An extended intelligent technique for multiple-attribute decision-making under single-valued neutrosophic sets and applications to sustainable development evaluation of rural ecotourism

Abstract

With the development of society and the times, the socialist modernization cause in China is constantly developing, and the improvement of the economy has also driven the development of rural ecotourism industry in China. The unique natural terrain and cultural customs in rural areas have promoted the continuous development of local ecotourism industry, which has had a positive impact on the economic development of rural areas and not only solved the employment problem of the majority of the population in rural areas. It can also bring a continuous stream of economic benefits and continuously improve the living standards of local residents. Rural ecotourism refers to a rural tourism activity that takes place in rural areas, supported by the agricultural industry, and attracted by the rural environment and typical rural ecotourism resources, with the concept of ecotourism. Rural ecotourism mainly takes the form of rural scenery, agricultural participation, and folk experience, integrating sightseeing experience, cognition, and tourism activities. The main principles of rural tourism development are agricultural support, people-oriented, ecological protection, community participation, income feedback, and ecological home construction. The sustainable development evaluation of rural ecotourism is a classical multiple-attribute decision-making (MADM) issue. In such paper, the generalized weighted Bonferroni mean (GWBM) operator and power average (PA) is constructed for MADM with single-valued neutrosophic sets (SVNSs). Then, the single-valued neutrosophic number power generalized BM (SVNNPGBM) operator is built and then the MADM decision methods are proposed based on the SVNNPGBM operator. Finally, an example for sustainable development evaluation of rural ecotourism and some comparative analysis were given to demonstrate the SVNNPGBM method.

Keywords

Multiple attribute decision making (MADM)single-valued neutrosophic sets (SVNSs)generalized weighted Bonferroni mean (GWBM) operator power average (PA)sustainable development evaluation of rural ecotourism

1. Introduction

With the proposal of China’s rural revitalization strategy, the development of rural ecotourism has become a key measure to achieve high-quality development in rural areas. Rural tourism can enable people living in cities to return to nature [1, 2, 3]. In recent years, the number of rural tourists has ushered in a new round of growth, bringing new opportunities for rural development. The implementation of rural revitalization strategy cannot be separated from the ecological environment, which is crucial for the development of rural revitalization [4, 5, 6]. Research has shown that the development of rural tourism contributes to a virtuous cycle of rural ecological environment and can effectively solve the problem of ecological environment deterioration in some rural areas of China. It is a new engine for high-quality rural revitalization in China. In the 1830s, rural ecotourism began to appear in some countries such as Europe and America, while in the 1990s, rural ecotourism gradually spread in China. At present, there is a wealth of research on rural ecotourism. Some scholars have shifted their research focus to rural ecotourism and analyzed the focus of rural ecotourism from different aspects, providing more reference for subsequent research [6, 7, 8]. Rural ecotourism is a tourism activity that takes into account the interests of tourism consumers and rural residents, and relies on the rural tourism environment for sustainable development. It is a new form of tourism that continuously promotes rural civilization and progress. Rural ecotourism is not just a form in the past, but should also fully utilize economic, cultural, and information technology means, continuously coordinate and promote internal factors of rural tourism, and use tourism to drive the continuous upgrading of traditional and backward industries in rural areas, creating more ecological value wealth for future generations [9, 10, 11]. With the continuous deepening of urbanization, the development of digitalization and new generation information technology, rural areas have achieved their own development. In recent years, more and more urban populations have gradually entered the countryside to enjoy the beautiful life brought by the ecological environment [12, 13, 14]. As of 2019, the number of rural tourism arrivals in China has reached 3.3 billion, and a number of characteristic towns with an output value exceeding 1 billion and characteristic industrial villages with an output value exceeding 100 million have been built. Rural tourism has become a key focus of China’s development and an important engine for the new generation of rural development. According to the development status of rural tourism in recent years compiled by the Ministry of Culture and Tourism and the Ministry of Commerce of China, it can be seen that the revenue of leisure and rural tourism in China has shown an upward trend, with a total of 800 billion yuan in 2018 [14, 15, 16]. The proportion of rural tourism revenue in China’s total tourism revenue is on the rise. Although it slightly declined in 2018 and 2019, the total revenue of rural tourism has increased compared to before. The ecological development of rural tourism meets the needs of the tourism industry and sustainable development of tourist destinations [17, 18]. In 1987, the World Commission on Environment and Development first proposed the concept of sustainable development in “Our Common Future”: sustainable development is a development that meets the needs of contemporary people without harming the ability of future generations to meet their needs. Sustainable development, as a new development concept, has emerged as the only development path that humanity must take in the future. However, behind the vigorous development of traditional rural tourism, there are various signs of unsustainable development: unscientific development and construction of scenic spots, overloaded opening of scenic spots to tourists, low quality tourists with weak environmental awareness, and a series of reasons, which have led to increasingly serious environmental pollution and ecological damage in tourist attractions, as well as the impact on the normal living order of local residents, It has greatly affected the sustainable development of the tourism industry and tourist destinations [19, 20, 21]. The ecological transformation of rural tourism emphasizes the protection of the environment, rational development of resources, and respect for the rights and interests of local residents during the tourism process [22, 23, 24]. Therefore, the ecological transformation of rural tourism has found the best connection point for the disconnection between ecological tourism in the true sense of rural development and the reality of rural tourism industry. It integrates the concept of ecological tourism in rural tourism, can effectively coordinate the relationship between rural tourism development and rural environmental protection, and solve the problem of coordinated development between tourism and ecological environmental protection. The sustainable utilization of tourism resources and the rational allocation of tourism revenue have been a series of long-term problems that have plagued the traditional rural tourism industry [25, 26, 27]. Therefore, it will inevitably become the only choice for the sustainable development of rural tourism. In short, ecotourism provides a new direction for the development of rural tourism, freeing it from the dilemma of damaging the environment and harming the interests of farmers [28, 29, 30]. Therefore, rural tourism should adhere to the path of ecological development, starting from various aspects such as culture, philosophy, planning, and management, allowing the wind of ecology to spread throughout every aspect of rural tourism, and achieving healthy and sustainable development of rural ecotourism.

