Multi-index grey target decision method based on adjustment coefficient

Abstract

With regard to multi-index decision-making problems, in which the index value of alternatives and weight are three-parameter interval grey number, a multi-index grey-target decision-making approach based on adjustment coefficient is proposed. First, considering that the resolution of index objective weights derived from entropy weighting method is not high, a normalized decision matrix method based on adjusting coefficient was proposed, and the optimizing entropy model was established to determine the index objective weights. Furthermore, the mean area method was applied to convert index subjective weights into real number, and the D-S (Dempster-Shafer) combination rule was utilized to integrate the subjective and objective weights into comprehensive weights. Finally, the optimizing solution was determined via the size of positive target distance. An example is presented to illustrate the suitability and effectiveness of the obtained methods.

Keywords

Three-parameter interval grey number adjustment coefficient grey-target decision-making D-S theory of evidence

1 Introduction

In literature [1] the multi-index decision-making problems, in which the rationality of the index weight directly affects the accuracy of decision results, the index weight is very important to the study. At present, a lot of methods can be used to determinate the index weight, which can be roughly divided into subjective empowerment method and objective empowerment method and comprehensive empowerment method. Subjective empowerment method mainly includes the Delphi method, expert investigation method, comparison matrix method and analytic hierarchy process (AHP), etc.; Objective empowerment methods mainly include the entropy value method, grey correlation analysis and multiple regression analysis, etc. In literature [2], attribute weights are determined by the superiority of attribute values; In literature [3], with the help of the deviations maximizing algorithm, the attribute weights based on the similarity of attribute values are proposed; In literature [4], the three types majorant operator of “rewarding good and punishing bad” is presented to standardize the index, optimization model is established to confirm the weight of each index by constructing the Lagrangian function. In literature [5], an improved approach is proposed to determining the weights of attributes based on grey incidence. In literature [6], multi-index decision-making problems that the index value of alternatives are three-parameter interval grey number is research, and a multi-index optimization model based on deviation degree is constructed to get the index weight. The resolution of objective weights of the index depending on objective empowerment method is poor and it can’t indicate the importance of each target accurately, and current researches on this issue are not common. Comprehensive empowerment methods is a kind of method which integrate subjective weight and objective weight by certain scientific method, however, how to integrate the subjective and objective weights is also a key point. As a method to fuse evidence, D-S evidence theory attracts many scholars’ attention since put forward. In literature [7], considering individual evidence can’t fully reflect the operation rules of things, how to promote the organic integration between different evidence was studied, and the evidence fusion results is used to characterize the nature of complicated things. In literature [8], in order to solve decision-making problems in which the index value of alternatives are interval number, the uncertain degrees of each index are determined and the mass functions of each alternatives are obtained, and information is fused by using the D-S combination rule. In literature [9], in order to solve conflict evidence fusion problems, the evidence coordinate weight factor is defined based on evidence similarity, and the distribution range of bull’s-eye distance ascertain model is established based on grey-target decision-making. In literature [10], the K-L information distance function is introduced to describe the conflict characteristics between evidences, and the application constraint of D-S theory’s synthetic rule is completed.

In literature [1] Grey target decision-making is a new way to solve the multiple index decision making problems, and it has become the research hotspot since presented by professor Deng. In literature [11], given the uncertainty and multi-stage of multi- attribute decision-making problems, a calculating model for the weights of the time sequence is constructed depending on grey entropy and the time temperature, and an objective function which aggregates target-center distances of all evaluated period is proposed. In literature [12], the comprehensive bull’s-eye distance is defined based on spatial analysis of bull’s-eye distance as a vector, and the decision model of grey target with both the positive bull’s-eye and the negative bull’s-eye is established. In literature [13], decision-making information is expanded to three parameters interval grey number, and subjective empowerment method and objective empowerment method and positive or negative the bull ‘s-eye spacing is taken into account, so three kinds of grey target decision model is constructed. In literature [14], with regard to classification decision-making problem under uncertainty, an interval gray target classification decision-making model is put forward.

