The Evaluation of Mineral Resources Development Efficiency Based on Hesitant Fuzzy Linguistic Approach and Modified TODIM

. The evaluation of mineral resources development efficiency is a typical multicriteria decision-making issue. Meanwhile, due to the limited existing technology, there might be subjectivity, ambiguity, and inaccuracy of the measurement of the evaluation index of mineral resources development efficiency. In this paper, we, considering the incomplete information, use the hesitant fuzzy linguistic approach to describe the psychological hesitation and ambiguity of the decision-maker in the actual evaluation process and then construct the general model of the development efficiency evaluation of the mineral resources by using the hesitant fuzzy linguistic terms sets and modified TODIM. Finally, this paper takes the Panxi area as an example to study the development efficiency of vanadium-titanium magnetite. The results show that the hesitant fuzzy linguistic multicriteria decision-making (MCDM) approach can be implemented to mineral resources evaluation and resources management.


Introduction
Zadeh [1] proposed the concept of fuzzy sets to describe the ambiguous relationship between different elements in 1965.The concept of fuzzy set has attracted the attention of scholars and has been applied quickly to decision theory.Lin et al. [2] applied the fuzzy theory to the management contract analysis.The study found that the fuzzy theory can avoid the subjective ambiguity caused by the information asymmetry, compared with the traditional binary probability theory.Based on the fuzzy set, Torra [3] first proposed the hesitant fuzzy sets, and Su et al. [4] applied the hesitant fuzzy sets to the multicriteria decision-making and sorted the alternatives by calculating the distance between the alternatives and the reference point.Rodriguez et al. [5] introduced the concept of hesitant fuzzy linguistic term sets on the basis of the hesitant fuzzy sets.Using the hesitant fuzzy linguistic terms sets, decisionmakers can evaluate the program attributes in a more flexible language.Ren et al. [6] argued that hesitant fuzzy sets can accurately describe the hesitation of the decision-makers and the ambiguity of evaluation caused by the limited information and that it can optimize the multicriteria decision-making more scientifically and reasonably [7].There are many scholars crowed in the direction of hesitant fuzzy linguistic MCDM.However, the management of mineral resources is full of fuzzification and randomness, which make it difficult to evaluate mineral resources development efficiency.To the authors' knowledge, there are few researches on mineral resource management implemented via the hesitant fuzzy linguistic MCDM approach.
The multicriteria decision-making refers to the sorting and selection problem of finite alternative with multiple attributes.For the multicriteria decision-making, scholars have proposed different decision-making methods.Among them, most decision-making methods are based on the expected utility theory, which assumes that the decisionmaker is completely rational and pursues maximum utility in the decision-making process.However, there is a certain deviation between the decision-maker's actual choice and the optimal choice due to the influence of the decisionmaker's cognitive ability, psychological state, and other factors.Kahneman and Tversky [8] put forward the prospect theory based on the assumption that the decision-maker is a "bounded rational person."The prospect theory suggests that decision-makers do not always pursue the maximum expected utility in the decision-making process but prefer to make the most satisfying decision.In 1992, Tversky and Kahneman [9] further promoted the prospect theory and proposed the cumulative prospect theory.The prospect theory and the cumulative prospect theory are widely used in the multiattribute decision-making issues as a typical behavior decision theory [10].Gomes and Lima [11] proposed the TODIM method on the basis of the prospect theory, and the TODIM method optimizes and sorts the alternatives by calculating the dominance of the alternatives relative to other alternatives.Wei et al. [12] conclude that the prospect theory can be used by revising impact caused by the maker's psychological behavior.Fan et al. [13] extended TODIM to the multicriteria decision-making, the attribute information of which includes clear numbers, interval numbers, and language terms.
Based on the study of the mineral resources development efficiency in the past, this paper researches the evaluation of the mineral resources development efficiency by using the hesitant fuzzy linguistic term sets and the modified TODIM method.Finally, this paper conducts a case study on vanadium-titanium magnetite resources development efficiency in the Panxi area.
The contributions of the research are as follows: (1) There are few researches on the evaluation method of the mineral resources development efficiency.This paper integrates the hesitant fuzzy linguistic approach and the modified TODIM method to evaluate the mineral resources development efficiency, which has practical significance.(2) Vanadium-titanium magnetite is a strategic resource.
There are few researches on the mineral resources development efficiency.This paper evaluates the development efficiency of vanadium-titanium magnetite in the Panxi area, which provides implication for other related researches.

Hesitant Fuzzy Linguistic Term Sets and TODIM
Decision-makers prefer to use linguistic terms in the process of decision-making.Using linguistic terms allows the decision-makers to describe program attributes flexibly and accurately.

