Linguistic neutrosophic Hamacher aggregation operators and the application in evaluating land reclamation schemes for mines

Land reclamation has become a significant way for the improvement of ecological environment in mining areas. When selecting the optimal land reclamation scheme, LNNs (linguistic neutrosophic numbers) are suitable to describe the complex fuzzy evaluation information through linguistic truth, indeterminacy and falsity membership degrees. Furthermore, the Hamacher aggregation operators are good tools to handle multi-criteria decision making problems. Accordingly, the aim of this paper is to extend Hamacher aggregation operators with LNNs and then build a decision making framework for evaluating land reclamation schemes in mining areas. First, new operational laws of LNNs based on Hamacher t-norm and t-conorm are defined. Then, several linguistic neutrosophic Hamacher aggregation operators, including the linguistic neutrosophic Hamacher weighted mean aggregation operators and linguistic neutrosophic Hamacher hybrid weighted mean aggregation operators are developed. Meanwhile, their desirable properties are proved. Thereafter, a method for decision making with linguistic neutrosophic information based on these operators is proposed to deal with complex decision problems. At last, the validity of this method is confirmed by an illustrative example of evaluating the land reclamation schemes in mining areas. In addition, the impact of the parameter in extended Hamacher aggregation operators is discussed. The merits of the proposed method are also highlighted by comparing with other decision making methods. The results show that the proposed linguistic neutrosophic Hamacher aggregation operators have great flexibility and advantages, and can provide powerful ways for the evaluation of land reclamation schemes.


