Model for evaluating the rural landscape design schemes with fuzzy number intuitionistic fuzzy information

Abstract

With the launch of new countryside construction plan, China has in the transition period of changing traditional rural landscape to the modern rural landscape. Destructive construction which leads to regional characteristics of urban construction in the loss, deterioration of ecological environment has radiated to the new countryside construction. Currently the research on the rural landscape design at an early stage, not on the new countryside construction in the formation of countryside landscape design specific guidance. Rebuilding the new countryside of regional features, protection and transmission of traditional culture and build harmony between man and nature imminent new socialist countryside, its theoretical research and practical guidance is of great significance. Vernacular landscapes as a regional history and culture carriers, the composition of the Vernacular landscape and its significance of mining, will help us protect the countrysides’ traditional culture, reshaping rural regional characteristics, the countryside landscape design will display a new vitality. In this paper, we study on the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. Firstly, we analyze several operations on the fuzzy number intuitionistic fuzzy sets. Afterwards, we design the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator. Then, we apply the FNIFHCA operator to solve multiple attribute decision making under the fuzzy number intuitionistic fuzzy environments. In the end, an example to estimate the rural landscape design schemes is proposed to test the effectiveness of the proposed method.

Keywords

Comprehensive evaluation fuzzy number intuitionistic fuzzy sets hamacher aggregation operators fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator rural landscape design

1 Introduction

With China’s rapid economic development, many questions arise, life become very fast, the ecological environment become deteriorate. These problems are particularly prominent in the city. Busy life and harsh living environment of modern urban city makes people looking forward to a beautiful rural countryside environment. Compared to urban, rural is rich in resources, the original ecological landscape, picturesque, glamour plus the different cultures and cities, for people living in a tense, crowded, complex, multi-polluted city, urban has great attraction. People’s travel sites transferred to the countryside gradually. When the economic, social, scientific and technological development achieved to a certain stage and a certain extent, rural tourism emerged. China’s vast territory, diverse natural landscape, rich in agricultural resources, rural folk and colorful makes it a unique condition in the development of rural tourism [1, 2]. However, the development of rural tourism villages must rely on the unique landscape environment, so there has been rural landscape build for the development of tourism. Shaping the rural landscape, we must integrated into multiple levels of ecological, economic, social, cultural, etc., in order to create a harmonious, comfortable and sustainable rural landscape system ultimately [3]. So the development and construction of rural landscape resources has become a much-needed rural construction practice discuss issues related to the ultimate success or failure of rural economic development. Rural landscape is not only the show of landscape and culture in the traditional sense, but also a measure of the quality of the local natural environment, human settlements standard environment and socio-economic development of the merits of the situation. Learn from the planning perspective, appropriate, clear and strong regional characteristics of urban and rural residents in the rural landscape is the goal; learn from the tourism point of view, combined with landscape planning and tourism planning more and meet the development needs of contemporary country; from economics perspective, rural tourism landscape planning is to adjust the industrial structure, economic development standpoint, is to build a harmonious society and promoting economic development sword [4, 5]. In the background of high-speed development of agricultural economy, the rapid rise of the green and eco-tourism and the quick establish of eco-agriculture landscape. Many papers uses landscape ecology, agricultural economics, tourism economics theory and the method of landscape ecological planning and ecological landscape design, and by collecting real-world experience to build rural landscape and collate, identify important values and basic principles of the rural landscape constructed for the various elements of the rural landscape of doing analysis, and the introduction of “ landscape ternary theory,” the idea of its specific technology, practice oriented, the development of rural tourism and leisure agriculture as the focus of exploration to determine the rural environmental protection methods, and ultimately made the process of protecting rural construction, build, develop specific measures to travel landscape, the rural economic development and environmental protection to achieve a state of equilibrium. Landscape promote tourism, tourism driven economy, the economy reacts to landscape construction, in order to achieve a real sustainable rural development [6 –8].

