Method for evaluating the eco-tourism environmental carrying capacity with hesitant fuzzy linguistic information

Abstract

Eco-tourism is a tourism style tourists to the nature and the history human landscape of tourism areas as the main Tour object to understand, appreciate and improve their self-cultivation and ecological protection for the purpose of tourism and the protection of the natural and ecological environment and improve the socio-economic environment and make positive contribution to tourism. As ecotourism emphasis on environmental protection, as well as the participation of local residents and the development of the educational function all over the world, but in the development of eco-tourism, many scenic spots under the banner of “eco-tourism” as a pretense, makes the development and construction of tourism projects hastily launched, which induces the worsening environment and makes tourism scale far exceeding the capacity of the ecological environment. Therefore, in order to better guide the development of eco-tourism, eco-tourism study on the environmental carrying capacity is a major significance. The aim of this paper is to investigate the hesitant fuzzy linguistic multiple attribute decision making problems based on the hesitant fuzzy linguistic sets and Geometric Bonferroni mean operator. Then, we have proposed the aggregation operator: dependent hesitant fuzzy linguistic geometric Bonferroni mean (DHFLGBM) operator. Then, we have utilized DHFLWGBM operator to tackle the multiple attribute decision making problems using hesitant fuzzy linguistic information. Finally, a practical example for evaluating the eco-tourism environmental carrying capacity is proposed to testify the method in this paper.

Keywords

Multiple attribute decision making hesitant fuzzy linguistic values dependent hesitant fuzzy linguistic weighted geometric Bonferroni mean (DHFLGWBM) operator assessment eco-tourism environmental carrying capacityis

1 Introduction

Behind in the high-speed economic growth at present, human economic activities led to the crisis of the ecological environment, Mode of economic growth Performance with high investment, high consumption, high emission characteristics, These lead to regional ecological environment and resources shortage, and bring about the regional ecological environment pollution and destruction. With the industrialization and urbanization in the western region, The ecological destruction and environmental pollution has gradually become an obstacle to the economic development of the western region, and the spillover effect of ecological environment problems pose a serious threat on the other area and even the whole country’s ecological environment, economic and social sustainable development. Therefore we should understand the mutual influence mechanism of the ecological environment and economic growth, through the optimization of ecological resources configuration, ecological environment and economic benefits of checks and balances in the western region, we build coordinated development mechanism reflecting resource scarcity and ecological environment of value evaluation, mining, sorting beneficial ingredients suitable for the coordinated development of ecological environment and economic growth, Through this way, the paper can provide scientific basis for sustainable development strategy and promote the construction of the ecological civilization in the western region.

The ecological environment of the city decides and restricts the quality and future trend of the urban development. However, the industrialization and rapid growth places great pressure to the ecological environment of the city. Minimizing the ecological costs from resources and energy consumption and effectively promoting the performance of the ecological environment management of the city to provide good environment supportability for the sustainable development of the city is the urgent task at present. As is known to all, the ecological environment management is a complicated systematic engineering. To solve the problems of ecological environment management can’t be simply understood as to control the pollution. After all, environmental problems are developmental problems. The key points of ecological environment management of the city are to coordinate the relationships between the city and economic growth. Finally, it shall optimize the economic structure and transform the mode of economic development. The assessment problems of eco-tourism environmental carrying capacity are the multiple attribute decision making problems [1 –14]. In this paper, we investigate the multiple attribute decision making problems for evaluating the eco-tourism environmental carrying capacity based on the Geometric Bonferroni mean operator [15 –17] utilizing the information of hesitant fuzzy linguistic information. Then, we have proposed the aggregation operators: dependent hesitant fuzzy linguistic geometric Bonferroni mean (DHFLGBM) operator. Then, we have utilized DHFLGBM operator to tackle the multiple attribute decision making problems using hesitant fuzzy linguistic information. In the end, a practical example for evaluating the eco-tourism environmental carrying capacity is proposed to test the performance of this method.