Introduction
Over years, the demand for mineral resources increases sharply due to the rapid industrial developments and global population growth [1,2]. At the same time, the large-scale exploitation of mineral resources has brought a lot of issues to the surrounding environment [3]. PLOS  Specifically, a large number of land resources are polluted and destroyed [4]. In order to protect ecological environment and achieve sustainable development of mining industry, it is urgent and vital to restore such damaged lands after the completion of mining operations. Mine land reclamation is the process of restoring the land that was destroyed during the mining exploitation to some degree [5]. On account of big differences in various mining area, a specific land reclamation scheme is required for a certain mine. Thus, numbers of multi-criteria decision making methods have been adopted to assess the land reclamation schemes in mine areas, like the AHP (analytical hierarchy process) [6], linear programming model [7], TOPSIS (technique for order preference by similarity to an ideal solution) [8], limit comprehensive conditions approach [9] and fuzzy comprehensive evaluation method [10]. However, the assessment information is expressed by real numbers in all these methods mentioned above. These crisp numbers cannot describe fuzziness and uncertainty in the process of evaluation [11]. Consequently, Zadeh [12] presented the fuzzy set to deal with fuzzy and uncertain information. Since then, numerous of fuzzy decision making methods based on fuzzy set and its extensions have been developed and widely applied in various fields [13,14,15,16,17]. In order to make comprehensive appraisal of mine land reclamation, Zhou et al. [18] proposed a modified VIKOR (vlsekriterijumska optimizacija i kompromisno resenje) method with triangle fuzzy numbers. On the other hand, considering that people often use linguistic phrases such as "bad" and "good" in the process of making assessments, the concept of linguistic term sets was put forward [19]. While there are still some imperfections of the representation of linguistic term sets. There is a hypothesis that the membership degree of a certain linguistic term is one, and the indeterminacy and inconsistency cannot be depicted by linguistic term sets [20,21].
Consequently, Fang and Ye [22] proposed LNNs (linguistic neutrosophic numbers) to deal with consistent, indeterminate and inconsistent linguistic information. After that, researchers paid great attentions to LNNs and suggested many linguistic neutrosophic decision making methods. Shi and Ye [23] presented a multi-attribute group decision making method based on the cosine similarity of LNNs; Liang et al. [24] adopted the TOPSIS approach with LNNs to assess the investment risk of metallic mines; Fan et al. [25] combined LNNs with Bonferroni averaging operators to handle linguistic neutrosophic information; Liu et al. [26] advised linguistic neutrosophic power Heronian averaging operators to aggregate LNNs; Liu and You [27] developed some Hamy averaging operators with LNNs to solve group decision problems; Liang et al. [28] extended the MULTIMOORA (multi-objective optimization by ratio analysis plus the full multiplicative form) method under linguistic neutrosophic circumstance.
The information aggregation operators have been always regarded as important and interesting topics in decision making areas. Garg and Nancy [29,30] defined some novel aggregation operators for neutrosophic numbers; Liu and Qin [31] extended the Maclaurin symmetric mean operators with LIFNs (linguistic intuitionistic fuzzy numbers) and discussed their application in multi-criteria decision making fields; Garg and Nancy [32] developed the priority aggregation operators to aggregate linguistic single-valued neutrosophic information; Garg and Kumar [33] proposed several aggregation operators of LIFNs to overcome the shortcomings of the existing operators; Garg and Nancy [34] discussed the prioritized Muirhead mean aggregation operator under neutrosophic set environment; Liu et al. [35] recommended the linguistic intuitionistic fuzzy partitioned Heronian means to tackle decision making issues.
These mentioned-above operators are all on the basis of the same type of operations, namely, the algebraic operational rules. Generally, another useful type of operations, the Hamacher t-norm and t-conorm, can be also utilized to model the intersection and union of various fuzzy sets [36]. A great number of extended Hamacher aggregation operators based on different fuzzy number situations have been developed [37,38,39]. Recently, Garg et al. [40] extended Hamacher t-norm and t-conorm with interval intuitionistic fuzzy numbers to deal with multi-criteria decision making problems; Gao et al. [41] presented some Hamacher prioritized operators for aggregating dual hesitant bipolar fuzzy numbers; Wu et al. [42] combined Hamacher aggregation operators with single-valued neutrosophic 2-tuple linguistic numbers; Li et al. [43] defined the multivalued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher operators; Tang and Meng [44] discussed the Hamacher aggregation operators under linguistic intuitionistic fuzzy environment. Nevertheless, the Hamacher aggregation operators have not been extended to LNNs so far.
Overall, the motivations of this paper can be summarized as follows.
1. The evaluation of land reclamation schemes is a complicated decision making problem with many uncertain factors. Nevertheless, evaluation information is just described by real numbers in the existing reclamation schemes selection methods, which may lead to information loss or distortion. Thus, for land reclamation schemes selection, finding a pertinent way to describe fuzzy or uncertain evaluation information is essential.
2. As a popular set, LNNs show great performance in expressing consistent, indeterminate and inconsistent evaluation information. Hence, LNNs may be a good choice to depict fuzzy assessment values for the selection of land reclamation schemes.
3. Considering that using information aggregation operators is a simple but powerful way to solve majority of decision making problems, this paper tends to define new aggregation operators under linguistic neutrosophic environment to address complex land reclamation schemes selection issues.
Consequently, a linguistic neutrosophic decision making method based on Hamacher aggregation operators is proposed to select the optimal land reclamation scheme in this paper. Compared with other methods, the advantages of the proposed work are three-fold. First, the proposed approach expresses decision making information using LNNs, which can describe the linguistic evaluation information more adequately. Second, the Hamacher aggregation operators with one parameter are more flexible and powerful in the information fusion and decision making process. Third, exploring some Hamacher aggregation operators for LNNs and applying them to model practical decision making problems are necessary and meaningful. The combination of LNNs and Hamacher aggregation operators can be much benefit for coping with complex decision making problems, like the selection of land reclamation schemes.
The novelties and contributions of this paper are listed as follows.
1. New operational rules, the Hamacher operational rules of LNNs are proposed. Based on them, the Hamacher aggregation operators are extended to LNNs and major properties are justified. They are general and flexible family of continuous triangular norms, and can be regarded as the generalization of Algebraic and Einstein t-norm and t-conorm.

