INTRODUCTION
Multicriteria decisionmaking (MCDM), as an important part of modern decisionmaking science, emphasizes on solving limited situation decisionmaking problem under multicriteria circumstance. Its theory and methods have been widely applied to social life, engineering design, system engineering and management science, etc. In real life, due to complexity of the external environment, ambiguity of objective things by themselves and limitations of human knowledge, there are many uncertainties in the decisionmaking process. Therefore, decisionmaking information in actual decisionmaking problems usually has such uncertainties as fuzziness, randomness or grayness, etc.
Grey stochastic MCDM problem has two characteristics of grayness and randomness. The relevant studies progresses slowly with relatively few studies results obtained. Present, studies of this aspect has attracted positive attention of experts and scholars all over the world, for instance, Yalcin et al. (2012) proposed a new financial performance evaluation approach to rank the companies of each sector in the Turkish manufacturing industry. For this purpose, a hierarchical financial performance evaluation model is structured based on the AFP and VFP maincriteria and their subcriteria. Krohling and de Souza (2012) proposed a hybrid approach combining prospect theory and fuzzy numbers to handle risk and uncertainty in MCDM problems. Wang et al. (2013) defined possibility degree of grey stochastic variable expectation, studied stochastic MCDM problem with weight not completely certain and with criterion value as interval grey number. Boran (2011) proposed the integration of intuitionistic fuzzy preference relation aiming to obtain weights of criteria and intuitionistic fuzzy TOPSIS method aiming to rank alternatives for dealing with imprecise information on selecting the most desirable facility location. Mousavi et al. (2013) developed a new fuzzy grey multicriteria group decision making model to solve evaluation and selection problems under uncertainty in reallife situations. Luo et al. (2008), based on relative membership degree of ideal matrix, explored risk multiple criteria group decisionmaking problem with weight information unknown and with criterion value as interval grey number. The abovementioned methods have provided some research ideas to solve MCDM problems. However, it can be found that there is relatively little study on stochastic MCDM problem with criterion value as extended grey number that considers criterion natural state. However, in the actual decisionmaking problems, it is relatively difficult for decision makers to accurately predict the occurrence probability of event or natural state. Thus, this study proposes the corresponding decisionmaking approach to meet the needs of such decisions.
MATERIALS AND METHODS
Preliminaries
Extended grey number: Grey number refers to number (Liu et al., 1999) only with approximate range known but not the exact value, which can effectively measure the grayness of things. In practice, the value of grey numbers is limited to a certain interval or a general set of numbers, usually denoted as "⊗".
Definition 1: Assume ⊗ is a grey number, D is a collection that covers ⊗, then:
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If D is an interval, then⊗ can be called interval grey number, denoted as∀⊗⇒d*∈ [a, b]or ⊗ = [a, b] 
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If D is a discrete set, then⊗ can be called discrete grey number, denoted as ∀⊗⇒d*∈D, D = {d_{1}, d_{2} ,..., d_{n}} or ⊗ = {d_{1}, d_{2},...,d_{n}} 
where, the value of interval grey number can be compared with probability degree of interval grey number. To better describe the grayness of decisionmaking information, extended grey number that combines discrete grey number and continuous grey number can be used (Dalalah et al., 2011).
Definition 2: If D is a set of a series of interval grey numbers, then ⊗ can be called extended grey number, denoted as:
Where:
Denote set of all extended grey numbers as R (⊗).
Extended grey number distance and expectation: Grey number distance describes the degree of separation between two grey numbers, which plays an important role in description of distance between criterion evaluation value and ideal value. In view of current study (Lin et al., 2008) on definition of interval grey number distance and considering that the theory does not fit extended grey number, this study gives definition of extended grey number distance.
Definition 3: If:
then Hausdorff distance between extended number ⊗_{1} and ⊗_{1} is (Wang and Wang, 2014):
Where:
is Hausdorff distance between ⊗_{1} and ⊗_{2}, ⊗x_{i} = [a_{i}, b_{i}], ⊗y_{j} = [c_{j }, d_{j}] (i = 1, 2, ..., n, j = 1, 2, ..., m). ∥⋅∥ represents any norm, such as L_{p}.
When ∥⋅∥ is L_{p}:
Thus obtain:
where, p = 1, 2, ..., l, l tends to + ∞.
Definition 4: Extended grey number random variable is a set of random variables composed of a limited number of different extended grey numbers ⊗, denoted as ξ(⊗). Its probability distribution is shown in Table 1, which can also be denoted with probability distribution function f (ξ(⊗)).
In Table 1, ⊗_{I} is the value of extended grey number random variable ξ(⊗) at occurrence of the ith state,
p_{i} = 1, n is the probability at occurrence of the ith state, which meets , n is the number of possible values for extended grey number random variables (Marques et al., 2011). Probability distribution function f (ξ (⊗)) is f (ξ(⊗) = ⊗_{I}) = p_{i}.
Table 1:  Probability distribution of extended grey number random variable ξ (⊗) 