The MADM is an important research area in modern management science [31, 32, 33, 34, 35, 36]. In recent years, many new advances have been made to enrich and improve the theory and methods of MADM [37, 38, 39, 40, 41]. Zadeh [42] constructed the fuzzy sets (FSs). Atanassov [43] structured the intuitionistic fuzzy sets (IFSs). Smarandache [44] structured the neutrosophic sets (NSs). Wang et al. [45] defined the Single valued neutrosophic sets (SVNSs). The sustainable development evaluation of rural ecotourism is a classical MADM issues [46, 47, 48, 49, 50] and the Single valued neutrosophic sets (SVNSs) could effectively depict the uncertain information during the sustainable development evaluation of rural ecotourism. However, in realistic MADM problems, due to the existence of input variables with the property of inter-influence and inter-dependence, which is not taken into account in the above studied integration operator [51, 52, 53, 54, 55]. To overcome this drawback, the Bonferroni Mean (BM) [56] operator is used as an integration operator with the property of being able to reflect the interrelationship between data. Subsequently, Yager [57] proposed the generalized BM (GBM) operator and applied it to MADM. Xia et al. [58] built the generalized WBM operator. In such paper, connected the generalized WBM operator [58] with PA operator [59] is designed for solving the MADM under SVNSs. The single-valued neutrosophic number power generalized BM (SVNNPGBM) operator operator is built and then the MADM methods are proposed based on SVNNPGBM operator. Eventually, an example for sustainable development evaluation of rural ecotourism and some comparative decision analysis were given. Thus, the main research aim of this paper is listed as follows: (1) In such paper, connected the generalized WBM operator with PA operator is designed for solving the MADM under SVNSs; (2) The single-valued neutrosophic number power generalized BM (SVNNPGBM) operator operator is built and then the MADM methods are proposed based on SVNNPGBM operator; (3) Eventually, an example for sustainable development evaluation of rural ecotourism and some comparative decision analysis were given.

In order to do so, the reminder of this paper proceeds. The SVNSs is concisely reviewed in Sec. 2. The SVNNPGBM operator is built in Sec. 3. The MADM based on the SVNNPGBM operator is built in in Sec. 4. An example application for sustainable development evaluation of rural ecotourism is given in Sec. 5. The conclusion is listed in Sec. 6.

2. Preliminaries

Wang et al. [45] structured the SVNSs.

Definition 1 [45]. The SVNSs $L A$ in $X$ is structured:

$\displaystyle LA=\{(x,LT_{A}(x),LI_{A}(x),LF_{A}(x))|{x\in X}\}$ (1)

with the truth-membership $LT_{A}(x)$ , indeterminacy-membership $LI_{A}(x)$ and falsity-membership $LF_{A}(x)$ , $LT_{A}(x)$ , $LI_{A}(x)$ , $LF_{A}(x)\in[{0,1}]$ , $0\leqslant LT_{A}(x)+LI_{A}(x)+LF_{A}(x)\leqslant 3$ .

The SVNN is structured as $LA=({LT_{A},LI_{A},LF_{A}})$ , where $LT_{A},LI_{A},LF_{A}\in[{0,1}]$ , and $0\leqslant LT_{A}+LI_{A}+LF_{A}\leqslant 3$ .

Definition 2 [60]. Let $LA=({LT_{A},LI_{A},LF_{A}})$ , the score value is:

$\displaystyle SV({LA})=\frac{({2+LT_{A}-LI_{A}-LF_{A}})}{3},SV({LA})\in[{0,1}].$ (2)

Definition 3 [60]. Let $LA=({LT_{A},LI_{A},LF_{A}})$ , the accuracy value is:

$\displaystyle AV({LA})=LT_{A}-LF_{A},AV({LA})\in[{-1,1}].$ (3)

Peng et al. [60] structured the order relation.

Definition 4 [60]. Let $LA=({LT_{A},LI_{A},LF_{A}})$ , $LB=({LT_{B},LI_{B},LF_{B}})$ , let $SV({LA})=\frac{({2+LT_{A}-LI_{A}-LF_{A}})}{3}$ and $SV({LB})=\frac{({2+LT_{B}-LI_{B}-LF_{B}})}{3}$ , and let $AV({LA})=LT_{A}-LF_{A}$ and $AV({LB})=LT_{B}-LF_{B}$ , if $SV({LA})<SV({LB})$ , $LA<LB$ ; if

(1) if $AV({LA})=AV({LB})$ , $LA=LB$ ; (2) if $AV({LA})<AV({LB})$ , $LA<LB$ .