2 Preliminaries

Assuming that S ={ s_i |i = 1, 2, ⋯ , t } refers to a linguistic word set utilizing odd cardinality. On the other hand, we suppose that s_i denotes a possible value for a specific variable, and then this value should follow the following features [18, 19]: ➀ The set is ranked: s_i > s_j, if i > j; ➁ There is the negation operator: neg (s_i) = s_j such that i + j = t + 1; ➂ Max operator: max(s_i, s_j) = s_i, if s_i ≥ s_j; ➃ Min operator: min(s_i, s_j) = s_i, if s_i ≤ s_j; Particularly, S is represented as follows. $\begin{matrix} S & = & {s_{1} = extremely poor, s_{2} = very poor, \\ s_{3} = poor, s_{4} = medium, s_{5} = good, \\ s_{6} = very good, s_{7} = extremely good} \end{matrix}$

Utilizing the linguistic term set [20 –26] and hesitant fuzzy set [27 –34], Lin et al. [35] design some important concepts and important operational rules which is corresponding to the hesitant fuzzy linguistic set.

Definition 1. [35] Given a fixed set X, then a hesitant fuzzy linguistic set (HFLS) on X is according to a function which is exploited to X returns a sunset of [0, 1]. In order to let the process much simpler, the HFLS can be represented by the following equation. $A = (〈 x, s_{θ (x)}, h_{A} (x) 〉 | x \in X)$ (1) where h_A (x) refers to a set of some values which are ranged from zero to one, which is represented as the possible membership degree of the element x ∈ X to the linguistic set s_θ(x). To be easier, we called a =〈 s_θ(x), h_A (x) 〉 a hesitant fuzzy linguistic element (HFLE) and A the set of all HFLEs.

Definition 2. [35] For a HFLE a =〈 s_θ(x), h_A (x) 〉, $s (a) = (\frac{1}{# h} \sum_{γ \in h} γ) s_{θ (x)}$ is named a score function of a, where # h refers to the number of the elements in h. Supposing that there are two HFLEs, which are a₁ and a₂, when s (a₁) > s (a₂), a₁ > a₂ is saisfied; if s (a₁) = s (a₂), a₁ = a₂ is satisfied.

In the following, Zhu et al. [15] studied on the geometric Bonferroni mean (GBM) which utilizing both the BM and the geometric mean (GM).

Definition 3. [15] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) refers to a collection of non-negative real numbers. Afterwards, the aggregation functions are defined as follows:

$\begin{matrix} {GBM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i} + {qa}_{j}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (2) is named the geometric Bonferroni mean (GBM) operator.

Then, Gu et al. [36] defined the hesitant fuzzy linguistic weighted Geometric Bonferroni mean (HFLWGBM) operator.

Definition 4. [36] Assuming that a_j = (s_{θ(a_j)}, h (a_j)) (j = 1, 2, ⋯ , n) represents a set of HFLEs, and p, q > 0, w = (w₁, w₂, ⋯ , w_n) ^T denotes the weight vector of a_j (j = 1, 2, ⋯ , n), in which w_j indicates the importance degree of a_j, satisfying w_j > 0 (j = 1, 2, ⋯ , n), and $\sum_{j = 1}^{n} w_{j} = 1$ . If