In actual decision problems, decision makers often fail to give specific values of the effect measure, considering the researches that the decision-making problems in which decision-making index and weight of index are three parameters interval grey Numbers is still rare, and taking into account the problem that the resolution of index objective weights derived from entropy weighting method is not high, the adjustment coefficient is given under three parameters interval grey Numbers. Using the D-S theory of evidence to integrate subjective empowerment method and objective empowerment method, the paper proposed a multi-index grey-target decision-making approach based on adjustment coefficient, and an example is presented to illustrate the suitability and effectiveness of the obtainedmethods.

2 Basic knowledge

Definition 1. [15] Assume a (⊗) is an interval grey number and the maximum probability value of the grey number is known, i.e. $a (\otimes) \in [\underline{a}, \tilde{a}, \bar{a}]$ , then a (⊗) is an three-parameter interval grey number, where $\tilde{a}$ is the maximum probability value of a (⊗) and is defined as the center of gravity. when $\tilde{a}$ is unknown, a (⊗) is considered as the usual interval grey number.

Using three-parameter interval grey numbers to making decisions is not only ensure the values range of interval grey numbers, but also emphasize the center of gravity which is the maximum probability value of a (⊗). By this way can we compensate for the shortcoming of “poor information” of grey numbers, so as to make the result of the decision-making nearer to the practical conditions.

Assume $a (\otimes) \in [\underline{a}, \tilde{a}, \bar{a}]$ , $b (\otimes) \in [\underline{b}, \tilde{b}, \bar{b}]$ are three-parameter interval grey numbers. By the operation definition of interval grey number in literature [1], the operation of three-parameter interval grey number is defined, such as $\begin{matrix} a (\otimes) + b (\otimes) \in [\underline{a} + \underline{b}, \tilde{a} + \tilde{b}, \bar{a} + \bar{b}] \\ a (\otimes) / b (\otimes) \in [min {\underline{a} / \underline{b}, \underline{a} / \bar{b}, \bar{a} / \underline{b}, \bar{a} / \bar{b}}, \\ max {\underline{a} / \underline{b}, \underline{a} / \bar{b}, \bar{a} / \underline{b}, \bar{a} / \bar{b}}] . \end{matrix}$ Definition 2. Set $a (\otimes) \in [\underline{a}, \tilde{a}, \bar{a}]$ and b (⊗) ∈ $[\underline{b}, \tilde{b}, \bar{b}]$ as three-parameter interval grey numbers, thus the $\begin{matrix} L (A, B) \\ = 3^{- 1 / 2} \sqrt{(\underset{-_{1}}{a} - \underset{-_{1}}{b})^{2} + ({\tilde{a}}_{2} - {\tilde{b}}_{2})^{2} + ({\bar{a}}_{3} - {\bar{b}}_{3})^{2}} \end{matrix}$ is distance of a (⊗) and b (⊗).

Definition 3. Set x _i, y _i ≥ 0, i = 1, 2, …, n, and $1 \geq \sum_{i = 1}^{n} x_{i} \geq \sum_{i = 1}^{n} y_{i}$ , then named $h (x, y) = \sum_{i = 1}^{n} x_{i} ln (x_{i} / y_{i})$ x = (x ₁, x ₂, … , x _n) as the relative entropy of y = (y ₁, y ₂, …, y _n).

The main characteristics of the relative entropy are as follow:

1) $\sum_{i = 1}^{n} x_{i} ln (x_{i} / y_{i}) \geq 0$ ;

2) The necessary and sufficient condition of $\sum_{i = 1}^{n} x_{i} ln (x_{i} / y_{i}) = 0$ is ∀_i, ∃x _i = y _i.

According to the characteristics above, we can see that the relative entropy can be used as the measure of the degree of agreement between two vectors when x and y are discrete attribution.

