Approach for multiple attribute decision-making with interval grey number based on Choquet integral

Abstract

With regard to the interaction between attributes which has influenced on decision results in Multiple attribute decision making, a multiple attribute decision making method of interval grey number based on degree of greyness and Choquet integral is presented. Degree of greyness of interval grey number is proved as a measure according to its definition, then discrete Choquet integral of degree of greyness of interval grey number is defined, and some of its properties are investigated. The weight of each attribute or attribute set is determined by Using degree of greyness of interval grey number and considering expert opinion, and the effect evaluation vector of each scheme are integrated based on Choquet integral. Finally, an example on Ning meng of the Yellow River disaster was also presented to illustrate the usefulness and effectiveness of the proposed method.

Keywords

Multiple attribute decision making interaction degree greyness Choquet integral

1 Introduction

Multiple Attribute Decision Making [1] is a system analysis method to study uncertain decision making problems, it aims to improve the decision-making process and find the optimal solution from a range of alternatives to meet certain requirements. Establishing decision attribute index system is one of the key issues affecting the rationality of decision results, so the decision attribute index system established should be representative, complete and independent; However, in practice independence is often difficult to achieve, so in most cases decision-makers ignore independence between the attribute index systems [2]. Correlations between attributes destroy the additive property of attribute weights, after weight summation, the conclusion obtained does not match with the reality. literature [3] showed that correlations often exist in practical decision-making problems, which becomes the bottleneck of decision analysis theory in application, and it also described the development of multi-attribute decision analysis theory based on correlation and analyzed the characteristic and hotspot of its research; literature [4] summarized correlations between attributes and analyzed their impact on decision-making results. Thus, research on multi-attribute decision making problems in which attributes are correlated is both theoretical and practical. literature [5] improved the traditional gray target decision-making methods using weighted Mahalanobis distance, and avoided influence of correlations, different dimensions and importance difference on decision-making outcomes and gray target transform incompatibility; literature [6] defined a generalized triangular fuzzy related average operator, which not only considers the importance among the elements and also reflects the correlation among the elements, and applied this operator in the multi-attribute decision making; literature [7] proposed two types of inducing intuition fuzzy related aggregation operators when expert attributes and weight attributes in multi-attribute group decision making problems are correlated; literature [8] given the inducing Choquet integral operator and studied group decision making problems in which the decision-makers have correlated fuzzy preferences for plans. Choquet capacity degree and its integral [9] was proposed by Choquet in 1953, Choquet integral considers the existence of mutual influence between attributes, that is the complementary or substitutable properties among attributes can be simulated by capacity u . Schmeidler [10] applied Choquet integral theory to decision theory for the first time, and so far Choquet integral has been widely used in many fields, such as data source selection, economic efficiency research and performance evaluation.

In practical management decision problems, it’s often difficult for decision-makers to give accurate attribute values due to the diversity and uncertainty of practical problems and limitations of human knowledge; as one of the means to handle multi-attribute uncertain problems, interval gray number [11] has become a hotspot of research since proposed by Professor Deng Julong. literature [12] proposed a method based on information restore operator to calculate the interval gray number sequence correlation degree, and established multi-index interval gray numbers correlation decision-making model; literature [13] proposed the concept of gray distance entropy, and established a grey distance entropy deviation maximization model of attribute value uncertain multi-attribute decision-making.

In summary, for multi-attribute decision making problems in which the attribute values are interval gray numbers and attributes are correlated, based on current research, in this paper, Choquet integral is applied to multi-attribute decision-making problems in which attribute values are interval gray numbers, and interval gray number multiple attribute decision-making method based on interval gray number and Choquet integral is proposed and used in the Ningxia and Inner Mongolia Yellow River ice disaster to prove the rationality and effectiveness of thismethod.