$\begin{array}{l} {HFLWGBM}_{w}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{p + q} \otimes_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(w_{i} a_{i})}^{p} \oplus {(w_{j} a_{j})}^{p}))}^{\frac{1}{n (n - 1)}} \\ = 〈 {(\frac{1}{p + q} \otimes_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(s_{w_{i} θ (a_{i})})}^{p} \oplus {(s_{w_{j} θ (a_{j})})}^{q}))}^{\frac{1}{n (n - 1)}}, \\ (\cup_{γ (a_{1}) \in h (a_{1}), γ (a_{2}) \in h (a_{2}), \dots,}_{γ (a_{n}) \in h (a_{n})} {{(\begin{array}{l} \frac{1}{p + q} \otimes_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(p γ (a_{i}))}^{w_{i}} \oplus {(q γ (a_{j}))}^{w_{j}}) \\ \otimes ({(p γ (a_{j}))}^{w_{j}} \oplus {(q γ (a_{i}))}^{w_{i}}) \end{array})}^{\frac{2}{n (n - 1)}}} 〉 \\ = 〈 {(\frac{1}{p + q} \otimes_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(s_{w_{i} θ (a_{i})})}^{p} \oplus {(s_{w_{j} θ (a_{j})})}^{q}))}^{\frac{1}{n (n - 1)}}, \\ (\cup_{γ (a_{1}) \in h (a_{1}), γ (a_{2}) \in h (a_{2}), \dots,}_{γ (a_{n}) \in h (a_{n})} {1 - {(1 - \prod_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(ε_{i, j}^{w})}^{\frac{2}{n (n - 1)}})}^{\frac{1}{p + q}}} 〉 \end{array}$ (3) then the ${HFLWGBM}_{w}^{p, q}$ is named as the hesitant fuzzy linguistic weighted Geometric Bonferroni mean (HFLWGBM) operator.

3 Dependent hesitant fuzzy linguistic weighted geometric Bonferroni mean (DHFLGWBM) operator

Definition 5. Let a_j = (s_{θ(a_j)}, h (a_j)) (j = 1, 2, ⋯ , n) denotes a set of HFLEs, the average value of the score function of a_j (j = 1, 2, ⋯ , n) is computed as $s (\bar{a}) = \frac{\sum_{j = 1}^{n} s (a_{j})}{n}$ (4)

Definition 6. Let a₁ and a₂ refer to two HFLEs on X ={ x₁, x₂, ⋯ , x_n }, then the distance between a₁ and a₂ is calculated as follows. $d (a_{1}, a_{2}) = | s (a_{1}) - s (a_{2}) |$ (5)

Definition 7. Let a_j = (s_{θ(a_j)}, h (a_j)) (j = 1, 2, ⋯ , n) be a set of HFLEs, and $s (\bar{a})$ the arithmetic mean of these hesitant fuzzy linguistic values.

Afterwards, the standard deviation of the hesitant fuzzy linguistic values is defined as follows. $σ = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} {| s (a_{1}) - s (a_{2}) |}^{2}}$ (6)

Definition 8. Let a_j = (s_{θ(a_j)}, h (a_j)) (j = 1, 2, ⋯ , n) mean a set of HFLEs, then we call

$\begin{matrix} sim (s (a_{j}), s (\bar{a})) & = & 1 - \frac{| s (a_{j}) - s (\bar{a}) |}{\sum_{j = 1}^{n} | s (a_{j}) - s (\bar{a}) |}, \\ j = 1, 2, \dots, n . \end{matrix}$ (7)

the similarity between the hesitant fuzzy linguistic values is defined as s (a_j) and the mean is represented as $s (\bar{a})$ .

In real-life situations, the hesitant fuzzy linguistic values a_j (j = 1, 2, ⋯ , n) often utilize the form of a set of n preference values generated by n different individuals. Several individuals can allocate unduly high or unduly low preference values to their preferred or repugnant objects. Therefore, we allocate very low weights to these “false” or “biased” opinions. Then, using Equation 3, we can define the HFHLWA weights as follows.