This paper studies on the problem of multiple attribute decision making using fuzzy number intuitionistic fuzzy information. Firstly, some operations on the fuzzy number intuitionistic fuzzy sets are discussed. Secondly, we propose the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator. Then, we apply the FNIFHCA operator to tackle multiple attribute decision making under the fuzzy number intuitionistic fuzzy environments. To do so, the rest of this paper is organized as follows. In Section 2, several important concepts which are related to fuzzy number intuitionistic fuzzy sets are given. In Section 3 we have developed the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator. In Section 4, the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator is utilized to design the model for multiple attribute decision making problems for evaluating the rural landscape design schemes with fuzzy number intuitionistic fuzzy information. In Section 5, an example for evaluating the rural landscape design schemes is proposed to test the performance of the proposed method. In Section 6, conclusions about this paper are given.

2 Preliminaries

Atanassov et al. [9, 10] introduced the basic ideas of intuitionistic fuzzy set (IFS), that refers to a generalization of fuzzy set [11]. In recent years, several researches have been proposed to study on the intuitionistic fuzzy sets [12 –33]. Liu et al. [31] discussed the fuzzy number intuitionistic fuzzy set (FNIFS). Next, wang et al. [32] presented the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator, and then they showed how to exploit this operator. Wang et al. [33] proposed several aggregation operators, such as the FNIFWG operator, the FNIFOWG operator and fuzzy number intuitionistic fuzzy hybrid geometric (FNIFHG) operator.

Definition 1. [31] Given a fixed set X ={ x₁, x₂, …, x_n }, An FNIFS $\tilde{A}$ over X denotes an object which is represented as follows: $\tilde{A} = {〈 x_{i}, {\tilde{t}}_{A} (x_{i}), {\tilde{f}}_{A} (x_{i}) 〉 | x_{i} \in X}$ (1) where ${\tilde{t}}_{A} (x_{i}) \subset [0, 1]$ and ${\tilde{f}}_{A} (x_{i}) \subset [0, 1]$ are tri-angular fuzzy numbers, and ${\tilde{t}}_{A} (x_{i}) = (a (x_{i}), b (x_{i}), c (x_{i})), X \to [0, 1]$ , ${\tilde{f}}_{A} (x) = (l (x_{i}), m (x_{i}), p (x_{i})), X \to [0, 1]$ , 0 ≤ c (x_i) + p (x_i) ≤ 1, ∀x ∈ X.

For convenience, let ${\tilde{t}}_{A} (x_{i}) = (a (x_{i}), b (x_{i}), c (x_{i})),$ ${\tilde{f}}_{A} (x) = (l (x_{i}), m (x_{i}), p (x_{i}))$ , so $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ and $\tilde{a} (x_{i})$ is defined as an fuzzy number intuitionistic fuzzy value (FNIFV).

Definition 2. [32] Suppose that $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ and $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})), (l (x_{j}), m (x_{j}), p (x_{j})) 〉$ be two FNIFVs, then

(1) $\tilde{a} (x_{i}) \oplus \tilde{a} (x_{j}) = 〈 (a (x_{i}) + a (x_{j}) - a (x_{i}) a (x_{j}), b (x_{i}) + b (x_{j}) - b (x_{i}) b (x_{j}), c (x_{i}) + c (x_{j}) - c (x_{i}) c (x_{j})), (l (x_{i}) l (x_{j}), m (x_{i}) m (x_{j}), p (x_{i}) p (x_{j})) 〉;$

(2) $λ \tilde{a} (x_{i}) = 〈 (1 - {(1 - a (x_{i}))}^{λ}, 1 - {(1 - b (x_{i}))}^{λ}, 1 - {(1 - c (x_{i}))}^{λ}), ({(l (x_{i}))}^{λ}, {(m (x_{i}))}^{λ}, {(p (x_{i}))}^{λ}) 〉, λ \geq 0 .$

Definition 3. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ be a FNIFV, a score function S of a FNIFV $\tilde{a} (x_{i})$ is defined as follows [32, 33]:

$\begin{matrix} S (\tilde{a} (x_{i})) & = & \frac{a (x_{i}) + 2 b (x_{i}) + c (x_{i})}{4} \\ - \frac{l (x_{i}) + 2 m (x_{i}) + p (x_{i})}{4}, \\ S (\tilde{a} (x_{i})) \in [- 1, 1] . \end{matrix}$ (2)

