Performance evaluation of urban ecological environment construction with interval-valued intuitionistic fuzzy information

Abstract

The development problem of small town is a present important issue in our country, concerning the urbanization process, agriculture and rural economy development and sustainable development of national economy directly. It is imperative to accelerate the construction of ecological environment in the small town under the national conditions of the increasing excessively of population, the serious shortage of resources, the environmental pollution, the ecology degradation and the great background of constructing the well-off society and the harmonious society comprehensively. Since the reform and opening-up, the construction of small town makes the enormous achievements, but the ecological environment problem in the urbanization process is serious day by day, and have already become the limiting factor of social progress. So the coordinated development of the economy, society and environment road is the necessary trend of small town development in the future. In this paper, we give some operations on the interval-valued intuitionistic fuzzy sets and further develop the interval-valued intuitionistic fuzzy Hamacher power weighted average (IVIFHPWA) operator. We also establish some desirable properties of this operator, such as idempotency, Boundedness and monotonicity. Then, we apply the IVIFHPWA operator to deal with multiple attribute decision making with interval-valued intuitionistic fuzzy information. Finally, a practical example for evaluating the performance of urban ecological environment construction is used to illustrate the developed procedures.

Keywords

Interval-valued intuitionistic fuzzy numbers operational laws interval-valued intuitionistic fuzzy Hamacher power weighted average (IVIFHPWA) operator urban ecological environment construction performance evaluation

1 Introduction

Atanassov and Gargov [1, 2] introduced the interval-valued intuitionistic fuzzy set (IVIFS). The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. Xu [3] proposed the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator, the interval-valued intuitionistic fuzzy ordered weighted averaging (IVIFOWA) operator and interval-valued intuitionistic fuzzy hybrid aggregation (IVIFHA) operator. Xu and Yager [4] developed some aggregation operators such as the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator and uncertain dynamic intuitionistic fuzzy weighted averaging (UDIFWA) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. Wei [5] developed some aggregation operators such as the dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. Wei and Zhao [6] investigated the induced interval-valued intuitionistic fuzzy correlated averaging (I-IVIFCA) operator and induced interval-valued intuitionistic fuzzy correlated geometric (I-IVIFCG) operator. Yu et al. [7] proposed some interval-valued intuitionistic fuzzy aggregation operators such as the interval-valued intuitionistic fuzzy prioritized weighted average (IVIFPWA) operator. The interval-valued intuitionistic fuzzy set has received more and more attention since its appearance [8 –25].

Urbanization is a major problem of China’s economic and social development. It is an important task during the 12th Five-Year Plan. In recent years, with in-depth development of China’s urbanization, the production and living conditions in rural areas have been improved significantly, so there is some effect of the urbanization in China. It can promote to build a well-off society all over the country and also help to speed the modernization of our country. However, for a long period, with the development of urbanization, people pay more attention to economic development but ignore the protection of the ecological environment; only concern about the economic benefits without considering about the environmental and social benefits. As the rapid development of urbanization process, the ecological environment suffered an enormous challenge in China’s rural area, so we should face more and more environmental crisis. The degradation of environment in the rural areas severely restricts the sustainable development of urbanization and also has negative influence on rural area’s economic development. It will not only bring adverse impact to people’s production and life and to the townships’ survival but also threaten to the whole country’s environmental security. As a major agricultural province of China, some provinces constantly explore the practice of rural urbanization and reform, so the level of urbanization there is in the first place of our country. Nevertheless, with rapid development of urbanization process, the ecological environment in rural areas has been seriously damaged. The environmental pollution has threatened to the agricultural products’ edible security; farmers’ health and also has slowed down the speed of the rural economic growth. Therefore, the government and society should pay attention to protect the rural ecological environment. Based on the above background,studying on the ecological and environmental protection issues during the urbanization of some Province is particularly important.