A decision making model based on the linguistic neutrosophic Hamacher aggregation
operators is constructed to tackle the problem of assessing land reclamation schemes. Like algebraic operators, Hamacher aggregation operators have the function of smooth approximation as well. Moreover, LNNs are good at expressing consistent and inconsistent qualitative information.
3. In the proposed method, not only the weights of criteria, but also the weights of ordered positions are under consideration.
4. The evaluation criteria of land reclamation schemes are recognized, and a case is studied to explain the calculation process and verify the practicality of our method. Furthermore, the features and strengths of the proposed method are demonstrated through sensitivity and comparison analyses.
The rest part of this paper is allocated as follows. Section 2 recalls some basic concepts and notations, such as the linguistic term set, LNNs, t-norm and t-conorm. Besides, the linguistic neutrosophic Hamacher operational laws are firstly defined. In Section 3, two main types of linguistic neutrosophic Hamacher operators are presented and some desirable properties are discussed. Thereafter, a method for decision making with linguistic neutrosophic information is recommended in Section 4. Section 5 gives an illustrative example of appraising land reclamation schemes in mine areas after the evaluation criteria system is constructed. In Section 6, the sensitivity and comparison analyses are respectively undertaken. Finally, concluding remarks are provided.

Preliminaries
In this section, general concepts of LNNs, t-norm and t-conorm are reviewed, and then new operational laws of LNNs based on Hamacher t-norm and t-conorm are introduced.

General concepts
Definition 1. [19] Suppose there are a group of linguistic variables t i (i = 0, 1,� � �,2h). For all i, j = 0, 1,� � �,2h, the followings are true: (1) When i < j, then t i < t j ; (2) when i = j, then t i = t j ; (3) when i > j, then t i > t j . And the operational rules of linguistic variables are: t i � t j = t i+j and r � t i = t ri (r � 0). When a collection of linguistic values exist, a discrete linguistic term set is denoted as T � = {t i |i = 0, 1,� � �,2h} (h > 0), and a continuous linguistic term set is expressed as Definition 2. [22] A LNN (linguistic neutrosophic number) is denoted as a ¼ ðt T a ; t I a ; t F a Þ, where t T a , t I a and t F a are three independent linguistic terms, which respectively represent true, hesitant and false membership degrees. Definition 3. [22] If T = {t i | i 2 [0,2s]} is a continuous linguistic term set and a ¼ ðt T a ; t I a ; t F a Þ is a LNN, then the score function of a is U(a) = (4s + T a − I a − F a )/6s, and the accuracy function of a is V(a) = (T a − F a )/2s. Definition 4. [45] Let b ¼ ðl T b ; l I b ; l F b Þ and c ¼ ðl T c ; l I c ; l F c Þ two LNNs, their order relationship can be described as follows: . Definition 5. [46]. The t-norm and t-conorm are the most general families of binary functions, which contain all types of the particular operators. Three special cases of t-norm and tconorm are itemized as follows: 1. Algebraic t-norm and t-conorm: 2. Einstein t-norm and t-conorm: 3. Hamacher t-norm and t-conorm: Note that when λ = 1, the Hamacher t-norm and t-conorm are respectively degenerated into Algebraic t-norm and t-conorm; and when λ = 2, the Hamacher t-norm and t-conorm are respectively degenerated into Einstein t-norm and t-conorm.

Operational laws of linguistic neutrosophic numbers based on Hamacher t-norm and t-conorm
Assume a conversion function is gðt i Þ ¼ i 2s and the corresponding converse function is g −1 (i) = t 2s×i , then the operational laws of them based on Hamacher t-norm and t-conorm are defined as follows: b � c ¼ ðg À 1 ðS Hl ðgðt T b Þ; gðt T c ÞÞÞ; g À 1 ðT Hl ðgðt I b Þ; gðt I c ÞÞÞ; g À 1 ðT Hl ðgðt F b Þ; gðt F c ÞÞÞÞ