$\begin{matrix} w_{j} & = & \frac{sim (s (a_{j}), s (\bar{a}))}{\sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))}, \\ j = 1, 2, \dots, n \end{matrix}$ (8)

That is to say w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ . By (2), we have $\begin{matrix} {DHFLWGBM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{p + q} \otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(\frac{a_{i} sim (s (a_{i}), s (\bar{a}))}{\sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))})}^{p} \oplus {(\frac{a_{j} sim (s (a_{j}), s (\bar{a}))}{\sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))})}^{q}))}^{\frac{1}{n (n - 1)}} \\ = 〈 {(\frac{1}{p + q} \otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\begin{matrix} {((\frac{s_{θ (a_{i})} sim (s (a_{i}), s (\bar{a}))}{\sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))}))}^{p} \\ \oplus {(\frac{s_{θ (a_{j})} sim (s (a_{j}), s (\bar{a}))}{\sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))})}^{q} \end{matrix}))}^{\frac{1}{n (n - 1)}}, \\ (\cup_{γ (a_{1}) \in h (a_{1}), γ (a_{2}) \in h (a_{2}), \dots, γ (a_{n}) \in h (a_{n})} {{(\begin{matrix} \frac{1}{p + q} \otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\begin{matrix} {(p γ (a_{i}))}^{sim (s (a_{i}), s (\bar{a})) / - \sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))} \\ \oplus {(q γ (a_{j}))}^{sim (s (a_{j}), s (\bar{a})) / - \sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))} \end{matrix}) \\ \otimes (\begin{matrix} {(p γ (a_{j}))}^{sim (s (a_{j}), s (\bar{a})) / - \sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))} \\ \oplus {(q γ (a_{i}))}^{sim (s (a_{i}), s (\bar{a})) / - \sum_{j = 1}^{n} sim (s (a_{j}), s (\bar{a}))} \end{matrix}) \end{matrix})}^{\frac{2}{n (n - 1)}}} 〉 \end{matrix}$ (9)

We define (9) a dependent hesitant fuzzy linguistic geometric Bonferroni mean (DHFLGBM) operator. Consider that the aggregated value of the DHFHLWA operator is independent of the ordering, hence, it is a neat operator.

4 Model for multiple attribute decision making with hesitant fuzzy linguistic information

In this section, we shall utilize the dependent hesitant fuzzy linguistic geometric Bonferroni mean (DHFLGBM) operator to multiple attribute decision making problems for evaluating the eco-tourism environmental carrying capacity with hesitant fuzzy linguistic information. Let A ={ A₁, A₂, ⋯ , A_m } denoting a discrete set of alternatives, and G ={ G₁, G₂, ⋯ , G_n } means the state of nature. Furthermore, when the decision makers can give some values for the alternative A_i under the state of nature G_j with respect to s_{θ
_ij} with anonymity, these values can be represented as a hesitant fuzzy linguistic values 〈s_{θ
_ij}, h_ij〉. Suppose that the decision matrix $H = {({\tilde{h}}_{ij})}_{m \times n} = {(〈 s_{θ_{ij}}, h_{ij} 〉)}_{m \times n}$ refers to the hesitant fuzzy linguistic decision matrix, where 〈s_{θ
_ij}, h_ij 〉 (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n) are in the form of HFLEs.

Next, we use the dependent hesitant fuzzy linguistic geometric Bonferroni mean (DHFLGBM) operator to the MADM problems for evaluating the eco-tourism environmental carrying capacity with hesitant fuzzy linguistic information.