Definition 5: Assume ξ (⊗) is an extended grey number random variable and then can be called expectation of extended grey random variable, denoted to be:
Grey stochastic multicriteria decisionmaking approach: For stochastic MCDM problem with criterion value as extended grey number, assume that A = {A_{1}, A_{2}, …, A_{m}} is a scheme set, B = {B_{1}, B_{2}, …, B_{n}} is a mutually independent set of criteria, criterion weight vector w = {w_{1}, w_{2}, …, w_{n}}, which satisfies , w_{j}≥0 (j = 1, 2, ..., n). Due to uncertainty of decisionmaking environment, solution has s kinds of natural state in various criteria, the state set of, which is θ = {θ_{1}, θ_{2}, ..., θ_{s}}. Denote the probability at occurrence of the tth state (t≤s) as P_{t}. The value of solution A_{i} at the jth criterion is extended grey number random variable u_{ij}, whose value at the t–th state is extended grey number denoted as:
and thus, obtain decision matrix (Nayagam et al., 2011).
When the various criteria weights are known, the decisionmaking approach is the best solution or sorting to determine solution set, whose decisionmaking procedure is as follows:
Step 1: 
Normalization approach of decision matrix. To eliminate the influence of criteria on decisionmaking results due to different dimensions, decision matrix R^{t} can be normalized (Li et al., 2007). In MCDM problems, the common types include efficiency and cost type. For efficiency criterion, the greater the value, the better, while for cost criterion, the smaller the value, the better 
Efficiency criterion value is:
Cost criterion value is:
Where:
Corresponding to various criteria, standardization decision matrix of s natural state is .
Step 2: 
Determine expectations. According to grey decision matrix G^{t} and probability P_{t }of natural state t, calculate expectation of each solution at various state based on definition 5 and thereby obtain expectation decision matrix 
Calculation formula of expectation is:
Step 3: 
Determine positive ideal solution and negative ideal solution 
Positive ideal solution A^{+}is:
Negative ideal solutionA‾ is:
Step 4: 
Calculate distance between various solution and positive, negative ideal solution 
The distance between A_{i} and A^{+} is:
The distance between A_{i} and A‾ is:
Where:
is the distance between ⊗r_{ij} and is the distance between ⊗r_{ij} and A_{j}^{}.
Step 5: 
Calculate relative closeness degrees K_{i} and sort the solution 
where, the smaller the value K_{i} is, the better the solution is.
RESULTS AND DISCUSSION
Ren and Gao (2010), for MCDM problem with criteria weight information incomplete and with criterion value as normally distributed random variables, proposed a stochastic MCDM based on interval arithmetic. Zhou et al. (2015) defined possibility degree and distance formula of extended grey number, studied uncertain MCDM problem with solution criterion value as extended grey number and proposed a multiple criteria decisionmaking approach with uncertain probability based on Hurwicz. The study results prove feasibility and effectiveness of this approach, from the computational analysis step and process, compared to method used in the reference literature (Krohling and de Souza, 2012; Wang et al., 2013; Ren and Gao, 2010), the study proposed approach can better meet practical needs, more in line with actual situation of MCDM problem and with stronger operability.
Illustrative example: The decision maker chooses information management systems providers from four optional companies (A_{1}, A_{2}, A_{3} and A_{4}). The decision maker evaluates each company from the four criteria: B_{1} is system reliability and adaptability, B_{2} is system flexibility, B_{3} is control ability, B_{4} is equipment cost. Under criterion B_{1}, B_{2}, B_{3}, solution corresponds to three different natural states. Natural state probability p = (0.3, 0.4 and 0.3), while B_{4} will not vary with state change. Each criterion weight vector given by decision makers is w = (0.1, 0.3, 0.4 and 0.2) (Zhou et al., 2015). In each state, evaluation information is given in the form of extended grey number random variable and its decisionmaking data is shown in Table 24. Determine best information system provider to be chosen by the decision maker.
Table 2:  Decision matrix R^{1} at good state 