Definition 5 [61]. Let $EA=({ET_{A},EI_{A},EF_{A}})$ and $EB=({ET_{B},EI_{B},EF_{B}})$ , then the normalized Hamming distance between $EA=({ET_{A},EI_{A},EF_{A}})$ and $EB=({ET_{B},EI_{B},EF_{B}})$ is defined:

$\displaystyle HD({EA,EB})=\frac{1}{3}({|{ET_{A}-ET_{B}}|+|{EI_{A}-EI_{B}}|+|{% EF_{A}-EF_{B}}|})$ (4)

Definition 6 [45]. Let $LA=({LT_{A},LI_{A},LF_{A}})$ and $LB=({LT_{B},LI_{B},LF_{B}})$ , the structured operations are:

$\displaystyle(1)\;LA\oplus LB=({LT_{A}+LT_{B}-LT_{A}LT_{B},LI_{A}LI_{B},LF_{A}% LF_{B}});$ $\displaystyle(2)\;LA\otimes LB=({LT_{A}LT_{B},LI_{A}+LI_{B}-LI_{A}LI_{B},LF_{A% }+LF_{B}-LF_{A}LF_{B}});$ $\displaystyle(3)\;\lambda LA=({1-({1-LT_{A}})^{\lambda},({LI_{A}})^{\lambda},(% {LF_{A}})^{\lambda}}),\lambda>0;$ $\displaystyle(4)\;({LA})^{\lambda}=({({LT_{A}})^{\lambda},({LI_{A}})^{\lambda}% ,1-({1-LF_{A}})^{\lambda}}),\lambda>0.$

Definition 7 [58]. Let $p,q,r\geqslant 0$ , $xa_{i}(i=1,2,3,\ldots,n)$ be positive numbers with weight $lw=(lw_{1},lw_{2},\ldots,lw_{n})^{T}$ , $lw_{i}\in[{0,1}]$ , $\sum\limits_{i=1}^{n}{lw_{i}}=1$ . The generalized WBM (GWBM) is defined:

$\displaystyle\text{GWBM}_{lw}^{p,q,r}(xa_{1},xa_{2},\ldots,xa_{n})=\left(\sum% \limits_{i,j,k=1}^{n}{lw_{i}lw_{j}lw_{k}}xa_{i}^{p}xa_{j}^{q}xa_{k}^{r}\right)% ^{1/(p+q+r)}$ (5)

3. SVNNPGBM operator

Then, Xu and Wang [62] defined the single-valued neutrosophic number generalized WBM (SVNNGWBM) operator based on the classical GWBM operator [58].

Definition 8 [62]. Let $s,t,r>0$ , $LA_{i}=(LT_{i},LI_{i},LF_{i})(i=1,2,\ldots,n)$ be a set of SVNNs with their weight $lw_{i}=(lw_{1},lw_{2},\ldots,lw_{n})^{T}$ , $\sum\limits_{i=1}^{n}{lw_{i}}=1$ . If

$\displaystyle\text{SVNNGWBM}_{lw}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})=\left(% \mathop{\oplus}\limits_{i,j,k=1}^{n}lw_{i}lw_{j}lw_{k}(LA_{i}^{s}\otimes LA_{j% }^{t}\otimes LA_{k}^{r})\right)^{1/(s+t+r)}=\left(\begin{array}[]{l}\left(1-% \mathop{\Pi}\limits_{i,j,k=1}^{n}(1-LT_{i}^{s}LT_{j}^{t}LT_{k}^{r})^{lw_{i}lw_% {j}lw_{k}}\right)^{1/(s+t+r)},\\ 1-\left(1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(1-(1-LI_{i})^{s}(1-LI_{j})^{t}(1-% LI_{k})^{r})^{lw_{i}lw_{j}lw_{k}}\right)^{1/(s+t+r)},\\ 1-\left(1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(1-(1-LF_{i})^{s}(1-LF_{j})^{t}(1-% LF_{k})^{r})^{lw_{i}lw_{j}lw_{k}}\right)^{1/(s+t+r))}.\\ \end{array}\right)$ (6)

then $\text{SVNNGWBM}_{lw}^{s,t,r}$ is named the SVNNGWBM operator.

The SVNNGWBM operator [62] has the three decision properties.

Property 1. (Idempotency). If $LA_{i}=LA=(LT,LI,LF)$ ,

$\displaystyle\text{SVNNGWBM}_{lw}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})=LA$ (7)

Property 2. (Monotonicity). Let $LA_{i}=(TT_{A_{i}},II_{A_{i}},FF_{A_{i}})$ , $LB_{i}=(LT_{B_{i}},LI_{B_{i}},LF_{B_{i}})$ . If $LT_{A_{i}}\leqslant LT_{B_{i}},LI_{A_{i}}\geqslant LI_{B_{i}},LF_{A_{i}}% \geqslant LF_{B_{i}}$ holds for all i, then

$\displaystyle\text{SVNNGWBM}_{lw}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})% \leqslant\text{SVNNGWBM}_{lw}^{s,t,r}(LB_{1},LB_{2},\ldots,LB_{n})$ (8)

Property 3. (Boundedness). Let $LA_{i}=(LT_{A_{i}},LI_{A_{i}},LF_{A_{i}})$ . If $LA^{+}=(\max_{i}(TT_{i}),\min_{i}(II_{i}),\min_{i}\linebreak(FF_{i}))$ , $LA^{-}=(\min_{i}(TT_{i}),\max_{i}(II_{i}),\max_{i}(FF_{i}))$ , then

$\displaystyle LA^{-}\leqslant\text{SVNNGWBM}_{lw}^{s,t,r}(LA_{1},LA_{2},\ldots% ,LA_{n})\leqslant LA^{+}$ (9)

Then, the single-valued neutrosophic number generalized power BM (SVNNPGBM) operator is built on the SVNNGWBM operator [62] and PA operator [59].