With the acceleration of urbanization of countryside, to produce a series of ecological environment problems in the course of urban construction Just think of the benefits of economic development without ecological environment of rapid economic growth brought about by considering issues, The ecological environment problems not only impede the construction of rural urbanization, but also seriously affected the normal life of residents living in rural areas, poor living conditions will affect people’s passion and mental state. The cause of these problems is the land use concept of strong, weak awareness of the legal concept of the protection of the peasant masses; the lack of rural enterprises; the extensive mode of economic growth; individual government department management capacity of thin; ecological and environmental protection system is not perfect etc. Many scholars analyze the interaction between rural and urban, rural urbanization and ecological environment of the ecological and environmental effects of stress, ecological and environmental constraints of rural environmental effects of stress, ecological and environmental constraints of rural urbanization etc. Based on the principle of integrity, sustainable principle, fair principle, the principle of human nature, the suitability principle, puts forward the path choice to solve the ecological environment construction, Mainly rely on progress of science and technology development of ecological towns; set up environment protection consciousness of the farmers; realize the transformation of township enterprises in the mode of economic growth, the development of circular economy; the establishment of ecological economic target responsibility system; establish and improve the ecological environment protection system. The construction of rural urbanization and ecological environment coordinated development. In this paper, we give some operations on the interval-valued intuitionistic fuzzy sets and further develop the interval-valued intuitionistic fuzzy Hamacher power weighted average (IVIFHPWA) operator. We also establish some desirable properties of this operator, such as boundedness, idempotency and monotonicity. Then, we apply the IVIFHPWA operator to solve the multiple attribute decision making with interval-valued intuitionistic fuzzy information. Finally, a practical example for evaluating the performance of urban ecological environment construction is used to illustrate the developed procedures.

2 Preliminaries

Atanassov and Gargov [1, 2] further introduced the interval-valued intuitionistic fuzzy set (IVIFS).

Definition 1. [1, 2] Let X be an universe of discourse, An IVIFS $\tilde{A}$ over X is an object having the form: $\tilde{A} = {〈 x, {\tilde{μ}}_{A} (x), {\tilde{ν}}_{A} (x) 〉 | x \in X}$ (1) where ${\tilde{μ}}_{A} (x) \subset [0, 1]$ and ${\tilde{ν}}_{A} (x) \subset [0, 1]$ are interval numbers, and $0 \leq sup ({\tilde{μ}}_{A} (x)) + sup ({\tilde{ν}}_{A} (x)) \leq 1$ , ∀ x ∈ X.

Definition 2. [3] Let $\tilde{a} = ([a, b], [c, d])$ be an interval-valued intuitionistic fuzzy number, a score function S can be represented as follows: $S (\tilde{a}) = \frac{a - c + b - d}{2}, S (\tilde{a}) \in [- 1, 1] .$ (2)

Definition 3. [3] Let $\tilde{a} = ([a, b], [c, d])$ be an interval-valued intuitionistic fuzzy number, an accuracy function H can be represented as follows: $H (\tilde{a}) = \frac{a + b + c + d}{2}, H (\tilde{a}) \in [0, 1] .$ (3)

The larger the value of $H (\tilde{a})$ , the more the degree of accuracy of the interval-valued intuitionistic fuzzy value $\tilde{a}$ .

Based on the score function S and the accuracy function H, in the following, Xu [3] give an order relation between two interval-valued intuitionistic fuzzy values.

Liu [26] developed some interval-valued intuitionistic fuzzy Hamacher aggregation operators.

Definition 4. [26] Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, ⋯ , n) be a collection of interval-valued intuitionistic fuzzy values, and let IVIFHWA: Qⁿ → Q, if $\begin{matrix} {IVIFHWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \oplus_{j = 1}^{n} (ω_{j} {\tilde{a}}_{j}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{j})}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - b_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{j})}^{ω_{j}}}], \\ [\frac{γ \prod_{j = 1}^{n} c_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} c_{j}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} d_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} d_{j}^{ω_{j}}}]) \end{matrix}$ (4)

where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ ,then IVIFHWA is called the interval-valued intuitionistic fuzzy Hamacher weighted averaging (IVIFHWA) operator.