Definition 4. [7] Set Θ as a frame of discernment, and if set functions that m : 2^Θ → [0, 1] (2^Θ is a power set of Θ) content with m (φ) =0 and ∑_A∈Θ m (A) =1, thus m is called as the basic probability assignment(Mass function), in which A focal element if m > 0; ∀A ⊆ Θ, m (A) is called as the basic probability assignment of A, bel (A) is called as the belief functions of A if ∀A ⊆ Θ as well as bel (A) = ∑_B⊆A m (B).

Corollary 1. [7] Set Bel₁ … , Bel_n as the belief functions on the same Θ, and m₁, …, m_n is its basic probability assignment, and if Bel ⊕ … ⊕ Bel_n is exist and the basic probability assignment is m, then ∀A ⊆ Θ, A ≠ φ $m (A) = K_{n} \sum_{A_{1} \cap \dots \cap A_{n} = A} m_{1} (A_{1}) \dots m_{n} (A_{n})$ in which, K_n = 1/∑_{A₁∩…∩A_n≠φ}m₁ (A₁) … m_n (A_n) is conflict coefficient, and the larger the K_n is, the more diversity of the preference information among each of the experts, this is the synthetic Dempster rule ofmultiple belief functions.

3 The Construction of decision model

Suppose the alternative set is A = {a ₁, a ₂, …, a _n}, index factor set is B = {b ₁, b ₂, …, b _n}, then the decision matrix is S = {u _ij = (a _i, b _j) |a _i ∈ A, b _j ∈ B} , u _ij (i = 1, 2, …, n, j = 1, 2, …, m) is the index value of alternative a _i under index b _j. The index value is not a precise number, but a three parameters interval gray number, so the index value of alternative a _i under index b _j is u _ij (i = 1, 2, …, n, j = 1, 2, …, m) $(0 \leq \underset{-_{ij}}{u} \leq {\tilde{u}}_{ij} \leq {\bar{u}}_{ij}, i = 1, 2, \dots, n, j = 1, 2, \dots, m)$ .The effectiveness evaluation vector of the alternative a _i is u _i = (u _i1 (⊗) , … , u _im (⊗)) (i = 1, 2, …, n), then the decision Matrices is: $U = (u_{ij})_{n \times m} = [\begin{matrix} u_{11} & u_{12} & \dots & u_{1 m} \\ u_{21} & u_{22} & \dots & u_{2 m} \\ \dots & \dots & ⋱ & \dots \\ u_{n 1} & u_{n 2} & \dots & u_{nm} \end{matrix}]$

3.1 Integration of index weight

Entropy is widely applied in decision analysis area in literature [16], entropy of index is calculated according to values of the different alternatives under same index, and then the objective weights of indexes are calculated. For the same alternative, the greater the difference of index value is, the smaller the entropy value, then the greater of the objective weights is. But certain problems still exist in using entropy to determine the index weight. In literature [17], the function -p log p (0 < p < 1) will be grow rapidly when p < 0.1, when p = 0.2 and p = 0.6 the function changes relatively flat, thus the objective weights obtained are over-concentrated to the average value and are poor of resolution,which can’t accurately report the importance of each index. In order to improve the resolution of the index weight and increase the degree of dispersion, R = (u _ij) _n×m can be processed as following:

For benefit-type target: $\underset{-_{ij}}{x} = {(\frac{\underset{-_{ij}}{u}}{\sum_{i = 1}^{n} {\bar{u}}_{ij}})}^{α}, {\tilde{x}}_{ij} = {(\frac{{\tilde{u}}_{ij}}{\sum_{i = 1}^{n} {\tilde{u}}_{ij}})}^{α}, {\bar{x}}_{ij} = {(\frac{{\bar{u}}_{ij}}{\sum_{i = 1}^{n} \underset{-_{ij}}{u}})}^{α}$ (1)

For cost-type target:

$\begin{matrix} \underset{-_{ij}}{x} = {(\frac{1}{{\bar{u}}_{ij}} / \sum_{i = 1}^{n} \frac{1}{\underset{-_{ij}}{u}})}^{α}, {\tilde{x}}_{ij} = {(\frac{1}{{\tilde{u}}_{ij}} / \sum_{i = 1}^{n} \frac{1}{{\tilde{u}}_{ij}})}^{α}, \\ \bar{x} = {(\frac{1}{\underset{-_{ij}}{u}} / \sum_{i = 1}^{n} \frac{1}{\bar{u}})}^{α} \end{matrix}$ (2) in which, α ≥ 1 is adjustment coefficient which can improve the resolution of objective weight, the greater α is, the more resolution of the index objective weight is; the smaller α is, the weight of each index tend to be average value.

Define 5. Set

$\begin{matrix} H_{ij} = - \frac{1}{ln (3)} \\ (\underset{-_{ij}}{x} ln (\underset{-_{ij}}{x}) + {\tilde{x}}_{ij} ln ({\tilde{x}}_{ij}) + {\bar{x}}_{ij} ln ({\bar{x}}_{ij})) \end{matrix}$ (3) is the entropy value of alternative a _i under index b _j.

Denote $ω_{ij} = H_{ij} / \sum_{j = 1}^{m} H_{ij}$ (4) as the objective weight of alternative a _i under index b _j, then objective weights for each alternative under index b _j is recorded as the following: $ω = [\begin{matrix} ω_{11} ω_{12} \dots ω_{1 m} \\ ω_{21} ω_{22} \dots ω_{2 m} \\ \dots \dots ⋱ \dots \\ ω_{n 1} ω_{n 2} \dots ω_{nm} \end{matrix}]$

Supposing ω _j is the objective weight of index b _j, in order to get higher resolution of index objective weight, a optimization model is established as follows: ${\begin{matrix} min W = \sum_{i = 1}^{n} \sum_{j = 1}^{m} ω_{j} log \frac{ω_{j}}{ω_{ij}} \\ s . t . \sum_{j = 1}^{m} w_{j} = 1, w_{j} \geq 0, (j = 1, 2, \dots, m) \end{matrix}$

Solving the above model, the optimal solution of the index weight vectors is get: $ω_{j}^{*} = Π_{i = 1}^{n} ω_{ij} / \sum_{j = 1}^{m} Π_{i = 1}^{n} ω_{ij}$ (5) $ω_{j}^{*}$ is standardized disposed and set $η_{j} = \frac{ω_{j}^{*}}{\sum_{j = 1}^{m} ω_{j}^{*}}$ (6) in which η = (η ₁, η ₂, …, η _m), η ₁ + η ₂ + … + η _m = 1, η _j ≥ 0, (j = 1, 2, …, m).

Index weight is not only related to index value given by decision makers, but also related to decision makers’ subjective preference of index, that is the subjective weight of index. Subjective weights of index given by decision makers is recorded as $λ_{j} \in [\underset{-_{j}}{λ}, {\tilde{λ}}_{j}, {\bar{λ}}_{j}]$ $(0 \leq \underset{-_{j}}{λ} \leq {\tilde{λ}}_{j} \leq {\bar{λ}}_{j}, j = 1, 2, \dots, m)$ , then λ _j is processed by using Average area of law in literature [18] $λ_{j}^{'} = \frac{\underset{-_{j}}{λ} + 2 {\tilde{λ}}_{j} + {\bar{λ}}_{j}}{4}$ (7)

Set $λ_{j}^{'}$ is as follow $η_{j}^{'} = \frac{λ_{j}^{'}}{\sum_{j = 1}^{m} λ_{j}^{'}}$ (8) in which, $η_{1}^{'} + η_{2}^{'} + \dots + η_{m}^{'} = 1$ , $η_{j}^{'} \geq 0 (j = 1, 2, \dots, m)$ .

