{0.8543 λ + 10 (1 - λ)} = \frac{2 - 1.7806 λ}{10 - 9.1457 λ}$ ,

$k_{4 c} = \frac{λ S_{4} + (1 - λ) P_{4}}{λ \max_{i} S_{i} + (1 - λ) \max_{i} P_{i}} = \frac{P_{4} + λ (S_{4} - P_{4})}{0.8543 λ + 10 (1 - λ)} = \frac{3.9658 - 3.587 λ}{10 - 9.1457 λ} .$

Ulteriorly, we can have the first partial derivative of k_1c, k_2c, k_3c, k_4c with the parameter λ, respectively,

$\frac{\partial k_{1 c}}{\partial λ} = \frac{- 0.9592}{(10 - 9.1457 λ)^{2}} < 0, \frac{\partial k_{2 c}}{\partial λ} = \frac{- 0.0462}{(10 - 9.1457 λ)^{2}} < 0,$

$\frac{\partial k_{3 c}}{\partial λ} = \frac{0.4854}{(10 - 9.1457 λ)^{2}} > 0, \frac{\partial k_{4 c}}{\partial λ} = \frac{0.4}{(10 - 9.1457 λ)^{2}} > 0$ .

Consequently, we can have conclusion that decision values of years of China’s REIS (x₁, x₂) decrease with the increasing of the parameter λ and years of China’s rare earth industry security (x₃, x₄) increase with the increasing of the parameter λ.

5.2.3 Effectiveness test

For illustrating the effectiveness of the explored method, it has been tested on certain test criteria [56].

Test Criterion 1: When the non-optimal alternative is replaced by another worst alternative, the effective decision making method should not alter the ordering of the optimal alternative (assuming that the relative importance of each decision criterion keeps the same).

Test Criterion 2: The given decision making method should keep to the transitive property.

Test Criterion 3: As the same DM problem and uniform DM method, the novel entire ordering of the alternatives should be the same as the original total ranking of the undecomposed issue when the decision making problem is decomposed into smaller problems after combination.

Next, we have employed three test criteria on our proposed neutrosophic soft DM method based on CoCoSo.

1) Effectiveness test by test Criterion 1

In this test, if we switch the falsity degrees and truth degrees with opposite indeterminacy degrees of years of China’s rare earth industry security x₁, x₂, x₄ (non-optimal) and x₃ (worse) in Table 5, then the transformed table is presented in Table 6. Based on above information, the developed CoCoSo method has been employed, the optimal year of China’s rare earth industry security remain x₅. Consequently, the developed algorithm is resultful under the test criterion 1.

Table 5
The neutrosophic soft matrix in Example 1

ɛ ₁ ɛ ₂ ɛ ₃ ɛ ₄ ɛ ₅ ɛ ₆

x ₁ (0.3,0.6,0.7) (0.9,0.1,0.2) (0.4,0.5,0.6) (0.5,0.5,0.5) (0.7,0.3,0.4) (0.7,0.3,0.4)

x ₂ (0.3,0.6,0.7) (0.8,0.2,0.3) (0.4,0.5,0.6) (0.5,0.5,0.5) (0.7,0.3,0.4) (0.8,0.2,0.3)

x ₃ (0.3,0.6,0.7) (0.8,0.2,0.3) (0.4,0.5,0.6) (0.4,0.5,0.6) (0.8,0.2,0.3) (0.7,0.3,0.4)

x ₄ (0.3,0.6,0.7) (0.8,0.2,0.3) (0.4,0.5,0.6) (0.6,0.4,0.5) (0.7,0.3,0.4) (0.8,0.2,0.3)

x ₅ (0.2,0.7,0.8) (0.8,0.2,0.3) (0.6,0.4,0.5) (0.6,0.4,0.5) (0.8,0.2,0.3) (0.8,0.2,0.3)