Definition 5. [26] Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, ⋯ , n) be a collection of interval-valued intuitionistic fuzzy values. An interval-valued intuitionistic fuzzy Hamacher ordered weighted averaging (IVIFHOWA) operator of dimension n is a mapping IVIFHOWA:Qⁿ → Q, that has an associated vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Furthermore, $\begin{matrix} {IVIFHOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \oplus_{j = 1}^{n} (w_{j} {\tilde{a}}_{σ (j)}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{w_{j}}}], \\ [\frac{γ \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}}]) \end{matrix}$ (5)

where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{α}}_{σ (j - 1)} \geq {\tilde{α}}_{σ (j)}$ for all j = 2, ⋯ , n.

3 Interval-valued intuitionistic fuzzy hamacher power weighted average (IVIFHPWA) operator

Yager [27] developed a nonlinear weighted average aggregation operator called power average (PA) operator, which can be defined as follows: $PA (a_{1}, a_{2}, \dots, a_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (a_{i})) a_{i}}{\sum_{i = 1}^{n} (1 + T (a_{i}))}$ (6) where $T (a_{i}) = \sum_{\begin{array}{l} j = i \\ j \neq i \end{array}}^{n} S u p (a_{i}, a_{j})$ , and Sup (a, b) is the support for a from b, which satisfies the following three properties:

Sup (a, b) ∈ [0, 1];

Sup (a, b) = Sup (b, a);

Sup (a, b) ≥ Sup (x, y), if |a - b| < |x - y|.

In this section, we shall propose the interval-valued intuitionistic fuzzy Hamacher power weighted average (IVIFHPWA) operator based on the power average [28 –32] operators and Hamacher operations [33 –35].

Definition 6. Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, ⋯ , n) be a collection of interval-valued intuitionistic fuzzy values, and let IVIFHPWA: Qⁿ → Q, if

$\begin{matrix} {IVIFHPWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \oplus_{j = 1}^{n} (\frac{ω_{j} (1 + T ({\tilde{a}}_{j})) {\tilde{a}}_{j}}{\sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} - \prod_{j = 1}^{n} {(1 - a_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} - \prod_{j = 1}^{n} {(1 - b_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}] \\ [\frac{γ \prod_{j = 1}^{n} {(c_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{j}))}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(c_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}, \\ \frac{γ \prod_{j = 1}^{n} {(d_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{j}))}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(d_{j})}^{ω_{j} (1 + T ({\tilde{a}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}}]) \end{matrix}$ (7) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ , and $T ({\tilde{α}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} ω_{i} S u p ({\tilde{α}}_{j}, {\tilde{α}}_{i})$ (8) and $Sup ({\tilde{α}}_{j}, {\tilde{α}}_{i})$ is the support for ${\tilde{α}}_{j}$ from ${\tilde{α}}_{i}$ , with the conditions:

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) \in [0, 1]$ ;

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) = Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j})$ ;

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) \geq Sup ({\tilde{α}}_{s}, {\tilde{α}}_{t})$ ,

if d (α_i, α_j) ≥ d (α_s, α_t), where d is a distance measure. Then IVIFHPWA is called the interval-valued intuitionistic fuzzy Hamacher power weighted average (IVIFHPWA) operator.

It can be easily proved that the IVIFHPWA operator has the following properties.

Theorem 1. (Boundedness) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ be a collection of IVIFN, and let ${\tilde{a}}^{-} = min_{j} {\tilde{a}}_{j}, {\tilde{a}}^{+} = max_{j} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq {IVIFHPWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (9)

Theorem 2. (Idempotency) If all ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ are equal, i.e. ${\tilde{a}}_{j} = \tilde{a}$ for all j, then ${IVIFHPWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (10)

Theorem 3. (Monotonicity) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ be two set of IV IF N s, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then

$\begin{matrix} {IVIFHPWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {IVIFHPWA}_{ω} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (11)

4 Performance evaluation of urban ecological environment construction with interval-valued intuitionistic fuzzy information