2 Basic knowledge

Definition 1. [11] Grey number which has both upper bounds $\bar{a}$ and lower bounds $\underline{a}$ is referred to as interval grey number, i.e. $a (\otimes) \in [\underline{a}, \bar{a}]$ Where $\underline{a} \leq \bar{a}$ . The difference between upper bounds and lower bounds of interval grey number is referred to as the length of interval grey number, i.e. $l_{a} = \bar{a} - \underline{a}$ .

Assume that $\otimes_{1} \in [\underline{a}, \bar{a}]$ , $\otimes_{2} \in [\underline{b}, \bar{b}]$ are interval grey number, then $\begin{matrix} \otimes_{1} + \otimes_{2} & \in & [\underline{a} + \underline{b}, \bar{a} + \bar{b}] \\ \otimes_{1} / \otimes_{2} & \in & [min {\underline{a} / \underline{b}, \underline{a} / \bar{b}, \bar{a} / \underline{b}, \bar{a} / \bar{b}}, \\ max {\underline{a} / \underline{b}, \underline{a} / \bar{b}, \bar{a} / \underline{b}, \bar{a} / \bar{b}}] \end{matrix}$

Definition 2. [14] Assume that $\otimes_{1} \in [\underline{a}, \bar{a}]$ , $\otimes_{2} \in [\underline{b}, \bar{b}]$ are interval grey number and let $l_{a} = \bar{a} - \underline{a},$ $l_{b} = \bar{b} - \underline{b}$ , then $p (a (\otimes) \geq b (\otimes)) = \frac{min {l_{a} + l_{b}, max (\bar{a} - \underline{b}, 0)}}{l_{a} + l_{b}}$ is defined as the possibility degree of interval grey number a (⊗) ≥ b (⊗).

Definition 3. [11] Assume that the emergence background or universe of grey number ⊗ is Ω, μ (⊗) is the measurement on number field of grey number ⊗, then $g^{\circ} (\otimes) = μ (\otimes) / μ (Ω)$ is referred to as degree of greyness of grey number ⊗.

Theorem 1. Assume that $\otimes_{1} \in [\underline{a}, \bar{a}]$ and $\otimes_{2} \in [\underline{b}, \bar{b}]$ are interval grey numbers, then

If ⊗₁ ⊂ ⊗ ₂, then g° (⊗ ₁) ≤ g° (⊗ ₂)

If ⊗₁ ∩ ⊗ ₂ = φ, then g° (⊗ ₁ ∪ ⊗ ₂) = g° (⊗ ₁) + g° (⊗ ₂);

If ⊗₁ ∩ ⊗ ₂ ≠ φ, then g° (⊗ ₁ ∪ ⊗ ₂) ≤ g° (⊗ ₁) + g° (⊗ ₂)

Proof.

From the definition of degree of greyness, when ⊗₁ ⊂ ⊗ ₂, g° (⊗ ₁) ≤ g° (⊗ ₂) is easily proved.

If ⊗₁ ∩ ⊗ ₂ = φ, $\begin{matrix} g^{\circ} (\otimes_{1} \cup \otimes_{2}) & = & \frac{μ (\otimes_{1} \cup \otimes_{2})}{μ (Ω)} \\ = & \frac{μ (\otimes_{1}) + μ (\otimes_{2})}{μ (Ω)} \\ = & \frac{μ (\otimes_{1})}{μ (Ω)} + \frac{μ (\otimes_{2})}{μ (Ω)} \\ = & g^{\circ} (\otimes_{1}) + g^{\circ} (\otimes_{2}) \end{matrix}$

If ⊗₁ ∩ ⊗ ₂ ≠ φ, $\begin{matrix} g^{\circ} (\otimes_{1} \cup \otimes_{2}) & = & \frac{μ (\otimes_{1} \cup \otimes_{2})}{μ (Ω)} \\ = & \frac{max {b_{1}, b_{2}} - min {a_{1}, a_{2}}}{μ (Ω)} \\ \leq & \frac{(b_{2} - a_{2}) + (b_{1} - a_{1})}{μ (Ω)} \\ \leq & g^{\circ} (\otimes_{1}) + g^{\circ} (\otimes_{2}) \end{matrix}$

According the theorems above, the degree of greyness of interval grey number satisfies the monotonicity and subadditivity, therefore degree of greyness of interval grey number can be thought of as a grey measurement.