Definition 9. Let $s,t,r>0$ , $LA_{i}=(LT_{i},LI_{i},LF_{i})(i=1,2,\ldots,n)$ be a set of SVNNs. If

where $T({LA_{a}})=\sum\limits_{\begin{subarray}{c}j=1\\ a\neq j\end{subarray}}^{m}{\textit{Sup}({LA_{a},LA_{j}})}$ , $\textit{Sup}({LA_{a},LA_{j}})$ is the decision support for $LA_{a}$ from $LA_{j}$ , with given conditions: (1) $\textit{Sup}({LA_{a},LA_{b}})\in[{0,1}]$ ; (2) $\textit{Sup}({LA_{b},LA_{a}})=\textit{Sup}({LA_{a},LA_{b}})$ ; (3) $\textit{Sup}\linebreak({LA_{a},LA_{b}})\geqslant\textit{Sup}({LA_{s},LA_{t}})$ , if $HD({LA_{a},LA_{b}})\leqslant HD({LA_{s},LA_{t}})$ , where $d$ is a distance.

The Theorem 1 is obtained.

Theorem 1. Let $s,t,r>0$ , $LA_{i}=(LT_{i},LI_{i},LF_{i})(i=1,2,\ldots,n)$ . The fused value is the SVNN and

$\displaystyle\text{SVNNPGBM}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})=\left(\begin% {array}[]{l}\mathop{\oplus}\limits_{i,j,k=1}^{n}\frac{({1+T({LA_{i}})})}{\sum% \limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T({LA_{j}})})}{\sum\limits_{j=1% }^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})}{\sum\limits_{k=1}^{n}{({1+T(% {LA_{k}})})}}\\ ({LA_{i}^{s}\otimes LA_{j}^{t}\otimes LA_{k}^{r}})\\ \end{array}\right)^{1/(s+t+r)}=\left({\begin{array}[]{l}\left({1-\mathop{\Pi}% \limits_{i,j,k=1}^{n}({1-LT_{i}^{s}LT_{j}^{t}LT_{k}^{r}})^{\frac{({1+T({LA_{i}% })})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T({LA_{j}})})}{\sum% \limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})}{\sum\limits_{k=1% }^{n}{({1+T({LA_{k}})})}}}}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-LI_{i})^{s% }(1-LI_{j})^{t}(1-LI_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-LF_{i})^{s% }(1-LF_{j})^{t}(1-LF_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}\\ \end{array}}\right)$ (11)

Proof:

$\displaystyle LA_{i}^{s}=(LT_{i}^{s},1-(1-LI_{i})^{s},1-(1-LF_{i})^{s});$ $\displaystyle LA_{j}^{t}=(LT_{j}^{t},1-(1-LI_{j})^{t},1-(1-LF_{j})^{t});$ (12) $\displaystyle LA_{k}^{r}=(LT_{k}^{r},1-(1-LI_{k})^{r},1-(1-LF_{k})^{r}).$

Thus,

$\displaystyle LA_{i}^{s}\otimes LA_{j}^{t}\otimes LA_{k}^{r}=\left({\begin{% array}[]{l}LT_{i}^{s}LT_{j}^{t}LT_{k}^{r},\\ 1-(1-LI_{i})^{s}(1-LI_{j})^{t}(1-LI_{k})^{r},\\ 1-(1-LF_{i})^{s}(1-LF_{j})^{t}(1-LF_{k})^{r}\\ \end{array}}\right)$ (13)

Thereafter,

$\displaystyle\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}% }\frac{({1+T({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T% ({LA_{k}})})}{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}({LA_{i}^{s}\otimes LA_% {j}^{t}\otimes LA_{k}^{r}})=\left({\begin{array}[]{l}1-({1-LT_{i}^{s}LT_{j}^{t% }LT_{k}^{r}}){\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})% }}\frac{({1+T({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+% T({LA_{k}})})}{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}},\\ \begin{array}[]{l}({1-({1-(1-LI_{i})^{s}(1-LI_{j})^{t}(1-LI_{k})^{r}})})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}},\\ \end{array}\\ \begin{array}[]{l}({1-({1-(1-LF_{i})^{s}(1-LF_{j})^{t}(1-LF_{k})^{r}})})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\\ \end{array}}\right)$ (14)

Furthermore,

$\displaystyle\mathop{\oplus}\limits_{i,j,k=1}^{n}\frac{({1+T({LA_{i}})})}{\sum% \limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T({LA_{j}})})}{\sum\limits_{j=1% }^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})}{\sum\limits_{k=1}^{n}{({1+T(% {LA_{k}})})}}({LA_{i}^{s}\otimes LA_{j}^{t}\otimes LA_{k}^{r}})=\left({\begin{% array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-LT_{i}^{s}LT_{j}^{t}LT_{k}^{% r}}){\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{(% {1+T({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}% })})}{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}},\\ \begin{array}[]{l}\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-LI_{i})^{s}(1-LI_{j}% )^{t}(1-LI_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}},\\ \end{array}\\ \begin{array}[]{l}\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-LF_{i})^{s}(1-LF_{j}% )^{t}(1-LF_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\\ \end{array}}\right)$ (15)

Therefore,

$\displaystyle\text{SVNNPGBM}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})=\left({% \begin{array}[]{l}\left({1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-LT_{i}^{s}LT_{% j}^{t}LT_{k}^{r}}){\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i% }})})}}\frac{({1+T({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac% {({1+T({LA_{k}})})}{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}}\right)^{1/(s+t% +r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-LI_{i})^{s% }(1-LI_{j})^{t}(1-LI_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-LF_{i})^{s% }(1-LF_{j})^{t}(1-LF_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}\\ \end{array}}\right)$ (16)

Hence, (11) is met.