In this section, based on the IVIFHPWA operator, we shall investigate the multiple attribute decision making problems for evaluating the performance of urban ecological environment construction with interval-valued intuitionistic fuzzy numbers. Let A = {A₁, A₂, ⋯ , A_m} be a discrete set of alternatives, and G = {G₁, G₂, ⋯ , G_n} be the set of attributes. Suppose that $\tilde{R} = ({\tilde{r}}_{ij})_{m \times n} = ([a_{ij}, b_{ij}], [c_{ij}, d_{ij}])_{m \times n}$ is the interval-valued intuitionistic fuzzy decision matrix, [a_ij, b_ij] ⊂ [0, 1], [c_ij, d_ij] ⊂ [0, 1], b_ij + d_ij ≤ 1, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n.

In the following, we apply the IVIFHPWA operator to multiple attribute decision making based on the IVIFHPWA operator with interval-valued intuitionistic fuzzy information. The method involves the following steps:

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the IVIFHPWA operator

$\begin{matrix} {\tilde{r}}_{i} & = & ([a_{i}, b_{i}], [c_{i}, d_{i}]) = IVIFHPWA ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) = \oplus_{j = 1}^{n} (\frac{ω_{j} (1 + T ({\tilde{r}}_{ij})) {\tilde{r}}_{ij}}{\sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}) \\ = & ([\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))} - \prod_{j = 1}^{n} {(1 - a_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))} - \prod_{j = 1}^{n} {(1 - b_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}], \\ [\frac{γ \prod_{j = 1}^{n} {(c_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{ij}))}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))} + (γ - 1) \prod_{j = 1}^{n} {(c_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}, \\ \frac{γ \prod_{j = 1}^{n} {(d_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{ij}))}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))} + (γ - 1) \prod_{j = 1}^{n} {(d_{ij})}^{ω_{j} (1 + T ({\tilde{r}}_{ij})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{r}}_{ij}))}}]), \\ i = 1, 2, \dots, m . \end{matrix}$ (12) to derive the collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternative, where $T ({\tilde{r}}_{i j}) = \sum_{\begin{matrix} k = 1 \\ k \neq j \end{matrix}}^{n} ω_{i} S u p ({\tilde{α}}_{i j}, {\tilde{α}}_{i k})$ .

Step 2. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the collective overall values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to rank all the alternatives A_i (i = 1, 2, ⋯ , m) and then to select the best one(s) (if there is no difference between two scores $S ({\tilde{r}}_{i})$ and $S ({\tilde{r}}_{j})$ , then we need to calculate the accuracy degrees $H ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{j})$ of the collective overall preference values ${\tilde{r}}_{i}$ and ${\tilde{r}}_{j}$ , respectively, and then rank the alternatives A_i and A_j in accordance with the accuracy degrees $H ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{j})$ .

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

5 Numerical example

The rapid urbanization leads to a series of eco-environmental problems, such as air pollution, urban heat island, water deterioration and decrease of biological diversity. This affects influence directly the state of regional ecological security and sustainable development. As the background of urban ecosystem, urban green space is a suitable way to balance the relationship between rapid urbanization and the eco-environmental problems. Landscape ecology that studies pattern, process and function offers a comprehensive way to solve these complicated ecological problems. Recently, multi-discipline researches on quantifying the pattern of urban green space and the following eco-environmental effects have become hot topics of urban ecological studies. As the most rapid urbanization area in China, Shanghai area’s landscape pattern and environment experience a tremendous change. In this section, we present an empirical case study of evaluating the performance of urban ecological environment construction. The project’s aim is to evaluate the best city from the different performance of urban ecological environment construction. The performance of urban ecological environment construction of four possible cities A_i (i = 1, 2, 3, 4) is evaluated. The expert group must take a decision according to the following four attributes: G₁ is the urban sewage treatment; G₂ is the urban forest coverage; G₃ is the town business carbon emissions management; G₄ is the development capability of urban ecological environment construction. The four possible cities A_i (i = 1, 2, 3, 4) are to be evaluated using the interval-valued intuitionistic fuzzy information by the decision maker under the above four attributes, as listed in the following matrix. $\begin{matrix} \tilde{R} & = & [\begin{matrix} ([0.5, 0.5], [0.5, 0.5]) & ([0.5, 0.7], [0.1, 0.2]) \\ ([0.1, 0.2], [0.5, 0.7]) & ([0.5, 0.5], [0.5, 0.5]) \\ ([0.3, 0.4], [0.2, 0.3]) & ([0.3, 0.4], [0.4, 0.5]) \\ ([0.4, 0.5], [0.2, 0.4]) & ([0.4, 0.6], [0.2, 0.3]) \end{matrix} \\ \begin{matrix} ([0.2, 0.3], [0.3, 0.4]) & ([0.2, 0.4], [0.4, 0.5]) \\ ([0.4, 0.5], [0.3, 0.4]) & ([0.2, 0.3], [0.4, 0.6]) \\ ([0.5, 0.5], [0.5, 0.5]) & ([0.1, 0.3], [0.2, 0.5]) \\ ([0.2, 0.5], [0.1, 0.3]) & ([0.5, 0.5], [0.5, 0.5]) \end{matrix}] \end{matrix}$