Let the attribute set of arbitrary multi-attribute decision making problems be X, for arbitrary T ⊆ X, g° (T) can be considered as the weight or importance of attribute sets, ∀S, T ⊆ X, if g° (T ∪ S) = g° (T) + g° (S), then it is independent among attribute sets. If g° (T ∪ S) < g° (T) + g° (S), then there are redundant associations between attribute sets.

Choquet capacity and integral [9] was proposed by Choquet in 1953. Choquet capacity is a non-additive measure and Choquet integral is a nonlinear integral on non-additive measure. Choquet capacity and Choquet integral generalizes the probability measure and mathematical expectation, which study problems in non-additive measures.

Definition 4. [12] Assume that N is a non-empty set, P (N) is the power set of N, u is a set function which defined on P (N), u : P (N) → [0, 1]. if u satisfies the following conditions:

u (φ) = 0, u (N) = 1;

∀S, T ⊆ N, S ⊆ T, then u (S) ≤ u (T)

then P (N) is referred to as the capacity defined on u.

Definition 5. [12] Assume that A = {x₁, x₂, ⋯ , x_n} is a non-empty set, p (A) is the power set of A, f is the non-negative function defined on A and u is the capacity defined on p (A), then the discrete Choquet integral of f over capacity u is $\begin{matrix} C (A) = \int fdu \\ = \sum_{i = 1}^{n} f (x_{(i)}) [u (A_{(i)}) - u (A_{(i + 1)})] \end{matrix}$

Wherein (i) is the transformation of vector f (x_(i)), which makes 0 ≤ f (x₍₁₎) ≤ f (x₍₂₎) ≤ ⋯ ≤ f (x_(n)), A_(i) ={ x_i, x_i+1, ⋯ , x_n } and A_(n+1) = 0.

As suggested above, the discrete Choquet integral of interval grey number on degree of greyness can be defined from the features of degree of greyness and the definitions of Choquet integral.

Definition 6. Assume that A ={ x₁, x₂, ⋯ , x_n } is an interval grey number sequence, p (A) is the power set of A, f is the non-negative function defined on A, g° (⊗) is the degree of greyness of interval grey number, then the discrete Choquet integral of f over degree of greyness g° (⊗) is $C (A) = \int fd (g^{\circ} (\otimes)) = \sum_{i = 1}^{n} f (x_{(i)}) [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})]$ Wherein (i) is the transformation of vector f (x_(i)), which makes 0 ≤ f (x₍₁₎) ≤ f (x₍₂₎) ≤ ⋯ ≤ f (x_(n)), A_(i) ={ x_i, x_i+1, ⋯ , x_n } and A_(n+1) = 0.

According to Definition 6 and Theorem 1, the discrete Choquet integral over degree of greyness g° (⊗) is a merging algorithm, in which the condition of redundant association among attributes is considered, namely the association between attributes can be modeled by degree of greyness g° (⊗).

From the algorithms of interval grey numbers and the discrete Choquet integral over degree of greyness g° (⊗), the properties of interval grey number discrete Choquet integral are as follows:

Property 1. (Idempotence) Assume that a_i (⊗) (i = 1, 2, ⋯ , n) is a set of interval grey number, of which the degree of greyness is g° (⊗). If $a_{1} (\otimes) = a_{2} (\otimes) = \dots = a_{n} (\otimes) = a (\otimes) = [\underline{a}, \bar{a}]$ , then $\begin{matrix} \int fd (g^{\circ} (\otimes)) \\ = \sum_{i = 1}^{n} f (x_{(i)}) [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] = a (\otimes) . \end{matrix}$