Distance Measure of Hesitant Fuzzy Linguistic Term Sets
Definition 2. Let   ,   ∈  be two linguistic variables; the definition of the deviation degree between   and   can be expressed as follows [15]: where 2 is the number of the linguistic terms in the set .
Liao et al. [16,17] extended the distance of the linguistic term set and proposed the distance of the hesitance fuzzy linguistic term set: 2.3.TODIM.Prospect theory holds that there are certainty effect, isolation effect, and reflection effect in the progress of decision-making.Decision-makers experience two stages when making decisions: first, the reference point is established.When the decision is better than the reference point, it is considered as "acquired", whereas it is regarded as "lost" when below the reference point.The second stage is to evaluate the expected total utility.The total utility of the decision-maker is described as where  represents the th attribute,  represents the number of evaluation indicators, and V(  ) reflects the prospect value gains or the prospect value loss; Kahneman and Tversky [8] give the prospect value function in the form of where   indicates the gains or losses related to reference point;  ( ∈ (0, 1)) and  ( ∈ (0, 1)) are the parameters which represent the risk attitude of the decision-maker;  is the loss aversion coefficient which means the risk aversion degree of the decision-maker.
When   ≥ 0, the weighting function is expressed as When   < 0, the weighting function is expressed as where   reflects the distribution of   .Kahneman and Tversky in their experiment found that  = 0.88,  = 0.88,  = 2.25,  = 0.61, and  = 0.69.
The TODIM method is designed to help people make effective decisions when they face uncertainty.The TODIM method can adjust the parameters in the calculation according to the decision-maker's risk preference to get the decision results that are in line with the decision makers' preference.Some scholars focus on the relationship between TODIM method and intuitionistic uncertain linguistic information.Liu and Teng [18] and Li et al. [19] extend the TODIM method to the multiple attribute group decision-making (MAGDM) with intuitionistic uncertain linguistic information.Lourenzutti and Krohling [20] generalize the Fuzzy-TODIM method to deal with intuitionistic fuzzy information.Some other researchers concern HFL-TODIM.Peng et al. [21] introduce a new comparison method and corresponding distance of multiset hesitant fuzzy elements (MHFEs) and propose a novel approach for MCGDM problems based on the traditional TODIM.Wang et al. [22] propose a likelihoodbased TODIM approach with multihesitant fuzzy linguistic information that solves MCDM problems in selecting and evaluating TPLSPs effectively.Zhang and Xu [23] extend the TODIM method, which is based on prospect theory and can effectively capture the decision-maker's psychological behavior, to solve this type of problems under hesitant fuzzy environment.Tan et al. [24] propose Choquetbased TODIM method to solve the hesitant fuzzy MCDM problems.
The steps of TODIM method are as follows.
Let  = { In the above equation,   is the evaluation of the attribute  of the alternative  and ∑  = 1.
The classic TODIM method can be described as follows.
Step 2. Calculate the relative weight   .
where   is the weight of reference criterion and   = max(  |  ∈ ).
Step 3. Calculate the dominance of each alternative   over each alternative   , according to where (  ,   ) expresses the contribution of the criterion   when comparing the alternative   with the alternative   ; the parameter  represents the attenuation factor of the losses.
Step 4. Obtain the overall value of the alternative   .

A Modified TODIM Approach Based on the Hesitant Fuzzy Linguistic Term Set and the Prospect Theory
According to Zhang and Xu [23], the modified TODIM approach can not only reflect the risk tendency of the decision-maker in the process of decision-making but also represent the fuzziness of the objects and the hesitant thought of the decision-maker.
Let  = { In this equation, ℎ  is the evaluation of the attribute  of the alternative  and ∑  = 1.
The dominance of each alternative   over each alternative   can be described as where * (ℎ  , ℎ  ) is the distance between ℎ  and ℎ  , and it can be expressed as where  ( ∈ (0, 1)) and  ( ∈ (0, 1)) are the parameters which represent the risk attitude of the decision-maker;  is the loss aversion coefficient which means the risk aversion degree of the decision-maker. *  is the decision weight function, and it depends on In the above two equations,  * +  denotes the weight function at the time of acquisition and  * −  denotes the weight function at the time of loss.
The overall prospect value of an alternative can be calculated by using