We have,

$\displaystyle 0\leqslant\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1% }^{n}({1-LT_{i}^{s}LT_{j}^{t}LT_{k}^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}\leqslant 1,$ $\displaystyle 0\leqslant 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,% k=1}^{n}({1-(1-LI_{i})^{s}(1-LI_{j})^{t}(1-LI_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}\leqslant 1,$ (17) $\displaystyle 0\leqslant 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,% k=1}^{n}({1-(1-LF_{i})^{s}(1-LF_{j})^{t}(1-LF_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}\leqslant 1.$

Thereafter,

$\displaystyle 0\leqslant\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1% }^{n}({1-LT_{i}^{s}LT_{j}^{t}LT_{k}^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}$ $\displaystyle{}+1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(% {1-(1-LI_{i})^{s}(1-LI_{j})^{t}(1-LI_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}$ (18) $\displaystyle{}+1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(% {1-(1-LF_{i})^{s}(1-LF_{j})^{t}(1-LF_{k})^{r}})\\ {\frac{({1+T({LA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({LA_{i}})})}}\frac{({1+T% ({LA_{j}})})}{\sum\limits_{j=1}^{n}{({1+T({LA_{j}})})}}\frac{({1+T({LA_{k}})})% }{\sum\limits_{k=1}^{n}{({1+T({LA_{k}})})}}}\\ \end{array}\right)^{1/(s+t+r)}\leqslant 3.$

Thereby finishing this proof.

The SVNNPGBM has the three decision properties.

Property 4. (Idempotency). If $LA_{i}=LA=(LT,LI,LF)$ ,

$\displaystyle\text{SVNNPGBM}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})=LA$ (19)

Property 5. (Monotonicity). Let $LA_{i}=(LT_{A_{i}},LI_{A_{i}},LF_{A_{i}})$ , $LB_{i}=(LT_{B_{i}},LI_{B_{i}},LF_{B_{i}})$ . If $LT_{A_{i}}\leqslant LT_{B_{i}},LI_{A_{i}}\geqslant LI_{B_{i}},LF_{A_{i}}% \geqslant LF_{B_{i}}$ holds for all i, then

$\displaystyle\text{SVNNPGBM}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{n})$ (20) $\displaystyle\leqslant\text{SVNNPGBM}^{s,t,r}(LB_{1},LB_{2},\ldots,LB_{n})$

Property 6. (Boundedness). Let $LA_{i}=(LT_{A_{i}},LI_{A_{i}},LF_{A_{i}})$ . If $LA^{+}=(\max_{i}(LT_{i}),\min_{i}(LI_{i}),\min_{i}\linebreak(LF_{i}))$ , $LA^{-}=(\min_{i}(LT_{i}),\max_{i}(LI_{i}),\max_{i}(LF_{i}))$ , then

$\displaystyle LA^{-}\leqslant\text{SVNNPGBM}^{s,t,r}(LA_{1},LA_{2},\ldots,LA_{% n})\leqslant LA^{+}$ (21)

4. The MADM method based on SVNNPGBM with SVNNs

Then, the SVNNPGBM method is built for MADM under SVNSs. Let $LX=\{{LX_{1},LX_{2},\ldots,LX_{n}}\}$ be attributes, $lw=\{{lw_{1},lw_{2},\ldots,lw_{n}}\}$ be weight of $LX_{j}$ . Let $LP=\{{LP_{1},LP_{2},\ldots,LP_{m}}\}$ be alternatives. $({lq_{ij}})_{m\times n}=({LT_{ij},LI_{ij},LF_{ij}})_{m\times n}$ is the SVNNmatrix. The SVNNPGBM method for MADM will be structured.

Step 1. Structure SVNN matrix $\textit{LXQ}=({lq_{ij}})_{m\times n}=({LT_{ij},LI_{ij},LF_{ij}})_{m\times n}$ .

$\displaystyle LQ=[{lq{}_{ij}}]_{m\times n}=\left[{{\begin{array}[]{*{20}c}{lq_% {11}}&{lq_{12}}&\ldots&{lq_{1n}}\\ {lq_{21}}&{lq_{22}}&\ldots&{lq_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {lq_{m1}}&{lq_{m2}}&\ldots&{lq_{mn}}\\ \end{array}}}\right]$ (22) $\displaystyle LQ=[{lq{}_{ij}}]_{m\times n}=({LT_{ij},LI_{ij},LF_{ij}})_{m% \times n}=\left[{{\begin{array}[]{*{20}c}{({LT_{11},LI_{11},LF_{11}})}&{({LT_{% 12},LI_{1},LF_{12}})}&\ldots&{({LT_{1n},LI_{1n},LF_{1n}})}\\ {({LT_{21},LI_{21},LF_{21}})}&{({LT_{22},LI_{22},LF_{22}})}&\ldots&{({LT_{2n},% LI_{2n},LF_{2n}})}\\ \vdots&\vdots&\vdots&\vdots\\ {({LT_{m1},LI_{m1},LF_{m1}})}&{({LT_{m2},LI_{m2},LF_{m2}})}&\ldots&{({LT_{mn},% LI_{mn},LF_{mn}})}\\ \end{array}}}\right]$ (23)

Step 2. Normalize the $LQ=[{lq_{ij}}]_{m\times n}$ to $\textit{NLQ}=[{\textit{nlq}_{ij}}]_{m\times n}$ .