Proof. Since $\sum_{i = 1}^{n} [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] = 1$ , when $a_{1} (\otimes) = a_{2} (\otimes) = \dots = a_{n} (\otimes) = a (\otimes) = [\underline{a}, \bar{a}]$ , then $\begin{matrix} \int fd (g^{\circ} (\otimes)) \\ = \sum_{i = 1}^{n} f (x_{(i)}) [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] \\ = a (\otimes) \sum_{i = 1}^{n} [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] = a (\otimes) . \end{matrix}$

Property 2. (Monotonicity) Assume that g° (⊗) is the degree of greyness of interval grey number, $a_{i} (\otimes) = [{\underline{a}}_{i}, {\bar{a}}_{i}]$ and $b_{i} (\otimes) = [{\underline{b}}_{i}, {\bar{b}}_{i}]$ are two sets of interval grey numbers, and a₁ (⊗) ≤ a₂ (⊗) ≤ ⋯ ≤ a_n (⊗), b₁ (⊗) ≤ b₂ (⊗) ≤ ⋯ ≤ b_n (⊗), a_i (⊗) ≥ b_i (⊗), then $\begin{matrix} C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \\ \geq C (b_{1} (\otimes), b_{2} (\otimes), \dots, b_{n} (\otimes)) \end{matrix}$

Proof. Since A_(i) ⊂ A_(i+1), then g° (A_(i)) ≥ g° (A_(i+1)). Assume λ_i = g° (A_(i)) - g° (A_(i+1)) ≥ 0, $l_{a_{i}} = \bar{a} - \underline{a},$ $l_{b_{i}} = \bar{b} - \underline{b}$ , then $\begin{matrix} l_{λ_{i} a_{i}} = λ (\bar{a} - \underline{a}) = λ_{i} l_{a_{i}} \\ l_{λ_{i} b_{i}} = λ_{i} (\bar{b} - \underline{b}) = λ_{i} l_{b_{i}} \end{matrix}$

And since a_i (⊗) ≥ b_i (⊗), then $\begin{matrix} p (λ_{i} a_{i} (\otimes) \geq λ_{i} b_{i} (\otimes)) \\ = \frac{min {λ_{i} l_{a} + λ_{i} l_{b}, max (λ_{i} \bar{a} - λ_{i} \underline{b}, 0)}}{λ_{i} l_{a} + λ_{i} l_{b}} \\ = λ_{i} p (a_{i} (\otimes) \geq b_{i} (\otimes)) \end{matrix}$ namely λ_ia_i (⊗) ≥ λ_ib_i (⊗), then

$\begin{matrix} C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \\ \geq C (b_{1} (\otimes), b_{2} (\otimes), \dots, b_{n} (\otimes)) \end{matrix}$

Property 3. (Boundedness) Assume that $a_{i} (\otimes) = [{\underline{a}}_{i}, {\bar{a}}_{i}] (i = 1, 2, \dots, n)$ is a set of interval grey numbers, g° (⊗) is the degree of greyness of interval grey numbers, let $\begin{matrix} a^{+} (\otimes) & = & [max_{i} {\underline{a}}_{i}, max_{i} {\bar{a}}_{i}], \\ a^{-} (\otimes) & = & [min_{i} {\underline{a}}_{i}, min_{i} {\bar{a}}_{i}], \end{matrix}$ then a^- (⊗) ≤ C (a₁ (⊗) , a₂ (⊗) , ⋯ , a_n (⊗)) ≤ a⁺ (⊗).