Case Study
Mineral resources, considered as an essential type of natural resources, provide a necessary material basis for the development of human society.Mineral resources are limited and nonrenewable [25].Improving the mineral resources development efficiency contributes considerably to building a resource-saving society [26].Du and Wang [27] study the mineral resources development efficiency in different regions of China, and the results show that the mineral resources development efficiency in China, generally, is lower than that of the developed countries.Yu et al. [28] use the two indexes of resource productivity and comprehensive utilization ratio to evaluate the mineral resources development efficiency of mineral mining enterprises in the Chengde area and pointed out that the low efficiency of mineral resources development in this place was caused by the backward production technology.Wei et al. [29] held that the development of mineral resources is a complex large system according to the system theory, and that when evaluating the mineral resources development efficiency, researchers should consider the cost, finance, development programs, and other evaluation indicators.
A report shows that as many as 54 kinds of mineral species and 9.393-billion-ton vanadium-titanium magnetizes exist in the Panxi area.The Panxi area is the second largest iron ore base in China and it has been developed for a long time, Therefore, it has great research value of mineral resources development efficiency evaluation because of the high level of mineral resources utilization.Nevertheless, because of the lack of effective measure methods in the current selection of indicators, which includes mining recovery rate, ore dressing recovery percentage, comprehensive utilization ratio, and other indicators, the low accuracy and a certain degree of ambiguity exist in the statistical results.A certain linguistic term may be not suitable to express the experts' linguistic opinions under certainty in such complex situation.The use of hesitant fuzzy linguistic term set is a direct and precise manner to represent uncertain linguistic information [30].

Summary of Vanadium-Titanium Magnetite Resources in the Panxi Region.
Vanadium-titanium magnetite is a type of ore containing iron, vanadium, titanium, and other elements which are mainly distributed in the Panxi and Chengde area [31].Located in the southwest of the Sichuan Province, the Panxi area is China's main metallogenic belt of vanadium-titanium magnetite.With a length of approximately 300 kilometers and a width (east-west) of about 20 kilometers, this belt, taking around 6000 square kilometers, is one of the world's major producing areas [32].A report about the exploration of the Panxi area shows that the reserves of vanadium-titanium magnetite are about 9.393 billion tons, of which the reserves of vanadium are about 23.48 million tons and the reserves of titanium resources are about 870 million tons.
Vanadium-titanium magnetite, considered as an important source of iron, contains vanadium, titanium, chromium, cobalt, nickel, platinum, and other components.At present, there are large mineral exploration enterprises.The production of vanadium-titanium magnet is over 100 million tons.The recovery of titanium from iron tailings is more than 1 million tons.The recovery rate is about 30%.As for the comprehensive utilization of vanadium, vanadium slag, high vanadium iron, vanadium oxide, and vanadium pentoxide have been explored.Vanadium-titanium magnetite is a kind of multiple-element symbiotic composite ore, which has extremely high comprehensive utilization value; however, the symbiotic elements are not exploited due to the technical and economic reasons [33].

Data Collection.
The data for this study are collected from the report of the Geological Survey Project of the China Geological Survey.The report contains four survey areas of the vanadium-titanium magnetite in the Panxi area (see Figure 1).
(2) Hongge Vanadium-Titanium Magnetite Mining Area.The Hongge mining area, the largest vanadium-titanium magnetite mining area in the Panxi area, is divided into eight sections: Anningcun, Baicao, Maanshan, Hongge, Xiushuihe, The output situation of the unit ore Unit ore revenue The unit ore quantity sales situation Zhongliangzi, Wanzitian, and Zhonggangou.Hongge mining area contains 4.8 billion tons of vanadium-titanium magnetite.
To conduct an illustrative example research, eight subareas are selected from the above four vanadium-titanium magnetite mining areas (see Table 1).

Evaluation Index System of
Vanadium-Titanium Magnetite.Luo et al. [34] suggested that the mineral resources development efficiency could be evaluated from four dimensions: the development and utilization level, environmental recovery, personnel efficiency, and utilization efficiency.The four dimensions are broken down into seven indicators (see Table 2).
The mining recovery rate, recovery percentage, and comprehensive utilization rate are the core contents of the mineral resources development efficiency.The mine geological environment restoration rate reflects the investment of mining enterprises to restore the environment of mining area.The per capita industrial output value indicates the economic efficiency of the staff in the survey area, and the unit mineral value and the unit ore revenue represent the profitability and deep processing degree of mineral resources.In this problem, the weighting vector is given as follows:  = (0.2, 0.2, 0.1, 0.1, 0.1, 0.15, 0.15)  .3.