In order to get a more objective comprehensive actual weight and make decision more justice, D-S combination rule is used to integrate the objective and subjective weight. Basic probability distribution of subjective index weight is $m_{1} (b_{j}) = η_{j}^{'}$ (j = 1, 2, …, m) basic probability distribution of objective index weight is m ₂ (b _j) = η _j (j = 1, 2, … , m), the comprehensive weight value m (b _j) = ξ _j (j = 1, 2, …, m) of each index is obtained by using D-S combinationrule.

3.2 The merits order of programs

Denote $\begin{matrix} u_{i} = (μ_{i 1} (\otimes), u_{i 2} (\otimes), \dots, u_{im} (\otimes)), \\ (i = 1, 2, \dots, n) \end{matrix}$ is the effect evaluation vector of alternative a _i. In order to eliminate dimension and increase the comparability, the effect evaluation is dealt with by the following range transformation formula:

For benefit-type target $\underset{-_{ij}}{x} = \frac{\underset{-_{ij}}{u} - \underset{-_{j}^{\nabla}}{u}}{{\bar{u}}_{j}^{*} - \underset{-_{j}^{\nabla}}{u}}, {\tilde{x}}_{ij} = \frac{{\tilde{u}}_{ij} - \underset{-_{j}^{\nabla}}{u}}{{\bar{u}}_{j}^{*} - \underset{-_{j}^{\nabla}}{u}}, {\bar{x}}_{ij} = \frac{{\bar{u}}_{ij} - \underset{-_{j}^{\nabla}}{u}}{{\bar{u}}_{j}^{*} - \underset{-_{j}^{\nabla}}{u}}$ (9)

For cost-type target $\underset{-_{ij}}{x} = \frac{{\bar{u}}_{j}^{*} - {\bar{u}}_{ij}}{{\bar{u}}_{j}^{*} - \underset{-_{j}^{\nabla}}{u}}, {\tilde{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - {\tilde{u}}_{ij}}{{\bar{u}}_{j}^{*} - \underset{-_{j}^{\nabla}}{u}}, {\bar{x}}_{ij} = \frac{{\bar{u}}_{j}^{*} - \underset{-_{ij}}{u}}{{\bar{u}}_{j}^{*} - \underset{-_{j}^{\nabla}}{u}}$ (10) in which, ${\bar{u}}_{j}^{*} = max_{1 \leq i \leq n} {{\bar{u}}_{ij}}$ , $\underset{-_{j}^{\nabla}}{u} = min_{1 \leq i \leq n} {\underset{-_{ij}}{u}}$ , set $r_{i} = (r_{i 1} (\otimes), r_{i 2} (\otimes), \dots, r_{im} (\otimes)), i = 1, 2, \dots n$ as the normalized effect evaluation vector of alternative a _i, where $r_{ij} \in (\underset{-_{ij}}{r}, {\tilde{r}}_{ij}, {\bar{r}}_{ij})$ is a three-parameter interval gray number on [0,1] and denotes evaluation information of alternative a _i under index b _j. Then the normalized decision matrix is get $R = [\begin{matrix} x_{11} x_{12} \dots x_{1 m} \\ x_{21} x_{22} \dots x_{2 m} \\ \dots \dots ⋱ \dots \\ x_{n 1} x_{n 2} \dots x_{nm} \end{matrix}]$

Definition 6. Set $z_{j}^{+} = max {(\underset{-_{ij}}{r} + {\tilde{r}}_{ij} + {\bar{r}}_{ij}) / 3 | 1 \leq i \leq n} (j = 1, 2, \dots, m)$ , and denote its effect evaluation as $[\underset{-_{ij}^{+}}{r}, {\tilde{r}}_{ij}^{+}, {\bar{r}}_{ij}^{+}]$ , then we consider

$\begin{matrix} z^{+} = {z_{1}^{+}, z_{2}^{+}, \dots, z_{m}^{+}} = {[\underset{-_{i 1}^{+}}{r}, {\tilde{r}}_{i 1}^{+}, \underset{-_{i 1}^{+}}{r}], \\ [\underset{-_{i 2}^{+}}{r}, {\tilde{r}}_{i 2}^{+}, {\bar{r}}_{i 2}^{+}], \dots, [\underset{-_{im}^{+}}{r}, {\tilde{r}}_{im}^{+}, {\bar{r}}_{im}^{+}]} . \end{matrix}$ (11) as the optimal effect vector of grey target decision, also named positive bull’s eye.

