Interval number is actual a collection of all real numbers on a closed interval. Interval number represents a kind of uncertainty, and it has a great potential for application in different fields, such as establish fuzzy portfolio model and multi-objective portfolio model [37] [38] . An interval number is denoted as, which is defined as follows [37] .

Definition 2.1. If is the upper bound of the range, is the lower range, we have:

Especially, the interval number is a real number when.

For two interval numbers, and, they are some properties as follows,

1) if and.

2).

3), and when,.

The interval decision matrix is key factors for investment decision. The interval decision is defined as below.

Definition 2.2. A decision matrix is composed of interval number, where

In investment decision, the attributes can be divided into two types: benefit type and cost type. For example, profit is benefit type, while risk is cost type. To eliminate the influence of different physical dimension for the decision result, the standardized original decision matrix is denoted as, where is defined as below,

or

where belongs benefit type, while belongs cost type.

D numbers theory is proposed by Deng [31] , which is the development of Dempster- Shafer theory of evidence. It is brief introduced as follows,

Definition 2.3. Let be a finite nonempty set, D number is a mapping D:, it satisfied with

where is an empty set and B is a subset of. From the definition, the elements of D numbers do not mutually exclusive and the sum of assessment can be less than one.

Definition 2.4. For a discrete set, where belongs to and if, for any and, a special form of D numbers can be expressed by:

or be represented simply as:

An example is given to describe the D numbers. For MADM, the set of is a frame of discernment. The assessment score belong the interval, an expert gives his evaluation in the frame of Dempster-Shafer theory, it is shown as below:

where, ,. The sum of equals to one, i.e., it means that information is complete. However, for another assessment, which is denoted as. Some information is incomplete since an expert has not full realization for this assessment [31] . Thus, the assessment is given as follows by D numbers method:

where, , , The set of are not a frame of discernment actually because the intersection between these elements is not empty. The sum value of equals to 0.9, i.e. the information is incomplete. Some rules of D numbers theory are given as follows [31] .

Definition 2.5. For a given D numbers, the overall assessment is defined as:

In Dempster-Shafer theory of evidence, two BPAs can be fused into a BPA. Similarity, in D numbers method, the fusing rules of two D numbers have been proposed [31] . It is defined as below,

Definition 2.6. Let and be two D numbers,

The combination of and denoted by, we have:

with

where:

where and.

According to the basic principle of information theory, information is the orderly degree of a measurement system. The entropy is a measurement of disorder degree in a system. The absolute value of them is equal, while their symbol is inverse. Therefore, many scholars use Entropy weight method to measure the weight. The smaller information entropy is, the higher weight is [8] [9] . Entropy weight method is introduced as below.

Definition 2.7. For MADM, any solution set, is a property set. Information entropy is defined as:

Using the formula (2) (3) (4), we have the standardized matrix. The matrix is normalized as, where

Then, attribute has information entropy

Specially, we have: when

According to information entropy, the attribute’s weight vector can be calculated as follows,

with

The comprehensive attribute values can be obtained as below,

According to the value of, the rank of investment decision is given. Finally, the MADM problem is solved according to the ranking.

In this section, a new MADA model for investment project is proposed. In this model, firstly, the uncertain information of investment decision is represented by using interval number and D numbers theory. The uncertain information is given as follows,

where

where is an interval number, is real number for the interval number of

the uncertainty information. We have: and. According to Equa-

tion (2), the decision matrix of uncertainty factors A is obtained. Secondly, using Equation (6), the corresponding matrix is obtained. Considering type of factors, the corresponding matrix must change into the standard matrix R. Thus, the standard matrix R is obtained according to Equations (3), (4) and (11) at third step. Lastly, the weights of uncertainty factors are calculated by using entropy weight method. According to Equation (14), the value of each decision is calculated. So, we make decision according to the value of each decision.

In this section, a numerical example is introduced for describing the proposed method. Supposing for promoting a new product, a company plans to choose a project from the four investment projects. These investment projects are denoted as, , and, respectively. Four attributes are considered in evaluating the four choices, they are amount investment (million dollar) denoted as, the expected net present value (million dollar) denoted as, the risk profit value (million dollar) denoted as and the risk loss value (million dollar) denoted as. In uncertainty theory, uncertain factors in many fields is determined by expert assessment measures. Then the company invites ten experts to give the assessments. In many previous studies, is a definite value, but in the study of uncertain information, mostly is blurred, so using interval number to re- present by. In D numbers, the value of is depended on the percent of experts who agree with. For example, for the attribute in investment project, seven experts think that the value of it is and two experts think that is. Another one don’t give any value. So we have. Similarly, all assessment are given by the combination of interval number and D numbers, which are shown as below,

where

For calculating the weight of each attribute, the process can be divided into four steps.

Step 1. Using the formula:, all the assessment are given by D numbers and shown as follows,

Step 2. We replace the D numbers in the matrix by the corresponding matrix is obtain as follows,

Step 3. The expected net present value and risk profit value are the benefit type. Amount of investment and risk loss value are the cost benefit attributes. Using the

formula (3) (4) (11), we can get the normalized matrix

Step 4. Using the formula (12) obtains the output information entropy of attribute.

According to Equation (13), the attribute’s weight vector can be calculated as follows,

Using Equation (14), each value of scheme can be obtained as follows,

So, we have. The rank of investment projects is. Thus, the second investment project is the best choice in four plans for company. In our methods, every uncertainty factor is represented by interval number. Furthermore, incomplete information is revealed D numbers method. For compare with previous methods, our proposed method is more flexible. It is accord with the actual circumstance in investment decision.