Evaluation of Vanadium-Titanium Magnetite
Because some experts' comments are discarded in the evaluation process, this paper uses the formula proposed by Zhu and Xu [35] to add elements to the evaluation sets and then calculate the distance between the evaluation sets.
In ( 16)  + and  − are the largest and smallest elements in the hesitant fuzzy linguistic term set , respectively. is the optimal parameter, which reflects the risk preference of the decision-makers.In general,  = 1/2.
Step 1.Using (16), we add the elements to the evaluation matrix, and the results are shown in Table 4.
Step 2. Use (14) to modify the weight of indicators.
Step 3.According to the evaluation matrix of Table 4, we can calculate the dominance matrix of mineral resource development efficiency of each survey area relative to other survey areas (see Table 5).Because different scholars have different estimates of the risk aversion coefficient, this paper uses  = 2.25 and  = 1 to calculate the dominance of alternatives and the results are shown in Tables 6 and 7.
The evaluation result of mineral resources development efficiency in the survey area is shown in Tables 8 and 9.
KF01, KF05, KF06, and KF07 are large survey areas.KF02, KF03, KF04, and KF08 are small survey areas.From the evaluation results, the mineral resources development efficiency of the large survey areas is higher than that of the small areas.It is found from the comparison between Tables 8  and 9 that the different risk tendencies of the decision-maker will slightly impact the sorting result of the survey area.
Tables 8 and 9 show the evaluation results under different risk preference.Among the 8 surveyed areas, KF01, KF05, KF06, and KF07 are all large-scale survey areas, and KF02, KF03, KF04, and KF08 are the small survey area.From the evaluation results of Tables 8 and 9, the development efficiency of mineral resources in large survey areas is generally higher than that of the small survey areas.This may be due to the fact that large survey areas generally invest more funds, and the equipment is more advanced.Hence the development efficiency of mineral resources is higher than that of the small-scale investigation area.
Comparing the evaluation results in Tables 8 and 9, we find that the evaluation results in Table 8 are not significantly different from the evaluation results in Table 9.Nevertheless, the rankings of the two large survey areas KF06 and KF07 are different, indicating that different risk preference coefficients have little effect on the final evaluation results.
Previous studies make general qualitative evaluations on the efficiency of mineral resources development [34].However, this article makes the evaluation results more accurate.

Conclusion
TODIM is a powerful tool for solving the MCDM problems under uncertainty and fuzziness.Therefore, in this paper, considering the incomplete information, we use the hesitant fuzzy linguistic approach to describe the psychological hesitation and ambiguity of the decision-maker in the actual evaluation process.Then we construct the general model by using the hesitant fuzzy linguistic terms sets and modified TODIM, to solve the MCDM problems with hesitant fuzzy information.The evaluation of mineral resources development efficiency is a complex and typical multicriteria decision-making issue.Meanwhile, due to the limited existing technology, there might be subjectivity, ambiguity, and inaccuracy of the measurement of the evaluation index of mineral resources development efficiency.The proposed method can be expected to be applicable to the mineral resources evaluation and resources management.
Based on the analysis of the factors affecting the development efficiency of vanadium and titanium magnets in the Panxi area, this paper establishes the evaluation index system from four dimensions, namely, development and utilization level, environmental recovery, personnel efficiency, and utilization efficiency.It then uses the hesitant fuzzy linguistic term sets and the TODIM method to conduct a case study.
Considering the ambiguity of the evaluation index of the mineral resources development efficiency, this paper uses the hesitant fuzzy linguistic term sets to evaluate the index.It, then, uses the hesitant mathematical approach to study the multicriteria decision-making, which accurately reflects the psychological hesitation of the decision-maker.The evaluation result accurately reflects the mineral resources development efficiency in the Panxi area.

Figure 1 :
Figure 1: Location of the evaluation areas.
Development Efficiency.Mineral resources development efficiency evaluation is the use of mathematical methods to integrate the information of indicators; however, the accuracy of indicators measure is limited.In this paper, the hesitant fuzzy linguistic term sets and the modified TODIM are used to evaluate the mineral resource development efficiency in the Panxi area.Three experts are invited to use the linguistic term set  = ( −3 = none,  −2 = very low,  −1 = low,  0 = medium,  1 = high,  2 = very high,  3 = perfect) to ,   is in mathematical terms of   = {⟨  , ℎ  (  )⟩ |   ∈ }, where ℎ  (  ) is a set of some values in the linguistic term set  and can be expressed as ℎ  (  ) = {  (  ) |   (  ) ∈ ,  = 1, . . ., } with  being the number of linguistic terms in ℎ  (  ).ℎ  (  ) denotes the possible degrees of linguistic variable   to the linguistic term set .For convenience, ℎ  (  ) is called the hesitant fuzzy linguistic element.

Table 1 :
The evaluation area of the Panxi area.

Table 2 :
Criteria for evaluating mineral resources development efficiency.The actual amount of ore produced in the mining area accounting for the geological reserves in the range Recovery percentage   The actual amount of the useful components in the selected ore accounting for the total amount in the selected ore Comprehensive utilization ratio   The comprehensive utilization degree of the main and coassociated mineral resources exploitation and utilization Mine geological environment restoration rate   Restoration situation of mining geological environment restoration Per capita industrial output value  io Per capita output situation Unit mineral value