Proof. Since $\sum_{i = 1}^{n} [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] = 1$ , then $\begin{matrix} C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \\ = \sum_{i = 1}^{n} a_{i} (\otimes) [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] \\ \leq \sum_{i = 1}^{n} max_{i} {\bar{a}}_{i} [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] \\ = max_{i} \bar{a} \\ C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \\ = \sum_{i = 1}^{n} a_{i} (\otimes) [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] \\ \geq \sum_{i = 1}^{n} min_{i} {\underline{a}}_{i} [g^{\circ} (A_{(i)}) - g^{\circ} (A_{(i + 1)})] \\ = min_{i} \underline{a} \end{matrix}$ therefore, $min_{i} \underline{a} \leq C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \leq max_{i} \bar{a}$ $\begin{matrix} p (C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \geq a^{-} (\otimes)) \\ = \frac{min {min_{i} \bar{a} - min_{i} \underline{a} + max_{i} \bar{a} - min_{i} \underline{a}, max (max_{i} \bar{a} - min_{i} \underline{a}, 0)}}{min_{i} \bar{a} - min_{i} \underline{a} + max_{i} \bar{a} - min_{i} \underline{a}} \\ \geq \frac{max_{i} \bar{a} - min_{i} \underline{a}}{max_{i} \bar{a} - min_{i} \underline{a} + max_{i} \bar{a} - min_{i} \underline{a}} \geq \frac{1}{2} \end{matrix}$

Namely, C (a₁ (⊗) , a₂ (⊗) , ⋯ , a_n (⊗)) ≥ a^- (⊗)is proved. Evidenced by the same token, C (a₁ (⊗) , a₂ (⊗) , ⋯ , a_n (⊗)) ≤ a⁺ (⊗). Thus, $\begin{matrix} a^{-} (\otimes) \leq C (a_{1} (\otimes), a_{2} (\otimes), \dots, a_{n} (\otimes)) \\ \leq a^{+} (\otimes) . \end{matrix}$

3 Approach for multiple attribute decision-making with interval grey number based on Choquet integral

Assume that the set of schemes is A ={ a₁, a₂, ⋯ , a_n }, the set of factors is B ={ b₁, b₂, ⋯ , b_m }, thus the decision matrixS = {u_ij = (a_i, b_j) |a_i∈ A, b_j ∈ B }, u_ij (i = 1, 2, ⋯ , n, j = 1, 2, ⋯ , m) is the evaluating value of scheme a_i about criterion b_j, which is wrote down as $u_{ij} \in [{\underline{u}}_{ij}, {\bar{u}}_{ij}] (0 \leq {\underline{u}}_{ij} \leq {\bar{u}}_{ij}, 2, \dots, n; j = 1, 2, \dots, m)$ , considering the effect is an interval grey number. The evaluating value of scheme a_i is wrote down as u_i = (u_i1 (⊗) , u_i2 (⊗) , ⋯ , u_im (⊗)) (i = 1, 2, ⋯ , n).

To eliminate dimensions and increase comparability, grey range analysis is adopted

For benefit-type target ${\underline{x}}_{ij} = \frac{{\underline{u}}_{ij} - {\underline{u}}_{j}^{\nabla}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}, {\bar{x}}_{ij} = \frac{{\bar{u}}_{ij} - {\underline{u}}_{j}^{\nabla}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}$ (1)

For cost-type target ${\underline{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - {\bar{u}}_{ij}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}, {\bar{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - {\underline{u}}_{ij}}{{\bar{u}}_{j}^{*} - {\underline{u}}_{j}^{\nabla}}$ (2)

where

{\bar{u}}_{j}^{*} = max_{1 \leq i \leq n} {{\bar{u}}_{ij}}

{\underline{u}}_{j}^{\nabla} = min_{1 \leq i \leq n} {{\underline{u}}_{ij}}

, then the normalized evaluating value of scheme a_i is

\begin{matrix} x_{i} & = & (x_{i 1} (\otimes), x_{i 2} (\otimes), \dots, x_{im} (\otimes)), \\ i = 1, 2, \dots n . \end{matrix}

where

x_{ij} \in ({\underline{x}}_{ij}, {\bar{x}}_{ij})

is the interval grey number on [0, 1], which shows the evaluating information of scheme a_i about criterion b_j. Thus, we get the normalized decision matrix of variousalternatives

X = [\begin{matrix} x_{11} & x_{12} & \dots & x_{1 m} \\ x_{21} & x_{22} & \dots & x_{2 m} \\ \dots & \dots & ⋱ & \dots \\ x_{n 1} & x_{n 2} & \dots & x_{nm} \end{matrix}]