Definition 7. Consider

$\begin{matrix} ɛ_{i}^{+} = 3^{- 1 / 2} {\sum_{j = 1}^{m} ξ_{j} [(\underset{-_{ij}}{x} - \underset{-_{ij}^{+}}{x})^{2} \\ + ({\tilde{x}}_{ij} - {\tilde{x}}_{ij}^{+})^{2} + ({\bar{x}}_{ij} - {\bar{x}}_{ij}^{+})^{2}]}^{1 / . 2} \end{matrix}$ (12) as the positive bull’s eye distance of the alternative a _i.

Sort the programs according to the order of $ɛ_{i}^{+}$ , and the smaller the $ɛ_{i}^{+}$ is, the better the alternative is.

3.3 Decision-making algorithms

Step 1. Deal with the R = (u _ij) _n×m according to formulas (1) and (2);

Step 2. Compute the objective weight ω _ij of alternative a _i under index b _j according to formulas (3) and (4);

Step 3. Compute the optimization model built by formula (5), and compute the objective weightη = (η ₁, η ₂, …, η _m) of index by formula (6);

Step 4. Compute the subjective weight $η^{'} = (η_{1}^{'}, η_{2}^{'}, \dots, η^{'})$ according to formulas (7) and (8);

Step 5. Using the D-S combination rule, we can obtain comprehensive weight m (b _j) = ξ _j (j = 1, 2, …, m) of each index;

Step 6. Normalize the decision-making matrix according to formulas (9) and (10), and obtain the positive bull’s eye and the distance of the positive bull’s eye according to formula (11) and (12), then the smaller the $ɛ_{i}^{+}$ is, the better the alternative is.

4 Case study

During the staff assessment and selection in a company, six appraisal indexs are refined: ideology and morality (b ₁), Work attitude (b ₂), Work style (b ₃), Cultural level and structure of knowledge (b ₄), leadership (b ₅), development capacity (b ₆). Here are 5 candidates and the value of every candidate is expressed in the form of three parameters interval number under each index (Table 1). Weight set and evaluation matrix is given by several experts, then calculate the data and get the normalized subjective weight vectors: λ _j = (λ ₁, λ ₂, …, λ ₆) ^T, and λ ₁ ∈ [0.15, 0.17, 0.20], λ ₂ ∈ [0.10, 0.14, 0.20], λ ₃ ∈ [0.16, 0.17, 0.20], λ ₄ ∈ [0.05, 0.08, 0.10], λ ₅ ∈ [0.18, 0.19, 0.20]. Sort of five candidates and determine the best candidates.

Step 1. According to the formulas (1) and (2), perform the processing for R = (u _ij) _n×m, and set α = 4.