Linguistic neutrosophic Hamacher weighted mean operators
Þ is a collection of LNNs on the linguistic term set T = {t i | i 2 [0,2s]}, and the weight vector is w = (w 1 , w 2 , � � �, w n ) where w i 2 [0,1] (i = 1,2,� � �,n) and w 1 + w 2 +� � �+w n = 1. Then the linguistic neutrosophic Hamacher weighted arithmetic mean (LNHWAM λ ) operator is 2s]} be a continuous linguistic term set, and d i ¼ ðt T i ; t I i ; t F i Þ is a set of LNNs, then the aggregated value by using LNHWAM λ operator is still a LNN, which is shown as follows: The proof of this theorem is provided in Appendix A.
2s]}, and the weight vector is w = (w 1 , w 2 , � � �, w n ) where w i 2 [0,1] (i = 1,2,� � �,n) and w 1 + w 2 +� � �+w n = 1. Then the linguistic neutrosophic Hamacher weighted geometric mean (LNHWAM λ ) operator is Theorem 2. Let T = {t i | i 2 [0,2s]} be a continuous linguistic term set, and d i ¼ ðt T i ; t I i ; t F i Þ is a set of LNNs, then the aggregated value by using the LNHWAM λ operator is still a LNN, which is shown as follows: Proof. Since the proof of Theorem 2 is similar to that of Theorem 1, it is omitted to save space.
Note that when λ = 1, the LNHWAM λ operator is degenerated to the linguistic neutrosophic Algebraic weighted arithmetic mean (LNHWGM 1 ) operator as follows: � and the LNHWGM λ operator is degenerated to the linguistic neutrosophic Algebraic weighted geometric mean (LNAWGM 1 ) operator as follows: When λ = 2, the LNHWAM λ operator is degenerated to the linguistic neutrosophic Einstein weighted arithmetic mean (LNEWAM 2 ) operator as follows: and the LNHWGM λ operator is degenerated to the linguistic neutrosophic Einstein weighted geometric mean (LNEWGM 2 ) operator as follows:

Property 1. (Commutativity) Given a group of LNNs
Proof. Based on Definition 7 and Definition 8, it is clear that the conclusion is true. Proof.

As gðt
Linguistic neutrosophic Hamacher aggregation operators and land reclamation schemes 2. As gðt I i Þ ¼ 3. Similarly, as gðt F i Þ ¼ Now, this property is proved.
are two arbitrary sets of LNNs, d 1 Proof.

Linguistic neutrosophic Hamacher hybrid weighted mean operators
If the weights of LNNs and ordered positions are taken into account at the same time, several hybrid weighted aggregation operators can be obtained as follows.
is the i-th smallest value of ω i d i , and two weight vectors are ω = (ω 1 , ω 2 ,� � � ω n ) and w = (w 1 , w 2 ,� � �, w n ) where ω i , w i 2 [0, 1] (i = 1, 2,� � �,n) and P n i¼1 o i ¼ P n i¼1 w i ¼ 1. Then the linguistic neutrosophic Hamacher hybrid weighted arithmetic mean (LNHHWAM λ ) operator is 2s]} is a continuous linguistic term set, and d i ¼ ðt T i ; t I i ; t F i Þ is a set of LNNs, then the aggregated value by using the LNHHWAM λ operator is still a LNN, which is shown as follows: 2s]} is a continuous linguistic term set, and d i ¼ ðt T i ; t I i ; t F i Þ is a set of LNNs, then the aggregated value by using the LNHHWGM λ operator is still a LNN, which is shown as follows: Note that when λ = 1, the LNHHWAM λ operator is degenerated to the linguistic neutrosophic Algebraic hybrid weighted arithmetic mean (LNAHWAM 1 ) operator as follows: and the LNHHWGM λ operator is degenerated to the linguistic neutrosophic Algebraic hybrid weighted geometric mean (LNAHWGM 1 ) operator as follows: When λ = 2, the LNHHWAM λ operator is degenerated to the linguistic neutrosophic Einstein hybrid weighted arithmetic mean (LNEHWAM 2 ) operator as follows: and the LNHHWGM λ operator is degenerated to the linguistic neutrosophic Einstein hybrid weighted geometric mean (LNEHWGM λ ) operator as follows: It is clear that the LNHOWAM λ , LNHOWGM λ , LNHOWAM λ , LNHOWAM λ , LNHOWAM λ and LNHOWAM λ operators also satisfy the commutativity, idempotency, boundary and commonotonicity properties. Since the proofs of these properties are similar to that for the LNHWAM λ operators and LNHWGM λ operators, in order to save space, they are left out in this paper.