Step 1. We exploit the decision information given in matrix H, and the DHFLWGBM operator $\begin{matrix} a_{i} = (〈 s_{θ_{i}}, h_{i} 〉) \\ {DHFLWGBM}_{w}^{p, q} (a_{i 1}, a_{i 2}, \dots, a_{in}) \\ = {(\frac{1}{p + q} \otimes_{\begin{matrix} t, j = 1 \\ t \neq j \end{matrix}}^{n} ({(\frac{a_{it} sim (s (a_{it}), s ({\bar{a}}_{i}))}{\sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))})}^{p} \oplus {(\frac{a_{ij} sim (s (a_{ij}), s ({\bar{a}}_{i}))}{\sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))})}^{q}))}^{\frac{1}{n (n - 1)}} \\ = 〈 {(\frac{1}{p + q} \otimes_{\begin{matrix} t, j = 1 \\ t \neq j \end{matrix}}^{n} ({(\frac{s_{θ (a_{it})} sim (s (a_{it}), s ({\bar{a}}_{i}))}{\sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))})}^{p} \oplus {(\frac{s_{θ (a_{jt})} sim (s (a_{jt}), s ({\bar{a}}_{i}))}{\sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))})}^{q}))}^{\frac{1}{n (n - 1)}}, \\ (\cup_{γ_{i 1} \in h_{i 1}, γ_{i 2} \in h_{i 2}, \dots, γ_{in} \in h_{in}} {{(\begin{matrix} \frac{1}{p + q} \otimes_{\begin{matrix} t, j = 1 \\ t \neq j \end{matrix}}^{n} (\begin{matrix} {(p γ_{it})}^{sim (s (a_{it}), s ({\bar{a}}_{i})) / - \sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))} \\ \oplus {(q γ_{ij})}^{sim (s (a_{jt}), s ({\bar{a}}_{i})) / - \sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))} \end{matrix}) \\ \otimes (\begin{matrix} {(p γ_{ij})}^{sim (s (a_{jt}), s ({\bar{a}}_{i})) / - \sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))} \\ \oplus {(q γ_{it})}^{sim (s (a_{it}), s ({\bar{a}}_{i})) / - \sum_{j = 1}^{n} sim (s (a_{jt}), s ({\bar{a}}_{i}))} \end{matrix}) \end{matrix})}^{\frac{2}{n (n - 1)}}} 〉 \end{matrix}$ to obtain the overall preference values a_i (i = 1, 2, ⋯ , m) of the city A_i.

Step 2. Computing the scores S (a_i) (i = 1, 2, ⋯ , m) of the hesitant fuzzy linguistic values a_i (i = 1, 2, ⋯ , m) to order the given the cities A_i (i = 1, 2, ⋯ , m) and the we can choose the optimal one.

Step 3. Order all the cities A_i (i = 1, 2, ⋯ , m) and choose the optimal using the equation S (a_i) (i = 1, 2, ⋯ , m).

Step 4. End.

5 Numerical example

With the increasing awareness of the importance of a healthy human living environment and with the implementation of the sustainable development strategy, eco-tourism has developed rapidly in the past few years and has become a very important trend in the development of tourism. How to coordinate the relationship between the traditional tourism and the ecological environment in the development of ecotourism, how to harmonize human activities with

the natural environment and how to realize the continuous use of tourism resources together with the sustainable tourism has received great attention from the academic field of tourism. We believe that a scientific definition of Environmental Carrying Capacity of Ecotourism (ECCE) and strict implementations are the keys to solving the existing problems. So far, the research on ECCE is rather limited. Most of the current research projects are focusing on forest parks, natural reserves and seaside tourist areas. In this section, we use a practical multiple attribute decision making problems to describe the utilization of the proposed methods. Assume that an organization plans to evaluate the eco-tourism environmental carrying capacity with hesitant fuzzy linguistic information. By collecting all possible information about eco-tourism environmental carrying capacity, the expert group selects 5 potential cities A_i (i = 1, 2, ⋯ , 5) as the candidate ones. The expert group selects four attributes to evaluate the eco-tourism environmental carrying capacity with hesitant fuzzy linguistic information: (1) social environmental carrying capacity (G₁), (2) natural environmental carrying capacity (G₂), (3) economic environmental carrying capacity (G₃); (4) visitor’ satisfaction (G₄). To avoid influence each other, the decision makers should be used to assess the 5 possible cities A_i (i = 1, 2, 3, 4, 5) under 4 attributes in anonymity and then the matrix $H = {({\tilde{h}}_{ij})}_{5 \times 4} = {(〈 s_{θ_{ij}}, h_{ij} 〉)}_{5 \times 4}$ is presented in Table 1.