Definition 4. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ be a FNIFV, an accuracy function H of a FNIFV $\tilde{a} (x_{i})$ is defined as follows [32, 33]: $\begin{matrix} H (\tilde{a} (x_{i})) \\ = \frac{(a (x_{i}) + 2 b (x_{i}) + c (x_{i})) + (l (x_{i}) + 2 m (x_{i}) + p (x_{i}))}{4}, \\ H (\tilde{a} (x_{i})) \in [0, 1] . \end{matrix}$ (3)

to evaluate the degree of accuracy of theFNIFV $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ , where $H (\tilde{a} (x_{i})) \in [0, 1]$ . The value of $H (\tilde{a} (x_{i}))$ increases when the accuracy of the FNIFV $\tilde{a} (x_{i})$ increases as well. Utilizing score function S and accuracy function H, the order relation between different fuzzy number intuitionistic fuzzy values is described in paper [32, 33].

3 Fuzzy number intuitionistic fuzzy Hamacher correlated average operators

In the following, we introduced some fuzzy number intuitionistic fuzzy Aggregation operator based on the operations of HIVFEs and Hamacher sum [34, 35].

Definition 5. [36] Let $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})),$ (l (x_j) , m (x_j) , p (x_j))〉 be a collection of FNIFEs, then the fuzzy number intuitionistic fuzzy Hamacher weighted average (FNIFHWA) operator is defined as follows: $\begin{matrix} FNIFHWA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \oplus_{j = 1}^{n} (ω_{j} \tilde{a} (x_{j})) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{j}))}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a (x_{j}))}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a (x_{j}))}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{j}))}^{ω_{j}} - \prod_{j = 1}^{n} (1 - γ_{j}) b {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b (x_{j}))}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{j}))}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - c (x_{j}))}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c (x_{j}))}^{ω_{j}}}), \\ (\frac{γ \prod_{j = 1}^{n} l {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l (x_{j})))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} l {(x_{j})}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} m {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m (x_{j})))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} m {(x_{j})}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} p {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p (x_{j})))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} p {(x_{j})}^{ω_{j}}}) 〉 \end{matrix}$ (4)

where ω = (ω₁, ω₂, …, ω_n) ^T refers to the weight vector of $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Definition 6. [36] Assume that $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})),$ ( l (x_j) , m (x_j) , p (x_j)) 〉 refers to a set of FNIFEs, then the fuzzy number intuitionistic fuzzy Hamacher ordered weighted average (FNIFHOWA) operator is defined as follows:

$\begin{matrix} FNIFHOWA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \oplus_{j = 1}^{n} (w_{j} \tilde{a} (x_{σ (j)})) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{σ (j)}))}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a (x_{σ (j)}))}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a (x_{σ (j)}))}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{σ (j)}))}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b (x_{σ (j)}))}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b (x_{σ (j)}))}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{σ (j)}))}^{w_{j}} - \prod_{j = 1}^{n} {(1 - c (x_{σ (j)}))}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c (x_{σ (j)}))}^{w_{j}}}), \\ (\frac{γ \prod_{j = 1}^{n} l {(x_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l (x_{σ (j)})))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} l {(x_{σ (j)})}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} m {(x_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m (x_{σ (j)})))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} m {(x_{σ (j)})}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} p {(x_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p (x_{σ (j)})))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} p {(x_{σ (j)})}^{w_{j}}}) 〉 \end{matrix}$ (5) where (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n), where $\tilde{a} (x_{σ (j - 1)}) \geq \tilde{a} (x_{σ (j)})$ for all j = 2, …, n, and w = (w₁, w₂, …, w_n) ^T is the aggregation-associated weight vector, and w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ should be satisfied.

With respect to real decision making problems, there is always some inter-dependent characteristics among the attributes. The independence axiom is not able to be satisfied.

Definition 7. [37] Assume that f is a positive real-valued function on X, and μ is a fuzzy measure for X. The discrete Choquet integral of f which is corresponding to μ is given as follows.

$C_{μ} (f) = \sum_{i = 1}^{n} f_{σ (i)} [μ (A_{σ (i)}) - μ (A_{σ (i - 1)})]$ (6) where (σ (1) , σ (2) , …, σ (n)) refers to a permutation of (1, 2, …, n), and the condition f_σ(i-1) ≥ f_σ(i) for all j = 2, …, n is satisfied, A_σ(k) ={ x_σ(j) | j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

In the following, we will discuss how to develop the Hamacher correlated aggregation operators with fuzzy number intuitionistic fuzzy information by utilizing the aggregation rule for FNIFs and Choquet integral.