A method for decision making with linguistic neutrosophic information
In this section, a method for decision making with linguistic neutrosophic information is suggested to handle linguistic neutrosophic decision making issues.

Problem description
A multi-criteria linguistic neutrosophic decision making problem can be described as follows: α = {α 1 , α 2 ,� � �, α m }, which is indexed by x and m � 2, is a collection of alternatives for experts. β = {β 1 , β 2 ,� � �, β n }, which is indexed by y and n � 2, is a group of criteria for each alternative. ω = (ω 1 , ω 2 ,� � � ω n ) and w = (w 1 , w 2 ,� � �, w n ) are respectively the criteria and ordered position weight vectors, and both of them are completely unknown for decision makers. D = (d xy ) m×n , which is provided by some decision makers, is an original evaluation matrix.
d xy ¼ ðt Td xy ; t Id xy ; t Fd xy Þ, which is in the form of LNNs, is the assessment value of alternative α x relating to criteria β y ; t Td xy , t Id xy and t Fd xy , which are acquired from a certain linguistic term , are three independent linguistic terms.

Decision making procedures
In order to pick out the ideal alternative, a linguistic neutrosophic decision making method based on Hamacher aggregation operators is presented. The decision making procedures are shown as follows.
Step 1: Normalize the original decision making matrix. For a certain evaluation matrix, when there are two types of criteria, namely, the benefit and cost criteria, they need to be changed into the same type for convenience. The transformation equation is Step 2: Determine the weights of criteria. The idea of criteria weight determination model is that when the deviation of evaluation values among all alternatives under a certain criteria is big, it means such criterion can greatly affect the ranking order. Thus, a large weight value ω y can be assigned to this criterion. The calculation equations can be expressed as 4s þ Te xy À Ie xy À Fe xy 6s À 4s þ Te zy À Ie zy À Fe zy 6s jÞ; ð64Þ Te xy À Fe xy 2s À Te zy À Fe zy 2s jÞ; ð65Þ where K y means the deviation of score function values among all alternatives under criterion β y , L y means the deviation of accuracy function values among all alternatives under criterion β y , and W y is the normalized weight value of criterion β y based on Eqs (64) and (65).
Step 4: Compute the comprehensive preference values. The LNHHWAM λ operator or LNHHWGM λ operator is suggested to compute the comprehensive preference value e x (x = 1, 2,� � �,m) of each alternative in matrix E.
Step 5: Obtain the optimal alternative. The ranking result can be obtained by calculating the score function and accuracy function values of e x .

Case study
In this section, a case is studied by adopting the presented method based on linguistic neutrosophic Hamacher aggregation operators to assess land reclamation schemes in mine area.

Project profile
The Kaiyang phosphate mine, which is located in Jinzhong Town, Guiyang City, Guizhou Province, China, is composed of Maluping, Qincaichong, Yongshaba and Shabatu sections. It has a history of nearly 60 years, and the proven phosphate reserve is 1.08 billion tons. With the increase of demand for phosphate rock, the mining intensity is rising greatly, and the annual output is up to 8 million tons. However, considerable land resources are destroyed due to the large-scale mining. Until the end of 2010, the cultivated area destroyed only by collapse of mountain exceeded 15 hm 2 [47]. Besides, a huge amount of waste residue is produced, which not only occupies large tracts of land, but also destroys ecological environment and induce geological disasters. Therefore, it is very important and urgent to select an optimal land reclamation scheme for Kaiyang phosphate mine to achieve sustainable development.

Evaluation criteria system
Determining the evaluation criteria is necessary and vital for the evaluation of land reclamation schemes. According to the specific characteristics of mine areas, four criteria are selected, including the technology, economy, environment and efficiency. More details are shown in Table 1.