Table 1
Hesitant fuzzy linguistic decision matrix

G₁ G₂ G₃ G₄

A₁ <s₂, (0.7,0.8)> <s₃,(0.2,0.5,0.6)> <s₄, (0.2,0.5)> <s₂, (0.2,0.4)>

A₂ <s₂, (0.4,0.7)> <s₂, (0.3,0.6,0.8)> <s₂, (0.3,0.4,0.5)> <s₃, (0.3,0.4)>

A₃ <s₅,(0.7, 0.9)> <s₅, (0.6,0.8)> <s₂,(0.2,0.3,0.7)> <s₂, (0.4,0.6)>

A₄ <s₃, (0.4,0.5)> <s₅, (0.4,0.6)> <s₂,(0.6,0.8)> <s₅, (0.3,0.4)>

A₅ <s₅, (0.8,0.9)> <s₂, (0.3,0.7)> <s₅,(0.4,0.5)> <s₃,(0.3,0.4,0.6)>

	G₁	G₂	G₃	G₄
A₁	<s₂, (0.7,0.8)>	<s₃,(0.2,0.5,0.6)>	<s₄, (0.2,0.5)>	<s₂, (0.2,0.4)>
A₂	<s₂, (0.4,0.7)>	<s₂, (0.3,0.6,0.8)>	<s₂, (0.3,0.4,0.5)>	<s₃, (0.3,0.4)>
A₃	<s₅,(0.7, 0.9)>	<s₅, (0.6,0.8)>	<s₂,(0.2,0.3,0.7)>	<s₂, (0.4,0.6)>
A₄	<s₃, (0.4,0.5)>	<s₅, (0.4,0.6)>	<s₂,(0.6,0.8)>	<s₅, (0.3,0.4)>
A₅	<s₅, (0.8,0.9)>	<s₂, (0.3,0.7)>	<s₅,(0.4,0.5)>	<s₃,(0.3,0.4,0.6)>

In the following, we utilize the approach developed to select the best city with hesitant fuzzy linguistic information.

Step 1. We utilize the decision information in matrix H, and the DHFLWGBM operator to obtain the overall preference values a_i of the city A_i (i = 1, 2, 3, 4, 5).

Step 2. Calculate the scores S (a_i) (i = 1, 2, 3, 4, 5) of the overall hesitant fuzzy linguistic values a_i (i = 1, 2, 3, 4, 5). $\begin{matrix} S ({\tilde{h}}_{1}) & = & s_{1 . 23}, S ({\tilde{h}}_{2}) = s_{3 . 35}, S ({\tilde{h}}_{3}) = s_{1 . 09} \\ S ({\tilde{h}}_{4}) & = & s_{2 . 82}, S ({\tilde{h}}_{5}) = s_{3 . 07} \end{matrix}$

Step 3. Rank all the five possible city A_i (i = 1, 2, 3, 4, 5) according to the scores S (a_i) (i = 1, 2, 3, 4, 5). The ordering of cities is as follows: A₂ ≻ A₅ ≻ A₄ ≻ A₁ ≻ A₃, and the best city A₂.

6 Conclusion

In this paper, we investigate the hesitant fuzzy linguistic multiple attribute decision making problems based on the hesitant fuzzy linguistic sets and Geometric Bonferroni mean operator. Then, we have proposed the aggregation operator: dependent hesitant fuzzy linguistic geometric Bonferroni mean (DHFLGBM) operator. Then, we have utilized DHFLWGBM operator to tackle the multiple attribute decision making problems using hesitant fuzzy linguistic information. Finally, a practical example for evaluating the eco-tourism environmental carrying capacity is proposed to testify the method in this paper. In the future, we shall continue working in the extension and application of the developed models to other application domains and other environments [37 –54].

Footnotes

Acknowledgments

The paper is supported the National Natural Science Foundation of China (Grant Nos. 41401146, 41630749, 41571150), Science and Technology Development Project in Jilin Province (201502040NY) and Jilin Province Education Department “the 13th Five-Year Plan” Social Science Research Project (JJKH20170140SK).

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