Some Interval-Valued Intuitionistic Fuzzy Dombi Heronian Mean Operators and their Application for Evaluating the Ecological Value of Forest Ecological Tourism Demonstration Areas

With China’s sustained economic development and constant increase in national income, Chinese nationals’ tourism consumption rate increases. As a major Chinese economic development engine, the domestic tourism industry has entered a transition period operation pattern featured by diversified products. Among them, as a new hot spot of the tourism industry in China, ecological tourism has enjoyed rapid development, with great potential. Thus, the ecological value evaluation of forest ecological tourism demonstration areas is very important to the domestic tourism industry. In this paper, we propose some Dombi Heronian mean operators with interval-valued intuitionistic fuzzy numbers (IVIFNs). Then, two MADM (multiple attribute decision making) methods are proposed based on IVIFWDHM (interval-valued intuitionistic fuzzy weighted Dombi Heronian mean) and IVIFWDGHM (interval-valued intuitionistic weighted Dombi geometric Heronian mean) operators. Finally, we gave an experimental case for evaluating the ecological value of forest ecological tourism demonstration area to show the proposed decision methods.

Based on the Dombi T-norm and T-conorm, we can give the operational rules of IVIFNs.

Proofs.
According to Property 1, we have: Therefore, Then, Property 3 is proved.

The IVIFWDHM Operator
In real MADM, it's very important to pay attention to attribute weights. Thus, we must define the interval-valued intuitionistic fuzzy weighted Dombi Heronian mean (IVIFWDHM) operator.
Then, we have: Similarly, we have: And, The proof is similar to Property 2 of IVIFDHM, therefore, it is omitted here.

The IVIFWDGHM Operator
In some practical MADM, it's very important to pay attention to attribute weights; we define the interval-valued intuitionistic weighted Dombi GHM (IVIFWDGHM) operator.

Numerical Example
The forest ecological tourism demonstration area has a variety of functions, which is an important part of the national ecological tourism demonstration area. It is also destined to the forest ecological tourism demonstration area of ecological value. The corresponding study of forest ecological tourism demonstration area of ecological value is important for the promotion of human welfare and social sustainable development. That is to say, the assessment of the ecological value of the forest ecological tourism demonstration area is one of the hot spots and key issues of the domestic and international ecology academic circles and the society. At present, the current urbanization, rapid industrialization, environmental pollution situations are grim, and there are many people living with less realistic conditions of these three heavy squeezes. This highlights that the research of forest ecological tourism demonstration area of ecological value has become even more urgent. The problems of evaluating the ecological value of forest ecological tourism demonstration areas are classical MADM problems [66][67][68][69][70][71][72][73][74][75]. Thus, we give an example to solve the MADM for evaluating the ecological value of forest ecological tourism demonstration areas with IVIFNs. There are five possible forest ecological tourism demonstration areas A i (i = 1, 2, 3, 4, 5), to been assessed. The experts use four evaluation attributes to assess five forest ecological tourism demonstration areas: (1) G 1 is the tourism and leisure value; (2) G 2 is the material production value; (3) G 3 is the scientific research and cultural value; (4) G 4 is the climatic regulation value. The five possible forest ecological tourism demonstration areas are to be assessed with IVIFNs (attributes weight w = (0.4, 0.2, 0.3, 0.1)), as shown in the Table 1. Then, we use the approach developed for selecting the best forest ecological tourism demonstration area.
Step 1. According to IVIFNs r ij (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4), we fuse all the IVIFNs r ij by IVIFWDHM (IVIFWDGHM) operator, to calculate the IVIFNs A i (i = 1, 2, 3, 4, 5) of the forest ecological tourism demonstration area A i . Let p = 2, q = 1,γ= 3, then the fused values are depicted in Table 2. Step 2. By Table 2, the score results of the forest ecological tourism demonstration areas are in Table 3. Step 3. By Table 3, the order of forest ecological tourism demonstration areas is listed in Table 4. The best forest ecological tourism demonstration area is A 3 . Table 4. Order of the forest ecological tourism demonstration areas.