The coherence evaluating vector of scheme a_i is wrote down as x_i = (x_i1 (⊗) , x_i2 (⊗) , ⋯ , x_im (⊗)), where $x_{ij} \in ({\underline{x}}_{ij}, {\bar{x}}_{ij})$ is interval grey number on [0, 1]. To select the optimal scheme from alternatives, the evaluating vectors of each scheme need to be integrated. Considering the impact of the interaction among each attribute to decision making result, we use discrete Choquet integral of degree of greyness of interval grey number to integrate the evaluating value of each scheme.

From the definition of discrete Choquet integral about degree of greyness g° (⊗), each component of coherence evaluating vector of each scheme needs to be ranked in order to integrate the evaluating value of each scheme. We use the possibility degree of interval grey number to rank each component of coherence evaluating vector of scheme a_i and the sorted coherence evaluating vector is $x_{i}^{'} = (x_{i 1}^{'} (\otimes), x_{i 2}^{'} (\otimes), \dots, x_{im}^{'} (\otimes))$ .

The attribute set B ={ b₁, b₂, ⋯ , b_m } of a certain multiple attribute decision making problem can be considered as its emergence background or universe Ω. Thus, we consider the attribute set B as a whole, namely taking μ (Ω) = 1, then from the definition of degree of greyness interval of grey number we get $g^{\circ} (\otimes) = μ (\otimes) / μ (Ω) = μ (\otimes)$

For ⊗ = { b_{i
₁}, b_{i
₂}, ⋯ , b_{i
_p} } ⊂ Ω, where p ≤ m, then μ (⊗) can be taken as the importance degree of chosen attributes {b_{i
₁}, b_{i
₂}, ⋯ , b_{i
_p} } given by experts.

From the above mentioned, the synthetic evaluating value of each alternative

$\begin{matrix} C (u_{i}) & = & \int fd (g^{\circ} (\otimes)) \\ = & \sum_{j = 1}^{m} x_{ij}^{'} (\otimes) [μ (B_{i}) - μ (B_{i + 1})] \end{matrix}$ (3)

C (a_i) is still an interval grey number. We use the possibility degree of interval grey number rank C (a_i), by which sort preferred schemes.

4 Example analysis

The Yellow River is the river where ice flood occurs most frequently in China, the most serious of which is Ningmeng segment, which is due to its special geographical location, hydrological and meteorological conditions and river characteristics [13]. The Yellow River overall is a large Chinese character of ji curvature in Inner Mongolia. Because the upstream flows through the Loess Plateau and the desert edge, the content of sand increase rapidly, force the fallen mud of the riverbed uplifting. Thus, the river turns narrow -deep to wide -shallow gradually, shoals and curves stack up in the channel and the slope slow. As a result, it is easy for the dam to jam in the ice in the carve of the channel or the narrow reach of the river when the river thawing.We choose the river width (b₁) in meter, the river depth (b₂) in meter, the levee distance (b₃) in meter to sort the possibility of ice flood in four sections, which are Bayangol (a₁), Sanhu River (a₂), Zhaojun Grave (a₃) and Baotou (a₄).

The river course characteristics

The river	The river	The levee
width	depth	distance
bayangol	[400,1200]	[3,5]	[2500,5200]
sanhuhekou	[250,800]	[3,5]	[2500,5200]
zhaojunfen	[200,600]	[7,9]	[2400,5000]
baotou	[200,800]	[6,8]	[900,5500]

The attribute value is standardized by grey range analysis and the standardized interval grey number evaluation matrix is $\begin{matrix} X (\otimes) = {(x_{ij} (\otimes))}_{4 \times 3} \\ = [\begin{matrix} [0.20, 1.00] [0.00, 0.33] [0.35, 0.93] \\ [0.05, 0.60] [0.00, 0.33] [0.34, 0.93] \\ [0.00, 0.40] [0.67, 1.00] [0.33, 0.89] \\ [0.00, 0.60] [0.50, 0.83] [0.00, 1.00] \end{matrix}] . \end{matrix}$