Step 2. Obtaining the objective weight of alternative a _i under index b _j according to the formulas of (3)and (4) $\begin{matrix} ω = [\begin{matrix} 0.1384 0.1695 0.1676 \\ 0.1958 0.1577 0.1606 \\ 0.1799 0.1416 0.1779 \to \\ 0.1637 0.1717 0.1492 \\ 0.1732 0.1757 0.1790 \end{matrix} \\ \begin{matrix} 0.1809 0.1556 0.1880 \\ 0.1570 0.1689 0.1600 \\ 0.1717 0.1646 0.1643 \\ 0.1563 0.1756 0.1835 \\ 0.1679 0.1672 0.1370 \end{matrix}] \end{matrix}$ Step 3. Establish the Optimization Model according to formula (5) and obtaining the subjective weight of index according to formula (6) $\begin{matrix} η_{1} & = & 0.1919, η_{2} = 0.1503, η_{3} = 0.1684, \\ η_{4} & = & 0.1686, η_{5} = 0.1672, η_{6} = 0.1636 \end{matrix}$ Step 4. Obtain the objective weight of index according to formulas (7) and (8) $\begin{matrix} η_{1}^{'} & = & 0.1382, η_{2}^{'} = 0.1667, η_{3}^{'} = 0.1321, \\ η_{4}^{'} & = & 0.1789, η_{5}^{'} = 0.1869, η_{6}^{'} = 0.1972 \end{matrix}$ Step 5. Obtain the comprehensive weight value of each index according to D-S combination rules $\begin{matrix} ξ_{1} & = & 0.1513, ξ_{2} = 0.1508, ξ_{3} = 0.1340, \\ ξ_{4} & = & 0.1815, ξ_{5} = 0.1882, ξ_{6} = 0.1942 \end{matrix}$ Step 6. Normalize the decision matrix according to formulas (9) and (10) and get the positive bull’s-eye $\begin{matrix} z^{+} (\otimes) = ([0.5263, 0.7894, 1.0],, \\ [0.5, 0.75, 1.0], [0.5385, 0.7692, 1.0], \\ [0.6667, 0.9167, 1.0], [0.7143, 0.8571, 1.0]) \end{matrix}$ Then calculate the bull’s-eye distance of the case $\begin{matrix} ɛ_{1}^{+} = 0.4239, ɛ_{2}^{+} = 0.3828, \\ ɛ_{3}^{+} = 0.5154, ɛ_{4}^{+} = 0.5288, ɛ_{5}^{+} = 0.5003 \end{matrix}$ then the order of the alternative: A ₂≻ A ₁ ≻ A ₅ ≻ A ₃ ≻ A ₄, that is A ₂ is the best alternative and theresult coincides with literature [19]. Comparing with the literature [19], the formula in my paper is simple,easier to understand, easy to operate and has broader practical application value. Using D-S evidence theory to integrate the objective and subjective weights, this is fully reflects the subjective and objective of the effective combination.

Different results will be obtained from formulas (1) and (2) depending on different values of α. Table 2 indicate: with the increasing α, the order of subjective weight of each index keep the same, but the resolution increase. Table 3 indicate: the order of comprehensive value of each alternative also keep the same, but the resolution keep stable. This instruction that the different values of α will not change the order of results and the resolution significantly, but the resolution of relative weights will increase and weight level is more effective, which is easier for decision makers to give suggestions.

5 Conclusions

In multiple attribute decision making problems, the decision-making information is usually grey. In situations that information is extremely poor, the opportunities to get values of gray numbers which are limited to a certain range are often unequal. Thus, the introduction of the three-parameter interval gray number in gray decisions is of great theoretical and practical significance. In order to deal with cases in which index value and weight are three-parameter interval greynumbers, considering that the resolution of index objective weights derived from entropy weighting method is not high, a normalized decision matrix method based on adjusting coefficient was proposed, and the optimizing entropy model was established to determine the index objective weights. The mean area method was applied to convert index subjective weights into real number, and the D-S combination rule was utilized to integrate the subjective and objective weights into comprehensive weights. As a result, with the increasing α, the order of subjective weight of each index keep the same, but the resolution increase; the order of comprehensive value of each alternative also keep the same.

Footnotes

Acknowledgments

This work was supported by National Natural science Foundation (71071077, 71371098, 71271086); Funding of Jiangsu Innovation Program for Graduate Education (CXZZ13_0183); the Fundamental Research Funds for the Central Universities(NC2012001); the Major Project of Emphasis Research Base for Philosophy and Social Science in Jiangsu Province(2012JDXM005).

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