Influence Analysis
The proposed methods have two independent parameters, p and q, which play an important role in the calculation of the results. Hence, different score values and orders may be derived when p and q change. Furthermore, the integer values of p and q in the range of 1-10 usually receive more attention in practical applications. We investigated the influences of p and q on the decision-making from the results of the IVIFWDHM operator and the IVIFWDGHM operator. Firstly, the different p and q are assigned in a certain order with p i , q j (i = 1, 2, · · · , 10; j = 1, 2, . . . , 10). The scores and ranking results of A i (i = 1, 2, 3, 4, 5) are given in Figure 1 and Figure 4. Then, the influence of q (or p) on the score is investigated from the result of A 3 , when the p value is fixed and q changes from 1 to 10. Details can be found in Figure 2 and Figure 5. Moreover, the influence of p + q on the score is investigated from the result of A 3 , when p + q changes from 2 to 20. Details can be found in Figure 3 and Figure 6.       According to Figures 1 and 4, we can conclude that different scores of alternatives can be derived according to different p and q. The differences between the maximum and minimum scores of A 1 to A 5 from the IVIFWDHM operator are 0.0243, 0.0265, 0.0326, 0.0132 and 0.0296, respectively, and the differences between the maximum and minimum scores of A 1 to A 5 from the IVIFWDHM operator are 0.0385, 0.0187, 0.0365, 0.0164 and 0.0212, respectively. It can be seen that the fluctuation range of the scores from the IVIFWDHM operator and the IVIFWDGHM operator are small. Different p and q values have little effect on the scores of the two methods for A 1 to A 5 , so the scores of IVIFWDHM operator and IVIFWDGHM operator are stable for different p and q values. However, A 1 to A 5 show some variation rules for different p and q values. The following takes the scores of A 3 from the IVIFWDHM operator and the IVIFWDGHM operator as examples to show the variation rules: (1) For the IVIFWDHM operator, Figure 2 shows that when the p value is fixed and the q value changes from 1 to 10, the fluctuation trend of the score is more complex, most of which has a decreasing trend; Figure 3 shows that when the p + q value changes from 2 to 20, the fluctuation range increases first and then decreases, and reaches its maximum when the p + q value equals 11, then the fluctuation range is from 0.0450 to 0.0776 when p + q value equals 11, which is the same as that of the scores about the 100 combinations of A 3 . But the average score of each group has little difference under a certain p + q value, the difference between the maximum average score and the minimum average score is 0.0070, while the fluctuation range is from 0.0458 to 0.0528, which is only 21.45% of the amplitude of the fluctuation range about the 100 combinations of A 3 . (2) For the IVIFWDGHM operator, Figure 5 shows that when the p value is fixed and the q value changes from 1 to 10, the fluctuation trend of the score is more complex, and the decreasing trend is dominant; Figure 6 shows that when p + q value changes from 2 to 20, the fluctuation range increases first and then decreases, and reaches its maximum when p + q value equals 11, then the fluctuation range is from 0.3050 to 0.3414, which is the same as that of the scores about the 100 combinations of A 3 . But the average score of each group has little difference under a certain p + q value, the difference between the maximum average score and the minimum average score is 0.0075, while the fluctuation range is from 0.3050 to 0.3414, which is only 20.46% of the amplitude of the fluctuation range about the 100 combinations of A 3 .   In this section, the influence of p and q on the scores are investigated for the IVIFWDHM operator and the IVIFWDGHM operator. Although results illustrate the regularity of the proposed method for the different p and q, different p and q values have little effect on the score values for the two methods . So the scores of the IVIFWDHM operator and the IVIFWDGHM operator are stable for different p and q values. When the scores of the subjects are similar, it is likely that the ranking of evaluation will change, but when there is a certain gap in the scores of the subjects, the ranking of evaluation will not change. Thus, the proposed methods are sufficient to solve practical MADM. Furthermore, the proposed methods show high robustness for information fusion in MADM.