Definition 8. Suppose that $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})), (l (x_{j}), m (x_{j}), p (x_{j})) 〉$ denotes FNIFEs, and μ be a fuzzy measure on X, next, the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator is defined as follows.

$\begin{matrix} FNIFHCA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) \tilde{a} (x_{σ (j)})) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - a (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - a (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - b (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - b (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - c (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - c (x_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}), \\ (\frac{γ \prod_{j = 1}^{n} l {(x_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l (x_{σ (j)})))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} l {(x_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} m {(x_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m (x_{σ (j)})))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} m {(x_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} p {(x_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p (x_{σ (j)})))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} p {(x_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}) 〉 \end{matrix}$ (7) where (σ (1) , σ (2) , …, σ (n)) refers to a arrange of (1, 2, …, n), where $\tilde{a} (x_{σ (j - 1)}) \geq \tilde{a} (x_{σ (j)})$ for all j = 2, …, n, A_σ(k) ={ x_σ(j) | j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

We can illustrate that the FNIFHCA operator satisfies the following theorems.

Theorem 1.(Idempotency) If all $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ are equal, i.e. $\tilde{a} (x_{j}) = \tilde{a} (x)$ for allj, then $FNIFHCA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \tilde{a} (x)$ (8)

Theorem 2. (Boundedness) Suppose that $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ be a set of FNIF Es, and let $\tilde{a} {(x_{j})}^{-} = min_{j} \tilde{a} (x_{j}), \tilde{a} {(x_{j})}^{+} = max_{j} \tilde{a} (x_{j})$

Then

$\begin{matrix} \tilde{a} {(x)}^{-} \leq FNIFHCA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ \leq \tilde{a} {(x)}^{+} \end{matrix}$ (9)

Theorem 3.(Monotonicity) Suppose that $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ and ${\tilde{a}}^{'} (x_{j}) (j = 1, 2, \dots, n)$ be 2 set of FNIF Es, if $\tilde{a} (x_{j}) \leq {\tilde{a}}^{'} (x_{j})$ , for allj, the following equation should be satisfied.

$\begin{matrix} FNIFHCA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ \leq FNIFHCA ({\tilde{a}}^{'} (x_{1}), {\tilde{a}}^{'} (x_{2}), \dots, {\tilde{a}}^{'} (x_{n})) \end{matrix}$ (10)

Theorem 4. (Commutativity) Assume that $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ and ${\tilde{a}}^{'} (x_{j}) (j = 1, 2, \dots, n)$ be two set of FNIF Es, then

$\begin{matrix} FNIFHCA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ = FNIFHCA ({\tilde{a}}^{'} (x_{1}), {\tilde{a}}^{'} (x_{2}), \dots, {\tilde{a}}^{'} (x_{n})) \end{matrix}$ (11) where ${\tilde{a}}^{'} (x_{j}) (j = 1, 2, \dots, n)$ refers to any permutation of $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ .

4 Model for evaluating the rural landscape design schemes with fuzzy number intuitionistic fuzzy information

Suppose that A ={ A₁, A₂, …, A_m } denotes a discrete set of alternatives, and G = {G₁, G₂, …, G_n} refers to the set of attributes, ω = (ω₁, ω₂, …, ω_n) is the weighting vector of the attribute G_j (j = 1, 2, …, n), where ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n} = 〈 (a_{ij}, b_{ij}, c_{ij}), (l_{ij}, m_{ij}, p_{ij}) 〉_{m \times n}$ is the fuzzy number intuitionistic fuzzy decision matrix, i = 1, 2, …, m, j = 1, 2, …, n.

Afterwards, we exploit the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator to MADM problems for estimating the rural landscape design schemes using fuzzy number intuitionistic fuzzy information.

(1) Use the decision information which are provided from the matrix $\tilde{R}$ , and the FNIFHCA operator. $\begin{matrix} {\tilde{r}}_{i} = FNIFHCA ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) = \oplus_{j = 1}^{n} (μ (A_{σ (j)}) - μ (A_{σ (j - 1)}) {\tilde{r}}_{σ (ij)}) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - a_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - b_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - c_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}), \\ (\frac{γ \prod_{j = 1}^{n} l_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l_{σ (ij)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} l_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} m_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m_{σ (ij)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} m_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} p_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p_{σ (ij)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} p_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}) 〉 \end{matrix}$ (12)

to calculate ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i.