$\displaystyle\textit{nlq}_{ij}=({\textit{NLT}_{ij},\textit{NLI}_{ij},\textit{% NLF}_{ij}})=\left\{{{\begin{array}[]{l}({LT_{ij},LI_{ij},LF_{ij}}),XL_{j}% \textit{ is a benefit criterion}\\ ({LF_{ij},1-LI_{ij},LT_{ij}}),XL_{j}\textit{ is a cost criterion}\\ \end{array}}}\right.$ (24)

Step 3. According to $\textit{NLQ}=[{\textit{nlq}_{ij}}]_{m\times n}$ , we can fuse all SVNNs $\textit{NLQ}=[{\textit{nlq}_{ij}}]_{m\times n}$ through SVNNPGBM operator to get the SVNNs $\textit{nlq}_{i}({i=1,2,\ldots,m})$ :

$\displaystyle\textit{nlq}_{i}=\text{SVNNPGBM}^{s,t,r}(\textit{nlq}_{i1},% \textit{nlq}_{i2},\ldots,\textit{nlq}_{in})=\left({\begin{array}[]{l}\mathop{% \oplus}\limits_{i,j,k=1}^{n}\frac{({1+T({\textit{nlq}_{if}})})}{\sum\limits_{i% =1}^{n}{({1+T({\textit{nlq}_{if}})})}}\frac{({1+T({\textit{nlq}_{ij}})})}{\sum% \limits_{j=1}^{n}{({1+T({\textit{nlq}_{ij}})})}}\cdot\frac{({1+T({\textit{nlq}% _{ik}})})}{\sum\limits_{k=1}^{n}{({1+T({\textit{nllq}_{ik}})})}}\\ ({\textit{nlq}_{if}^{s}\otimes\textit{nlq}_{ij}^{t}\otimes\textit{nlq}_{ik}^{r% }})\\ \end{array}}\right)^{1/(s+t+r)}=\left({\begin{array}[]{l}\left(\begin{array}[]% {l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-\textit{NLT}_{if}^{s}\textit{NLT}_{% ij}^{t}\textit{NLT}_{ik}^{r}})\\ {\frac{({1+T({\textit{nlq}_{if}})})}{\sum\limits_{i=1}^{n}{({1+T({\textit{nlq}% _{if}})})}}\frac{({1+T({\textit{nlq}_{ij}})})}{\sum\limits_{j=1}^{n}{({1+T({% \textit{nlq}_{ij}})})}}\frac{({1+T({\textit{nlq}_{ik}})})}{\sum\limits_{k=1}^{% n}{({1+T({\textit{nllq}_{ik}})})}}}\\ \end{array}\right)^{1/(s+t+r)},\\ \begin{array}[]{l}({1-({1-(1-\textit{NLI}_{if})^{s}(1-\textit{NLI}_{ij})^{t}(1% -\textit{NLI}_{ik})^{r}})})\\ {\frac{({1+T({\textit{nlq}_{if}})})}{\sum\limits_{i=1}^{n}{({1+T({\textit{nlq}% _{if}})})}}\frac{({1+T({\textit{nlq}_{ij}})})}{\sum\limits_{j=1}^{n}{({1+T({% \textit{nlq}_{ij}})})}}\frac{({1+T({\textit{nlq}_{ik}})})}{\sum\limits_{k=1}^{% n}{({1+T({\textit{nllq}_{ik}})})}}},\\ \end{array}\\ \begin{array}[]{l}({1-({1-(1-\textit{NLF}_{if})^{s}(1-\textit{NLF}_{ij})^{t}(1% -\textit{NLF}_{ik})^{r}})})\\ {\frac{({1+T({\textit{nlq}_{if}})})}{\sum\limits_{i=1}^{n}{({1+T({\textit{nlq}% _{if}})})}}\frac{({1+T({\textit{nlq}_{ij}})})}{\sum\limits_{j=1}^{n}{({1+T({% \textit{nlq}_{ij}})})}}\frac{({1+T({\textit{nlq}_{ik}})})}{\sum\limits_{k=1}^{% n}{({1+T({\textit{nllq}_{ik}})})}}}\\ \end{array}\\ \end{array}}\right)$ (25)

Step 4. Obtain the $SV({\textit{nlq}_{i}}),AV({\textit{nlq}_{i}})({i=1,2,\ldots,m})$ .

$\displaystyle SV({\textit{nlq}_{i}})=\frac{({2+\textit{NLT}_{i}-\textit{NLI}_{% i}-\textit{NLF}_{i}})}{3},AV({\textit{nlq}_{i}})=\textit{NLT}_{i}-\textit{NLF}% _{i}$ (26)

Step 5. Rank the $LP_{i}({i=1,2,\ldots,m})$ through $SV({\textit{nlq}_{i}})$ , $AV({\textit{nlq}_{i}})$ .