Comparative Analysis
We compare the IVIFWDHM and IVIFWDGHM operators with the IVIFWA operator [64], the IVIFWG operator [4], the gray relational analysis method [47] and correlation coefficient [76]. The results are given in Table 5.

Comparative Analysis
We compare the IVIFWDHM and IVIFWDGHM operators with the IVIFWA operator [64], the IVIFWG operator [4], the gray relational analysis method [47] and correlation coefficient [76]. The results are given in Table 5.

Comparative Analysis
We compare the IVIFWDHM and IVIFWDGHM operators with the IVIFWA operator [64], the IVIFWG operator [4], the gray relational analysis method [47] and correlation coefficient [76]. The results are given in Table 5. Table 5. Order of the tourism scenic spots.

Methods Order
IVIFWA operator [64] A 3 > A 1 > A 4 > A 2 > A 5 IVIFWG operator [4] A 3 > A 1 > A 2 > A 4 > A 5 Gray Relational Analysis Method [47] A 3 > A 5 > A 1 > A 2 > A 4 Correlation Coefficient [76] A 3 > A 1 > A 2 > A 4 > A 5 From the above analysis, we get the same best forest ecological tourism demonstration areas, while the four methods' orders are slightly different. However, the existing methods with IVIFNs don't consider the interrelationship among the arguments. Our proposed IVIFWDHM and IVIFWDGHM operators consider the interrelationship among aggregated arguments.
Xu and Chen [77] defined some Bonferroni mean for aggregating the IVIFNs. However, these Bonferroni mean for aggregating the IVIFNs only consider the relationship information between two arguments, and do not consider the relationship information among more than two arguments.

Conclusions
Traditional mass tourism attaches much importance to economic profits, while it is intended to meet the aesthetic needs of people. However, behind the high-speed development of tourism, there are difficulties in solving the relationship between man and the nature with the problems aroused in the ecological environment and resources in tourist spots. Eco-tourism is the result of advocating a harmonious coexistence between human beings and the nature, which also indicates both a new concept of tourism, and the ecological conceptions reflected in recreation behaviors of tourists. It advocates such ideas as the harmonious coexistence of man and the nature, and enjoying the nature without destroying the environment, which essentially derive from a concept of humans going back to the nature. Superficially, it comes from people's attention to "the exterior" of traditional mass tourism, while philosophically, it suggests people's awakening to environmental ethics. Tourist theories are becoming more mature with a change of paradigm, in which tourism is developing from the activities of a privileged minority, to a popular mass behavior, observed at present. Essentially, we perceive eco-tourism to be a kind of ecological culture based on the recognition of man's relationship with the nature, and, that we have entered a new tourism paradigm under the guidance of eco-ethics. Furthermore, it reflects on the ideas of the traditional man-oriented mass tourism and corrects people's misunderstanding about tourist resources and the ecological environment. In this paper, we investigated MADM with IVIFNs. Then, we utilized HM and Dombi operations to design some HM operators with IVIFNs: IVIFDHM operator, IVIFWDHM operator, IVIFDGHM operator and IVIFWDGHM operator. The main characteristic of these proposed operators were studied. Then, we employed the IVIFWDHM and IVIFWDGHM operators to propose two models for MADM problems with IVIFNs. Finally, a real experimental case for evaluating the ecological value of forest ecological tourism demonstration area was used to show the developed approach. In the subsequent studies, the extension and application of IVIFNs need to be studied in many other uncertain environments [78][79][80][81][82][83][84] and other applications [85][86][87][88][89][90].
Author Contributions: L.W., G.W., J.W. and C.W. conceived and worked together to achieve this work, L.W. compiled the computing program by Excel and analyzed the data, L.W. and G.W. wrote the paper. All authors have read and agreed to the published version of the manuscript.