ɛ ₇ ɛ ₈ ɛ ₉ ɛ ₁₀ ɛ ₁₁ ɛ ₁₂

x ₁ (0.6,0.4,0.5) (0.4,0.5,0.6) (0.8,0.2,0.3) (0.7,0.3,0.4) (0.9,0.1,0.2) (0.4,0.5,0.6)

x ₂ (0.6,0.4,0.5) (0.4,0.5,0.6) (0.8,0.2,0.3) (0.7,0.3,0.4) (0.9,0.1,0.2) (0.4,0.5,0.6)

x ₃ (0.6,0.4,0.5) (0.4,0.5,0.6) (0.8,0.2,0.3) (0.7,0.3,0.4) (0.9,0.1,0.2) (0.3,0.6,0.7)

x ₄ (0.6,0.4,0.5) (0.4,0.5,0.6) (0.8,0.2,0.3) (0.7,0.3,0.4) (0.9,0.1,0.2) (0.4,0.5,0.6)

x ₅ (0.7,0.3,0.4) (0.5,0.5,0.5) (0.9,0.1,0.2) (0.8,0.2,0.3) (0.8,0.2,0.3) (0.5,0.5,0.5)

	ɛ ₁	ɛ ₂	ɛ ₃	ɛ ₄	ɛ ₅	ɛ ₆
x ₁	(0.3,0.6,0.7)	(0.9,0.1,0.2)	(0.4,0.5,0.6)	(0.5,0.5,0.5)	(0.7,0.3,0.4)	(0.7,0.3,0.4)
x ₂	(0.3,0.6,0.7)	(0.8,0.2,0.3)	(0.4,0.5,0.6)	(0.5,0.5,0.5)	(0.7,0.3,0.4)	(0.8,0.2,0.3)
x ₃	(0.3,0.6,0.7)	(0.8,0.2,0.3)	(0.4,0.5,0.6)	(0.4,0.5,0.6)	(0.8,0.2,0.3)	(0.7,0.3,0.4)
x ₄	(0.3,0.6,0.7)	(0.8,0.2,0.3)	(0.4,0.5,0.6)	(0.6,0.4,0.5)	(0.7,0.3,0.4)	(0.8,0.2,0.3)
x ₅	(0.2,0.7,0.8)	(0.8,0.2,0.3)	(0.6,0.4,0.5)	(0.6,0.4,0.5)	(0.8,0.2,0.3)	(0.8,0.2,0.3)
	ɛ ₇	ɛ ₈	ɛ ₉	ɛ ₁₀	ɛ ₁₁	ɛ ₁₂
x ₁	(0.6,0.4,0.5)	(0.4,0.5,0.6)	(0.8,0.2,0.3)	(0.7,0.3,0.4)	(0.9,0.1,0.2)	(0.4,0.5,0.6)
x ₂	(0.6,0.4,0.5)	(0.4,0.5,0.6)	(0.8,0.2,0.3)	(0.7,0.3,0.4)	(0.9,0.1,0.2)	(0.4,0.5,0.6)
x ₃	(0.6,0.4,0.5)	(0.4,0.5,0.6)	(0.8,0.2,0.3)	(0.7,0.3,0.4)	(0.9,0.1,0.2)	(0.3,0.6,0.7)
x ₄	(0.6,0.4,0.5)	(0.4,0.5,0.6)	(0.8,0.2,0.3)	(0.7,0.3,0.4)	(0.9,0.1,0.2)	(0.4,0.5,0.6)
x ₅	(0.7,0.3,0.4)	(0.5,0.5,0.5)	(0.9,0.1,0.2)	(0.8,0.2,0.3)	(0.8,0.2,0.3)	(0.5,0.5,0.5)