Step 6. End.

5. The empirical example and comparative analysis

5.1 An empirical example

Rural tourism refers to using rural areas as a tourist destination, attracting tourists with their native beauty, simplicity, and sense of return. The main audience of rural tourism is urban residents, who can stay away from the hustle and bustle of the city, come closer to nature, return to nature, and experience the beauty of the fields. However, with the rapid development of rural tourism in China, more and more problems have emerged, such as repetitive development of tourism products, low quality, environmental damage, and damage to the interests of farmers, making rural tourism face development bottlenecks. Ecological tourism is a high-end tourism project. Based on the actual situation of rural tourism in China, most rural tourism cannot be upgraded to pure ecological tourism. There are two specific reasons: firstly, rural tourism operators often lack advanced management concepts, development and operation are often not scientific enough, and there is a lack of understanding of the importance of environmental protection, As a result, most tourism projects fall far short of the requirements of ecotourism in terms of management and operation. In addition, ecotourism also puts forward requirements for tourists to protect the environment. Tourists should take on the responsibility of protecting the environment during the tourism process, but most tourists tend to forget about the mountains and rivers while visiting, and neglect their responsibility for environmental protection. Secondly, ecotourism also has restrictions on the number of tourists. Only Clique can be called ecotourism, and rural tourism is usually large in scale. In summary, although rural tourism cannot be transformed into pure ecological tourism, it can retain the characteristics of rural areas, supplemented by ecological characteristics, and follow the development model of rural ecological tourism. This allows tourists to experience the local customs while feeling the atmosphere of environmental protection and education, making rural tourism a comprehensive project that integrates entertainment, tourism, and ecological experience. While increasing farmers’ income, it achieves the goal of protecting the environment. This is a rural tourism development model that is in line with China’s national conditions. The sustainable development evaluation of rural ecotourism is a classical MADM issue. In this paper, an empirical application of sustainable development evaluation of rural ecotourism is given through SVNNPGBM method. There are five rural natural ecological landscape resource points are evaluated their embedded computers performance. In order to assess five rural natural ecological landscape resource points fairly, the experts give their information with the four decision attributes: ⟀ LX1 is the rural tourism resource system; ⟁ LX2 is the rural environmental system cost; ⟂ LX3 is the rural socio-economic system; ⟃ LX4 is the rural management service system. Evidently, LX2 is the cost, others are the benefit. Then, the SVNNPGBM method is applied to MADM for solving the sustainable development evaluation of rural ecotourism with SVNNs. The SVNNPGBM method involves the decision steps as below:

Step 1. Set up the SVNNmatrix $LQ=({lq_{ij}})_{5\times 4}$ as in Table 1.

Table 1
SVNN matrix

	LX ${}_{1}$	LX ${}_{2}$	LX ${}_{3}$	LX ${}_{4}$
LP ${}_{1}$	(0.43, 0.87, 0.24)	(0.41, 0.40, 0.37)	(0.15, 0.44, 0.45)	(0.12, 0.20, 0.33)
LP ${}_{2}$	(0.47, 0.18, 0.34)	(0.38, 0.37, 0.46)	(0.12, 0.17, 0.50)	(0.21, 0.32, 0.20)
LP ${}_{3}$	(0.40, 0.17, 0.38)	(0.43, 0.37, 0.50)	(0.20, 0.42, 0.40)	(0.60, 0.20, 0.40)
LP ${}_{4}$	(0.44, 0.12, 0.37)	(0.37, 0.50, 0.24)	(0.44, 0.28, 0.22)	(0.37, 0.23, 0.48)
LP ${}_{5}$	(0.35, 0.39, 0.45)	(0.43, 0.40, 0.41)	(0.36, 0.38, 0.74)	(0.23, 0.40, 0.36)

Table 2

The $\textit{NLQ}=[{\textit{nlq}_{ij}}]_{5\times 4}$

	LX ${}_{1}$	LX ${}_{2}$	LX ${}_{3}$	LX ${}_{4}$
LP ${}_{1}$	(0.43, 0.87, 0.24)	(0.37, 0.60, 0.41)	(0.15, 0.44, 0.45)	(0.12, 0.20, 0.33)
LP ${}_{2}$	(0.47, 0.18, 0.34)	(0.46, 0.63, 0.38)	(0.12, 0.17, 0.50)	(0.21, 0.32, 0.20)
LP ${}_{3}$	(0.40, 0.17, 0.38)	(0.50, 0.63, 0.43)	(0.20, 0.42, 0.40)	(0.60, 0.20, 0.40)
LP ${}_{4}$	(0.44, 0.12, 0.37)	(0.24, 0.50, 0.37)	(0.44, 0.28, 0.22)	(0.37, 0.23, 0.48)
LP ${}_{5}$	(0.35, 0.39, 0.45)	(0.41, 0.60, 0.43)	(0.36, 0.38, 0.74)	(0.23, 0.40, 0.36)

Table 3

The $\textit{nlq}_{i}({i=1,2,\ldots,5})$ by SVNNPGBM operator ( $s=t=r=1$ )