(2) Compute the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the overall fuzzy number intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to sort all the alternatives A_i (i = 1, 2, …, m) and choose the optimal one.

(3) Sort all the alternatives A_i (i = 1, 2, …, m) and choose the optimal one according to $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

(4) End.

5 Numerical example

In the year of 2003, the Chinese central government initiated a series of policies to forge “Integrated Urban-rural Development”. In 2008, a new policy document was issued to allow the circulation of rural land contracted management right. It allows the transfer of the right to households within and beyond the rural community via a valid procedure. This represents a major step taken for Integrated Urban-rural Development. Since then, many regions’ urban-rural planning has led to big changes in layout and functionality of their rural space. The modernization of agriculture, the urbanization and recreational transition of the rural area, result in homogenization and loss of regional landscape characters in many rural areas. Chinese rural landscape has main characters of high heterogeneity, naturalness and strong human influence. Rather than being regarded as merely a consequence of rural development, rural landscape should be a drive to promote rural development. A win-win development model between traditional landscape inheritance and social economic development can be established. The goal is to promote Integrated Rural Development, so the rural area is economically optimized, suitable for living and for tourism, protected environmentally, and features landscapes with strong regional character and cultural heritage. To achieve Integrated Rural Development, rural development needs to be guided by landscape planning, combining the strategies to develop modern agriculture, rural tourism and rural settlement. After discussing some developed countries’ rural area functionality and landscape characters, in this part, we provide an empirical case study for evaluating the rural landscape design schemes. To test five possible rural landscape design schemes, the expert group should make a decision in terms of: (1) G₁ is the artistic quality of rural landscape design schemes; (2) G₂ is the ecological features of rural landscape design schemes; (3) G₃ is sociality of rural landscape design schemes; (4) G₄ is the scientific quality of rural landscape design schemes. The five possible rural landscape design schemes A_i (i = 1, 2, …, 5) should be calculated via the fuzzy number intuitionistic fuzzy numbers by the decision maker using the given attributes, the decision matrices as illustrated in $\tilde{R} = {({\tilde{r}}_{i j})}_{5 \times 4}$ : $\begin{matrix} \tilde{R} & = & [\begin{matrix} 〈 (0.1, 0.2, 0.3), (0.6, 0.7, 0.7) 〉 & 〈 (0.4, 0.5, 0.6), (0.3, 0.3, 0.4) 〉 \\ 〈 (0.2, 0.3, 0.4), (0.4, 0.5, 0.6) 〉 & 〈 (0.3, 0.4, 0.4), (0.3, 0.4, 0.5) 〉 \\ 〈 (0.1, 0.2, 0.2), (0.6, 0.7, 0.8) 〉 & 〈 (0.2, 0.3, 0.4), (0.5, 0.5, 0.6) 〉 \\ 〈 (0.3, 0.4, 0.5), (0.3, 0.4, 0.4) 〉 & 〈 (0.6, 0.6, 0.6), (0.4, 0.4, 0.4) 〉 \\ 〈 (0.5, 0.5, 0.6), (0.3, 0.3, 0.4) 〉 & 〈 (0.3, 0.3, 0.3), (0.4, 0.5, 0.6) 〉 \end{matrix} \\ \begin{matrix} 〈 (0.4, 0.4, 0.5), (0.4, 0.4, 0.5) 〉 & 〈 (0.3, 0.4, 0.5), (0.3, 0.4, 0.4) 〉 \\ 〈 (0.2, 0.3, 0.4), (0.4, 0.5, 0.6) 〉 & 〈 (0.1, 0.1, 0.2), (0.6, 0.7, 0.8) 〉 \\ 〈 (0.5, 0.6, 0.7), (0.3, 0.3, 0.3) 〉 & 〈 (0.1, 0.1, 0.1), (0.7, 0.8, 0.9) 〉 \\ 〈 (0.3, 0.4, 0.4), (0.4, 0.5, 0.6) 〉 & 〈 (0.2, 0.3, 0.4), (0.5, 0.6, 0.6) 〉 \\ 〈 (0.5, 0.5, 0.5), (0.3, 0.3, 0.3) 〉 & 〈 (0.1, 0.2, 0.2), (0.6, 0.7, 0.8) 〉 \end{matrix}] \end{matrix}$

In the following, we use the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator to MADM problems for evaluating the rural landscape design schemes with fuzzy number intuitionistic fuzzy information.