Table 6

The transformed neutrosophic soft matrix

	ɛ ₁	ɛ ₂	ɛ ₃	ɛ ₄	ɛ ₅	ɛ ₆
x ₁	(0.7,0.4,0.3)	(0.2,0.9,0.9)	(0.6,0.5,0.4)	(0.5,0.5,0.5)	(0.4,0.7,0.7)	(0.4,0.7,0.7)
x ₂	(0.7,0.4,0.3)	(0.3,0.8,0.8)	(0.6,0.5,0.4)	(0.5,0.5,0.5)	(0.4,0.7,0.7)	(0.3,0.8,0.8)
x ₃	(0.7,0.4,0.3)	(0.3,0.8,0.8)	(0.6,0.5,0.4)	(0.6,0.5,0.4)	(0.3,0.8,0.8)	(0.4,0.7,0.7)
x ₄	(0.7,0.4,0.3)	(0.3,0.8,0.8)	(0.6,0.5,0.4)	(0.5,0.6,0.6)	(0.4,0.7,0.7)	(0.3,0.8,0.8)
x ₅	(0.2,0.7,0.8)	(0.8,0.2,0.3)	(0.6,0.4,0.5)	(0.6,0.4,0.5)	(0.8,0.2,0.3)	(0.8,0.2,0.3)
	ɛ ₇	ɛ ₈	ɛ ₉	ɛ ₁₀	ɛ ₁₁	ɛ ₁₂
x ₁	(0.5,0.6,0.6)	(0.6,0.5,0.4)	(0.3,0.8,0.8)	(0.4,0.7,0.7)	(0.2,0.9,0.9)	(0.6,0.5,0.4)
x ₂	(0.5,0.6,0.6)	(0.6,0.5,0.4)	(0.3,0.8,0.8)	(0.4,0.7,0.7)	(0.2,0.9,0.9)	(0.6,0.5,0.4)
x ₃	(0.5,0.6,0.6)	(0.6,0.5,0.4)	(0.3,0.8,0.8)	(0.4,0.7,0.7)	(0.2,0.9,0.9)	(0.7,0.4,0.3)
x ₄	(0.5,0.6,0.6)	(0.6,0.5,0.4)	(0.3,0.8,0.8)	(0.4,0.7,0.7)	(0.2,0.9,0.9)	(0.6,0.5,0.4)
x ₅	(0.7,0.3,0.4)	(0.5,0.5,0.5)	(0.9,0.1,0.2)	(0.8,0.2,0.3)	(0.8,0.2,0.3)	(0.5,0.5,0.5)

2) Effectiveness test by test Criterion 2

In this test, we resolve the given issue into 10 sub-issues ({x₄, x₅} , {x₂, x₄} , {x₁, x₂} , {x₁, x₃} , {x₂, x₅}, {x₁, x₄} , {x₂, x₃} , {x₁, x₅} , {x₃, x₅} , {x₃, x₄}). Moreover, we can obtain the ranking of 10 sub-issues when the same decision making algorithm based on CoCoSo has been employed. The ranking results are x₅ ≻ x₄, x₄ ≻ x₂, x₂ ≻ x₁, x₁ ≻ x₃, x₅ ≻ x₂, x₄ ≻ x₁, x₂ ≻ x₃, x₅ ≻ x₁, x₅ ≻ x₃ and x₄ ≻ x₃, which is in line with the total ranking x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃. In other words, it keeps to the transitive property. Consequently, the presented algorithm is resultful under the test criterion 2.

3) Effectiveness test by test Criterion 3

Based on the transitive property (Criterion 2), we can achieve the overall ranking of the years of China’s REIS is x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃, which is equal that of the original order. Consequently, the presented method is resultful under the test criterion 3.

5.3 Comparison analysis with some existing DM methods

To further verify the effectiveness of the proposed method, some illustrative examples are explored by using some existing neutrosophic soft DM methods (Dey et al. [23], Peng and Liu [22], Riaz et al. [21] and Zavadskas et al. [57]).

For a better comparison with some selected existing decision making methods [21 –23], we set σ = 0.5 in [23], q = 2, ξ = 0.5 in [22]. Moreover, the weight strategies are relied on their original use-pattern (subjective weight [57], objective weight [23], combined weight [22] and without weight [21]). The comparison results are given in Fig. 4 and Table 7 by employing the Example 1.

Fig. 4

The comparison with some existing methods.

Table 7

Ranking results from diverse decision making methods from Example 1

Methods	Ranking	Optimal alternative
Dey et al. [23]	x₅ ≻ x₁ ≻ x₄ ≻ x₂ ≻ x₃	x ₅
Peng and Liu (EDAS) [22]	x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃	x ₅
Peng and Liu (Similarity measure) [22]	x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃	x ₅
Peng and Liu (Level soft set) [22]	x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃	x ₅
Riaz et al. [21]	x₅ ≻ {x₁, x₂} ≻ x₄ ≻ x₃	x ₅
Zavadskas et al. (WASPAS) [57]	x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃	x ₅
CoCoSo (Proposed)	x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃	x ₅

From Fig. 4, we can find that the some existing decision making methods [22 , 57] can distinguish the optimal alternative and obtain the corresponding diacritical ranking to some extent. Nevertheless, they have low discrimination, which almost tend to a line that doesn’t fluctuate much. Moreover, the decision results of all alternatives of one existing decision making method [21] have huge differentiation. In other words, it has good discrimination degree, which fit the proposed algorithm.