Alternatives	$\textit{nlq}_{i}({i=1,2,\ldots,5})$
LP ${}_{1}$	(0.5016, 0.4326, 0.3226)
LP ${}_{2}$	(0.7398, 0.2216, 0.3337)
LP ${}_{3}$	(0.5326, 0.5043, 0.3224)
LP ${}_{4}$	(0.6537, 0.5362, 0.3161)
LP ${}_{5}$	(0.6953, 0.6218, 0.2319)

Table 4

The $SV({\textit{nlq}_{i}})({i=1,2,\ldots,5})$

Alternatives	$SV({\textit{nlq}_{i}})({i=1,2,\ldots,5})$	Order
LP ${}_{1}$	0.5775	5
LP ${}_{2}$	0.7280	1
LP ${}_{3}$	0.6091	4
LP ${}_{4}$	0.6213	3
LP ${}_{5}$	0.6359	2

Table 5

Results of different methods

Methods	Order	The optimal choice	The worst choice
SVNNWA [60]	$LP_{2}>LP_{5}>LP_{4}>LP_{3}>LP_{1}$	$LP_{2}$	$LP_{1}$
SVNNWG [60]	$LP_{2}>LP_{5}>LP_{3}>LP_{4}>LP_{1}$	$LP_{2}$	$LP_{1}$
SVNN-MABAC [63]	$LP_{2}>LP_{5}>LP_{4}>LP_{3}>LP_{1}$	$LP_{2}$	$LP_{1}$
SVNN-TOPSIS [64]	$LP_{2}>LP_{5}>LP_{4}>LP_{3}>LP_{1}$	$LP_{2}$	$LP_{1}$
SVNNPGBM operator	$LP_{2}>LP_{5}>LP_{4}>LP_{3}>LP_{1}$	$LP_{2}$	$LP_{1}$

Step 2. Normalize $LQ=({lq_{ij}})_{5\times 4}$ to $\textit{NLQ}=[{\textit{nlq}_{ij}}]_{5\times 4}$ (See the Table 2).

Step 3. Obtain the $\textit{nlq}_{i}({i=1,2,\ldots,5})$ by utilizing SVNNPGBM operator (Table 3).

Step 5. Obtain the $SV({\textit{nlq}_{i}})({i=1,2,\ldots,5})$ (Table 4).

Step 6. From Table 4, the order is $LP_{2}>LP_{5}>LP_{4}>LP_{3}>LP_{1}$ , and the best rural natural ecological landscape resource point is $LP_{2}$ .

5.2 Compare analysis

The SVNNPGBM operator is made comparison with SVNNWA and SVNNWG operator [60], SVNN-MABAC method [63] and SVNN-TOPSIS method [64]. The results are recorded in Table 5.

Obtained from Table 5, it is evident that the optimal rural natural ecological landscape resource point is $LP_{2}$ , while the worst rural natural ecological landscape resource point is $LP_{1}$ in most decision situations. In other words, the ranking results of these five methods are slightly different. Different methods can effectively solve the MADM problem from different angles. The proposed SVNNPGBM operator has the following advantages: the proposed SVNNPGBM can effectively capture the intrinsic connection between attributes in MADM problems, and the decision maker can change the parameters in the operator according to his own decision preference, risk attitude and other subjective consciousness in the decision process, so that the decision result can reach the satisfaction of the decision maker.

6. Conclusion

The ecological transformation of rural tourism emphasizes environmental protection, rational development of resources, and respect for the rights and interests of local residents during the tourism process. Therefore, the ecological transformation of rural tourism has found the best connection point for the disconnection between ecological tourism in the true sense of rural development and the reality of rural tourism industry. It integrates the concept of ecological tourism in rural tourism, can effectively coordinate the relationship between rural tourism development and rural environmental protection, and solve the problem of coordinated development between tourism and ecological environmental protection. The sustainable utilization of tourism resources and the rational allocation of tourism revenue have been a series of long-term problems that have plagued the traditional rural tourism industry. Therefore, it will inevitably become the only choice for the sustainable development of rural tourism. The sustainable development evaluation of rural ecotourism is a classical MADM issues. In such paper, the generalized WBM operator and PA operator is designed for solving the MADM under SVNSs. The SVNNPGBM operator is built and then the MADM methods are proposed based on the SVNNPGBM operator. Eventually, an example for sustainable development evaluation of rural ecotourism and some comparative decision analysis are given to demonstrate the SVNNPGBM method. Thus, the main research contribution of this paper is outlined as follows: (1) In such paper, connected the generalized WBM operator with PA operator is designed for solving the MADM under SVNSs; (2) The single-valued neutrosophic number power generalized BM (SVNNPGBM) operator operator is built and then the MADM methods are proposed based on SVNNPGBM operator; (3) Eventually, an example for sustainable development evaluation of rural ecotourism and some comparative decision analysis were given.

In the future, the theoretical decision framework will be improved by merging several rankings resulting from different MADM techniques with metamethods. In addition, the established processes for ranking rural natural ecological landscape resource points will be modified to allow for estimating uncertainty by employing a range of fuzzy set types. In addition, we want to develop new hybrid methods for sustainable development evaluation of rural ecotourism by combining algorithms for calculating weights with MADM methodology.

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An extended intelligent technique for multiple-attribute decision-making under single-valued neutrosophic sets and applications to sustainable development evaluation of rural ecotourism

Abstract

Keywords

1. Introduction

2. Preliminaries

5.1 An empirical example

Table 1 SVNN matrix

6. Conclusion

References

Table 1
SVNN matrix