To integrate evaluating vector of each scheme using discrete Choquet integral of interval grey number greyscale, we first need to sort each component of coherence evaluating vector of each scheme. The possibility degree of interval grey number is used in this paper to resort each component of coherence evaluating vector of scheme a_i. The sorted coherence evaluating vector of each scheme is $\begin{matrix} x_{1}^{'} = {[0.35, 0.93], [0.20, 1.00], [0.00, 0.33]} \\ x_{2}^{'} = {[0.34, 0.93], [0.05, 0.60], [0.00, 0.33]} \\ x_{3}^{'} = {[0.67, 1.00], [0.33, 0.89], [0.00, 0.40]} \\ x_{4}^{'} = {[0.00, 1.00], [0.50, 0.83], [0.00, 0.60]} . \end{matrix}$

In a certain multiple attribute decision making problem, the attribute set can be considered as a whole, namely μ (Ω) = 1, then g° (⊗) = μ (⊗)/μ (Ω) = μ (⊗) is obtained. Thus, μ (⊗) can be considered as the weight of attributes {b_{i
₁}, b_{i
₂}, ⋯ , b_{i
_p} }. We invite relevant experts to evaluate each selected attribute and the weight of attribute set and the results are as follows, $\begin{matrix} μ (φ) = 0, μ ({x_{1}}) = 0.4, μ ({x_{2}}) = 0.3 \\ μ ({x_{3}}) = 0.5, μ ({x_{1}, x_{2}}) = 0.7, \\ μ ({x_{1}, x_{3}}) = 0.8, μ ({x_{2}, x_{3}}) = 0.6 \\ μ ({x_{1}, x_{2}, x_{3}}) = 1 . \end{matrix}$

The information of evaluating vector of each scheme is integrated using Equation (3), and the synthetic evaluating values of each scheme are $\begin{matrix} C (a_{1}) = [0.184, 0.78], C (a_{2}) = [0.124, 0.62] \\ C (a_{3}) = [0.264, 0.717], C (a_{4}) = [0.15, 0.79] . \end{matrix}$

The synthetic evaluating values of each scheme are sorted using possibility degree of interval grey number, the results are as follows $C (a_{3}) \underset{0.507}{≻} C (a_{1}) \underset{0.51}{≻} C (a_{4}) \underset{0.586}{≻} C (a_{2}),$ namely the section a₂ is the most easily jammed in ice.

According to the above results, for the sorting of scheme a₁ and a₂, from the data perspective, with the same of the river depth and the levee distance, the river width of a₁ is longer than a₂, so the scheme a₁ is superior to the a₂, so the sorting is reasonable in this article. For other sort, similar analysis can be carried out.

With the features of long duration, suddenly happening and rising and operation rule hard to grasp, the ice flood in the Yellow River sometimes cause more intensive calamity than that of storm flood. Thus, we can determine which section is easier to jam in the ice according to the feature of the river and prevent floods early before the river thawing period. The relationship among the river is considered using Choquet integral in the paper, which makes the decision-making results more reasonable.

5 Conclusions

In the multi-attribute decision making, establishing an index system of decision attributes is one of the key issues that affect the rationality of decision results. With regard to the interaction between attributes which has influenced on decision results in Multiple attribute decision making, a multiple attribute decision making method of interval grey number based on degree of greyness and Choquet integral is presented in this article. With the rapid development of China’s social economy and steady growth of national economy of relevant regions or provinces, losses caused by ice disaster will be increasing. With regard to the issue, considering the relationship among the features of the river channel by using Choquet integral, we can determine which section is easier to jam in the ice according to the feature of the river and prevent floods early to reduce economic loss combined with the local condition.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation (71271086); The brainstorm project of key scientific of Henan province science Department (142102310123); Funded project of Key research for Colleges and universities in Henan province (15A630005).

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