Table 3: 
Decision matrix R^{2} at moderate state 

Table 4: 
Decision matrix R^{3} at poor state 

Table 5:  Normalized decision matrix G^{1} at good state 

Table 6: 
Expectation decision matrix G 

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In the above criteria, system reliability and adaptability, flexibility and control ability belong to efficiency criteria, cost of equipment belongs to cost criterion. According to formula 3 and 4, normalize decision matrix R^{1} and obtain normalized decision matrix G^{1}, as shown in Table 5 
Similarly, for normalized decision matrix G^{2}, G^{3 }at moderate or poor constractible state, due to limited space, its operation process will not be repeated.
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According to natural state probability p = (0.3, 0.4 and 0.3) and operation rule of extended grey number, calculate expectation with formula 5 and obtain expectation decision matrix G = {⊗r_{ij}}_{3×3}, the result of which is shown in Table 6 
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According to formula 6 and 7, calculate the positive ideal solution and negative ideal solution 
A^{+ }= ([0.6625, 0.8625], [0.8461, 1.0000], [0.8802, 0.9557], [0.5789, 0.6471])
A^{‾} = ([0.2375, 0.4625], [0.3513, 0.4477], [0.5221, 0.7093], [0.8462, 1.0000])
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According to the formula 8 and 9, calculate the distance between various solutions and positive, negative ideal solution, respectively, knowing that each criterion weight vector w = (0.1, 0.3, 0.4 and 0.2) 
The distance between A_{i} and A^{+} is:
The distance between A_{i} and A‾ is:
• 
Calculate relative closeness degree based on formula 10 
K_{1} = 0.4449, K_{2} = 0.3733, K_{3} = 0.5082, K_{4} = 0.3934
Thereby, obtain K_{2}<K_{4}<K_{1}<K_{3}, so sorting result of various solutions is as follows: A_{2}>A_{4}>A_{1}>A_{3}. Therefore, it can be known that best provider of information management system is A_{2}. The result is basically consistent with conclusion of literature (Zhou et al., 2015), which proves feasibility and effectiveness of this study, from the computational analysis step and process, it can be seen that, compared to method used in the reference literature, the proposed approach can better meet practical needs, more in line with actual situation of MCDM problem and with stronger operability.
CONCLUSION
For grey stochastic MCDM problem with criterion value as extended grey number, the study provides Hausdorff distance formula of extended grey number, proposes grey stochastic MCDM study based on Hausdorff distance, discusses in detail its implementation steps and verifies feasibility and rationality of the proposed approach with sample calculation analysis. The decisionmaking approach proposed in this study is very effective for dealing with decisionmaking problem with both extended grey number and randomness. The solution sorting process takes full account of natural state probability corresponding to various criteria and enhances scientific and rationality of the study. The decisionmaking study has good value in application promotion and actual decisionmaking and can be widely applied in the fields of project evaluation, supply chain management and investment decision.
ACKNOWLEDGMENTS
This study was supported by the Scientific Studies Fund of Hunan Provincial Education Department (No. 14C0184), by the Hunan Province Philosophy and Social Science Foundation (No. 14YBA065). Supported by the construct program of the key discipline in Hunan province.