Selecting optimal land reclamation scheme
After preliminary investigation, managers in a mine intend to make appraisals among four land reclamation schemes and then select the optimal alternative. The linguistic term set 6]} is utilized, where t 0 = very low, t 1 = low, t 2 = a little low, t 3 = medium, t 4 = a little high, t 5 = high and t 6 = very high. Then, the original evaluation matrix provided by experts is shown in Table 2. Subsequently, the linguistic neutrosophic decision making method suggested in Section 4 is applied to obtain the ranking result of these schemes.
Step 1: Normalize the original decision making matrix.
Since β 2 belongs to the cost criterion whereas other three indicators are benefit criteria, β 2 is transferred by Eq (63). Then, the normalized decision making matrix is attained (See Table 3).
Step 3: Calculate the weights of ordered positions. Assume g xy = w y e xy with λ = 0.1 is used to obtain the weighted linguistic neutrosophic decision making matrix (See Table 4), then according to Eq (67), the weights of ordered positions are calculated as: w 1 � 0.17, w 2 � 0.18, w 3 � 0.35 and w 4 � 0.30.
Step 4: Compute the comprehensive preference values. According to Eq (56), the LNHHWAM λ operator with λ = 1 is utilized to aggregate each row of the normalized matrix E. Thereafter, the comprehensive preference values of Table 1. Evaluation criteria of land reclamation schemes.

Benefit/ Cost
Descriptions Technology β 1 Benefit It indicates the technological feasibility of schemes, which includes local climate, terrain and soil conditions.
Economy β 2 Cost It indicates the predicted economic cost, which includes materials, labor, equipment and maintenance fees.
Environment β 3 Benefit It indicates the comprehensive environment effect of schemes, which includes land fertility, species diversity and ecological environment quality.
Efficiency β 4 Benefit It indicates the expected efficiency of schemes, which includes labor intensity and project duration.
Step 5: Obtain the optimal alternative. The land reclamation in Kaiyang phosphate mine is indicated in Fig 1. By using this optimal land reclamation scheme, the mutual benefits of economy and environment are attained. On the basis of the actual conditions, the proposed linguistic neutrosophic Hamacher aggregation operators can be adopted to assess the land reclamation schemes reliably and efficiently.

Discussions
In this section, the meaning of parameter λ in Hamacher aggregation operators is clarified by sensitivity analyses. Furthermore, the effectiveness and advantages of the proposed method are demonstrated with comparison analyses.

Sensitivity analyses
In subsection 5.2, when the LNHHWAM λ operator with λ = 1 is used, the ranking order α 1 � α 4 � α 2 � α 3 is derived. In this subsection, the influence of parameter λ on aggregation values and ranking orders is explored with the same example.
When different λ values are provided in Step 4 of the proposed method, the comprehensive preference values and ranking orders with the LNHHWAM λ operator and LNHHWGM λ operator are itemized in Tables 5 and 6, respectively.
As shown in Table 5, it is clear that various aggregation results are obtained by the LNHHWAM λ operator with diverse values of λ. As a result, the final ranking orders are changed when different λ values are chosen. Besides, it can be seen that: when 0 < λ < 1, the ranking result is α 4 � α 1 � α 2 � α 3 ; and when λ � 1, the ranking result is As demonstrated in Table 6, it is true that the aggregation values by the LNHHWGM λ operator with distinct λ values are diverse. Hence, the ultimate rankings are altered when unequal λ values are assigned. Moreover, it can be seen that: when 0 < λ < 0.5, the ranking order is α 4 � α 1 � α 3 � α 2 ; when λ = 0.5, the ranking order is α 4 � α 1 � α 2 � α 3 ; and when λ > 0.5, the ranking order is α 4 � α 2 � α 1 � α 3 . However, it seems that the λ values have no impact on the selection of the optimal alternative, because the best scheme is always α 4 no matter what the value of λ is.
As indicated in Fig 2, it can be seen that the values of score function calculated by the LNHHWAM λ operator go down accordantly with the increase of λ values. Whereas, the values of the score function go up accordantly with the growth of λ values when the LNHHWGM λ operator is used in Fig 3. Reasons for this phenomenon may be twofold. One is that the score  function values increase as t Te xy , yet decrease as t Ie xy and t Fe xy in the argument values. Another important cause is that the Hamacher t-norm family monotonically rises with the intensification of λ, but the Hamacher t-conorm family monotonically declines with the increase of λ [42]. Moreover, an interesting event is that when the same values of arguments and λ are allocated, the values of score function calculated by the LNHHWGM λ operator are invariably larger than those by the LNHHWAM λ operator.
In fact, different λ value means dissimilar operational laws and aggregation operators. Generally, the values of the parameter λ = 1 or λ = 2 can be adopted, linguistic neutrosophic Algebraic aggregation operators or linguistic neutrosophic Einstein aggregation operators is derived in this case. Moreover, the selection of λ values in different aggregation operators can reflect the risk preferences of decision makers [42]. That is to say, in the LNHHWAM λ operator, a big λ value implies the risk aversion or the pessimism of experts, and a small λ value indicates the risk loving or the optimism of experts. On the contrary, a completely opposite inference is drawn in the LNHHWG0 λ operator. Thus, specialists can determine the value of λ in a certain linguistic neutrosophic Hamacher aggregation operator according to their experience, risk attitudes or reality. To be specific, if the decision maker is a risk lover, λ = 1 with LNHHWAM 2 operator and λ = 2 with LNHHWGM 2 operator are suggested. In contrast, if the decision maker is a risk averter, λ = 2 with LNHHWAM 2 operator and λ = 1 with LNHHWGM 1 operator are suggested.