Step 1. Assume that the fuzzy measure of attribute of G_j (j = 1, 2, …, n) and attribute sets of G. $\begin{matrix} μ (G_{1}) = 0.20, μ (G_{2}) = 0.25, μ (G_{3}) = 0.18 \\ μ (G_{4}) = 0.25, μ (G_{1}, G_{2}) = 0.45, \\ μ (G_{1}, G_{3}) = 0.50, μ (G_{1}, G_{4}) = 0.60, \\ μ (G_{2}, G_{3}) = 0.55, μ (G_{2}, G_{4}) = 0.35, \\ μ (G_{3}, G_{4}) = 0.48, μ (G_{1}, G_{2}, G_{3}) = 0.70, \\ μ (G_{1}, G_{2}, G_{4}) = 0.80, μ (G_{1}, G_{3}, G_{4}) = 0.78, \\ μ (G_{2}, G_{3}, G_{4}) = 0.82, μ (G_{1}, G_{2}, G_{3}, G_{4}) = 1.00 \end{matrix}$

Step 2. We utilize the decision information given in matrix $\tilde{R}$ , and the FNIFHCA operator to obtain the overall preference values ${\tilde{r}}_{i}$ of the rural landscape design schemes A_i (i = 1, 2, …, 5). $\begin{matrix} {\tilde{r}}_{1} = 〈 (0.397, 0.483, 0.536), (0.248, 0.327, 0.363) 〉 \\ {\tilde{r}}_{2} = 〈 (0.234, 0.316, 0.419), (0.333, 0.435, 0.541) 〉 \\ {\tilde{r}}_{3} = 〈 (0.209, 0.276, 0.368), (0.419, 0.497, 0.604) 〉 \\ {\tilde{r}}_{4} = 〈 (0.387, 0.488, 0.545), (0.277, 0.368, 0.392) 〉 \\ {\tilde{r}}_{5} = 〈 (0.378, 0.449, 0.461), (0.312, 0.357, 0.445) 〉 \end{matrix}$

Step 3. Compute the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, 5)$ of the overall fuzzy number intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ $\begin{matrix} S ({\tilde{r}}_{1}) = 0.168, S ({\tilde{r}}_{2}) = - 0.113 \\ S ({\tilde{r}}_{3}) = - 0.227, S ({\tilde{r}}_{4}) = 0.129 \\ S ({\tilde{r}}_{5}) = 0.062 \end{matrix}$

Step 4. Rank all the rural landscape design schemes A_i (i = 1, 2, 3, 4, 5) according to the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, 5)$ of the overall fuzzy number intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ : A₁ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₃, and thus the most desirable rural landscape design schemes is A₁.

6 Conclusion

With the continuous developing of Chinese economy, how to develop the rural landscape has gradually become the attention focus of scholars. The insufficient awareness of the prospects for the development of rural landscape led to the unreasonably rural landscape planning. Under the current situation, especially the rapid development of urbanization, the current landscape towns and villages suffering a badly impact, rural landscape gradually being swallowed by urbanization, which produced a tremendous negative impact on natural environment and cultural resources. So, it is particularly crucial to seeking new ways of development of rural landscape to protect to the natural resources and the economic value of rural human resources and improve rural living environment. Particularly, we try to tackle the multiple attribute decision making problems using fuzzy number intuitionistic fuzzy information. We discuss some operations on the fuzzy number intuitionistic fuzzy sets in advance. Afterwards, we design the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator. Furthermore, we apply the FNIFHCA operator to solve multiple attribute decision making based on the fuzzy number intuitionistic fuzzy environments. At last, to test the effectiveness of the proposed method, we provide an example about the rural landscape design. In the future, we will extend the proposed models to some other fields with other novel decision and evaluation ideas [38 –54].

Footnotes

Acknowledgments

This paper is supported by the Project of Natural Science Foundation of Fujian Province (Grant No. 2016J01732).

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