When it comes to specific ranking (Table 7), we will find that the ranking differences [21, 23] can be put down to two reasons. One is that both approaches are approximate comparisons [21] of the evaluation values, and the original evaluation values are not used to the greatest extent. The other is that the special weight information (w₈ = 0) obtained by entropy method [23] can directly result in diverse final rankings. If we take original weight or proposed integrated weight, the final ranking results are consistent with Peng and Liu [22] and the proposed approach. Moreover, the proposed CoCoSo method has better discrimination compared with [22].

Based on the above analysis, the primary preponderance of the developed algorithm can be concluded as three parts.

This paper firstly uses neutrosophic soft sets for dealing years of China’s rare earth industry security evaluation, which cannot be employed in the existing neutrosophic soft decision making approaches.

The developed integrated weight can efficaciously and simultaneously reflect the subjective weight preference information given by experts and the objective weight preference information carried by the decision data.

The developed method has better discrimination compared with some existing neutrosophic soft decision making methods.

6 Conclusion

This paper explored the revised CoCoSo method under the neutrosophic soft environment and proposed the NSS-CoCoSo method for handling the issue of years of China’s REIS evaluation. The key innovations and contributions can be concluded as follows: Five dimensions of REIS are identified and then prioritized against twelve different criteria relevant to structure, organization, layout, policy and ecological aspects of industrial security; The integrated weight model is developed, which can efficaciously and simultaneously reflect subjective factor and objective factor (CRITIC); The new neutrosophic soft DM method based on CoCoSo is presented, which can possess a strong discriminating power.

However, the proposed decision making method consumes a certain amount of time complexity and space complexity, and lacks the self-adjusting weight mechanism.

In the future, we have the new thoughts as follows: The developed neutrosophic soft CoCoSo method just deal the problem of single DM. We will take the multiple DMs into consideration, i.e., group decision making (GDM). The splendid CoCoSo method will be employed in managing the issue of years of China’s rare earth industry security evaluation under different uncertain environment [58 –79] The integrated weight under neutrosophic soft environment is an interesting topic. The concept of self-paced learning [80] will be brought into the integrated weight and may produce better decision results. Moreover, the application of medical diagnosis [78] also will adopt the thought of integrated weight.

Footnotes

Acknowledgments

Our work is sponsored by the National Natural Science Foundation of China (No. 61806213), Humanities and Social Science Fund of Ministry of Education of China (No. 18YJCZH054), Natural Science Foundation of Guangdong Province (Nos. 2018A030307033, 2018A0303130274), Social Science Foundation of Guangdong Province (No. GD18CFX06), Special Innovation Projects of Universities in Guangdong Province (No. KTSCX205).

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A decision-making framework for China’s rare earth industry security evaluation by neutrosophic soft CoCoSo method

Abstract

Keywords

1 Introduction

2 Literature review

2.1 A review on the evolution of CoCoSo method

2.2 A review on the evolution of CRITIC method

2.3 A review on the evolution of NSS theory

3 Preliminaries

4.1 The description of decision making issue

Table 1 Neutrosophic soft set (ϝ , E) ɛ 1 ɛ 2 ⋯ ɛ n x 1 ϝ (ɛ1) (x1) ϝ (ɛ2) (x1) ⋯ ϝ (ɛ n ) (x1) x 2 ϝ (ɛ1) (x2) ϝ (ɛ2) (x2) ⋯ ϝ (ɛ n ) (x2) ⋮ ⋮ ⋮ ⋱ ⋮ x m ϝ (ɛ1) (x m ) ϝ (ɛ2) (x m ) ⋯ ϝ (ɛ n ) (x m )

4.2.1 Determine objective weight: CRITIC method

5.1 Criteria for REIS

Footnotes

Acknowledgments

References

Table 1
Neutrosophic soft set (ϝ , E)

ɛ ₁ ɛ ₂ ⋯ ɛ _n

x ₁ ϝ (ɛ₁) (x₁) ϝ (ɛ₂) (x₁) ⋯ ϝ (ɛ_n) (x₁)

x ₂ ϝ (ɛ₁) (x₂) ϝ (ɛ₂) (x₂) ⋯ ϝ (ɛ_n) (x₂)

⋮ ⋮ ⋮ ⋱ ⋮

x _m ϝ (ɛ₁) (x_m) ϝ (ɛ₂) (x_m) ⋯ ϝ (ɛ_n) (x_m)