In the following, we apply the IVIFHPWA operator to multiple attribute decision making for evaluating the performance of urban ecological environment construction with interval-valued intuitionistic fuzzy information. The method involves the following steps:

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the IVIFHPWA operator with weighting vector w = (0.20, 0.30, 0.30, 0.2), we obtain the overall preference values ${\tilde{r}}_{i}$ of the different cities A_i (i = 1, 2, 3, 4). $\begin{matrix} {\tilde{r}}_{1} & = & ([0.401, 0.514], [0.220, 0.395]) \\ {\tilde{r}}_{2} & = & ([0.432, 0.679], [0.172, 0.280]) \\ {\tilde{r}}_{3} & = & ([0.220, 0.428], [0.377, 0.497]) \\ {\tilde{r}}_{4} & = & ([0.416, 0.655], [0.183, 0.322]) \end{matrix}$

Step 2. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, 3, 4)$ of the overall interval-valued intuitionistic fuzzy values ${\tilde{r}}_{i} (i = 1, 2, 3, 4)$ $\begin{matrix} S ({\tilde{r}}_{1}) & = & 0.150, S ({\tilde{r}}_{2}) = 0.329 \\ S ({\tilde{r}}_{3}) & = & - 0.113, S ({\tilde{r}}_{4}) = 0.283 \end{matrix}$

Step 3. Rank all the cities A_i (i = 1, 2, 3, 4) in accordance with the scores $S ({\tilde{r}}_{i}) (i = 1, 2, 3, 4)$ of the preference values ${\tilde{r}}_{i} (i = 1, 2, 3, 4)$ : A₂ ≻ A₄ ≻ A₁ ≻ A₃, and thus the most desirable city is A₂.

6 Conclusion

With the development of economic, Chinese constructions of roads, highway, railway and urbanization are making rapid progress. It forms the traffic network and solves traffic problems basically. But in some small towns, deficiencies of environmental awareness and environmental protection measures in road construction are making bad effect on the life of residents and the development of the town. There is few law or standard of environmental protection in small towns’ road construction. In this paper, we give some operations on the interval-valued intuitionistic fuzzy sets and further develop the interval-valued intuitionistic fuzzy Hamacher power weighted average (IVIFHPWA) operator. We also establish some desirable properties of this operator, such as idempotency, Boundedness and monotonicity. Then, we apply the IVIFHPWA operator to deal with multiple attribute decision making with interval-valued intuitionistic fuzzy information. Finally, a practical example for evaluating the performance of urban ecological environment construction is used to illustrate the developed procedures. In the future, we shall continue working in the extension and application of the developed operators to other domains [36 –50].

Footnotes

Acknowledgments

The work was supported by the China Postdoctoral Science Foundation funded project (2015M571973) and Social science planning project in Fujian Province (FJ2015B194).

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