Comparison analyses
To the best of our knowledge, several major decision making methods with regard to LNNs have been suggested in literature [22][23][24][25][26][27][28]. In this subsection, in order to justify the validity and superiority of the proposed method, in-depth analyses are made through comparison with these existent approaches.
Part I: Theoretical comparative analyses In this Part, the theoretical comparison results among different decision making methods in terms of numbers of parameter, computational complexity, whether capture relationships between arguments, whether consider the criteria weights objectively and whether consider the weights of ordered position are described in Table 7.
The detailed theoretical comparative analyses are listed as follows. 1. Compare with other aggregation operators in literature [22,25,26,27] First, the linguistic neutrosophic arithmetic or geometric mean operators [22] are special cases of the proposed linguistic neutrosophic Hamacher aggregation operators. Hence, the Hamacher aggregation operators are more general. In addition, the relationship between inputs can be reflected in this method. Second, the calculation of our method is relatively simpler than others [25,26,27]. Third, unlike the Bonferroni averaging operators [25] and power Heronian averaging operators [26], only one parameter needs to be determined in the proposed method. Thus, our method is more convenient and feasible.
2. Compare with classical decision making methods in literature [23,24,28] In general, the basic ideas of dissimilar decision making methods are different. All of them have their respective characteristics and highlights. In literature [23], the weight vector of criteria is known in advance, while an objective determination model is provided in this paper. In literature [28], the computational complexity is too high. In addition, the interrelations between arguments cannot be captured in literature [23,24]. Compared with them [23,24,28], the biggest advantage of the proposed method is that the weights of ordered positions are taken into consideration.
Part II: Numerical comparative analyses In this Part, numerical ranking results among different decision making methods in terms of ranking orders and difference degrees are depicted in Table 8.
It can be seen that dissimilar ranking orders are obtained when different decision making methods are implemented. In order to determine the optimal ranking order, an aggregation method proposed by Jahan et al. [48] is adopted as follows.
Step 1: Calculate the numbers of times a land reclamation scheme is assigned to different ranks, as shown in Table 9. Take α 1 as an example, α 1 has five times a ranking of 1, two times a ranking of 2 and once a ranking of 3 and 4.

Schemes
Step 3: Determine the optimal ranking order. On the basis of Table 10, a linear programming model is built as Eq (68). By solving this linear programming problem, the best ranking order is determined as For clarity, the ranking orders by using different methods are portrayed in Fig 4. It is true that the optimal ranking orders and the ranking results of the proposed approach in this paper are the same. That is to say, our method is more suitable to cope with the evaluation of land reclamation schemes for mines to some extent.