Grey stochastic multi-criteria decision-making approach based on expected probability degree



==Abstract==

After definition of the discrete grey stochastic variable and its expected value, the expected probability degree is defined. For multi-criteria decision-making problems, in which the criteria weights are incompletely certain and the criteria values of alternatives are in the form of grey stochastic variables, a grey stochastic multi-criteria decision-making approach is proposed. In this method, the evaluation value of each alternative under each criterion can be transformed to comprise the expected probability degree judgment matrix, based on which, a non-linear programming model can be enacted. In the end, the genetic algorithm is used to solve the model to attain the criteria weights, and the ranking of alternatives can be produced consequently. The feasibility and validity of this approach are illustrated by an example.

==Keywords==

Grey stochastic variable ; Expected probability degree ; Alternative similarity scale ; Multi-criteria decision-making ; Genetic algorithm

==1. Introduction==

Multi-criteria decision-making theories and methods have become one of the most active subjects in many disciplines, such as decision-making science, system engineering, management and logistics, etc. Since the rapid development of society and economics, the fuzziness of the decision-making environment has been realized by more and more people, in addition to the complexity and uncertainty existing commonly in decision-making problems. Hence, information should be presented by fuzziness, randomness and uncertainty in a real decision-making process. So far, there has been considerable research unto multi-criteria decision-making problems, which are in the form of the above three types of uncertain decision-making information, and, meanwhile, some studies have also focused on problems with multiple kinds of uncertainty. The multi-criteria decision-making problems, in which criteria values have randomness and fuzziness simultaneously are discussed in  [[#br000005|[1]]] , [[#br000010|[2]]]  and [[#br000015|[3]]] . Problems with criterion values taking the form of grey and fuzziness in the meantime are discussed in  [[#br000020|[4]]] , [[#br000025|[5]]]  and [[#br000030|[6]]] . However, there has been relatively little research conducted on multi-criteria decision-making problems with criteria value in the form of grey and randomness at the same time. In  [[#br000035|[7]]] , the complementary problems of grey theory and random theory are studied, showing that some methods for random problems contain the thoughts of grey methods and concepts, while grey problems can be also realized and solved from the randomness perspective. The objective function can be established by minimization of the comprehensive weighed distances of alternatives, which are to the positive alternative and the negative alternative. To solve that function, a method is provided to deal with some multi-criteria decision-making problems, in which the criterion value is in the form of a random variable. However, it only takes into consideration the determining of the criteria value in  [[#br000040|[8]]] . As for the grey risk decision-making problems with uncertain criteria weights, the two-base-point method is proposed by applying the thought of a positive and negative ideal point to the grey stochastic domain in  [[#br000045|[9]]] . Considering the dynamic hybrid multi-attribute decision making problems under risk, in which the weights are uncertain and criteria values take the form of precise numbers and interval numbers, as well as language-type fuzzy numbers at the same time, an approach, based on grey matrix relative degree, is provided to solve them in  [[#br000050|[10]]] . The main thought of this method is transforming the risk judgment matrix to a risk-free judgment matrix, and then ranking the alternatives using the grey interval relative analysis method. According to the characteristics of multi-criteria decision making problems under risk, a reasonable solution is given to these problems by introducing the concept of probability preference in  [[#br000055|[11]]] . Even though the above-mentioned research points out the problems existing in grey stochastic, there are some kinds of preliminary studies on these issues. Since there is little research on grey stochastic multi-criteria decision-making problems, and these kinds of problems can be more objective in describing some decision-making situations in reality, they are worthy of more attention. Thus, in this paper, a method is given to meet the demand of practical decision-making.      

The rest of this paper is organized as follows. Section  [[#s000010|2]]  gives the definition of a grey number, grey stochastic variable and expected probability degree. Section  [[#s000015|3]]  proposes the grey stochastic multi-criteria decision-making method, based on expected probability degree, and illustrates the procedures in detail. Section  [[#s000020|4]]  applies the proposed method to a practical example to explain its rationality and effectiveness. Section  [[#s000025|5]]  is the conclusion.      

<span id='s000010'></span>
==2. Grey stochastic variable and expected probability degree==

==Definition 1 [[#br000060|[12]]] .                                                         ==

The grey number can be defined as the number with a general range, but the exact value of this number cannot be known. In application, the grey number is an uncertain number which takes the value in a scope or a particular number set. It can be denoted as <math display="inline">\otimes </math> .

An interval grey number can be defined as the grey number with the lower limit, <math display="inline">a^L</math> , and the upper limit, <math display="inline">a^U</math> . The interval grey number can be denoted as <math display="inline">\otimes \in [a^L,a^U]</math> .      

Assume interval grey numbers, <math display="inline">\otimes _1\in [a^L,a^U],a^L<a^U,\otimes _2\in [b^L,b^U],b^L<b^U</math> . The interval grey numbers’ operational rules can be defined as follows  [[#br000060|[12]]] :
* <math display="inline">\otimes _1+\otimes _2\in [a^L+b^L,a^U+b^U]</math> ,                          
* <math display="inline">-\otimes _1\in [-a^U,-a^L]</math> ,                          
* <math display="inline">\otimes _1-\otimes _2=\otimes _1+(-\otimes _2)\in [a^L-b^U,a^U-b^L]</math> ,                          
* <math display="inline">k\times \otimes \in [ka^L,ka^U]</math> , where <math display="inline">k</math>  is a positive real number.

==Definition 2.                     ==

If the stochastic variables are countable values in the form of interval grey numbers, and the corresponding probability with regard to each value can be attained, then, this kind of variable can be defined as the discrete grey stochastic variable, which is called a grey stochastic variable, for short, in this paper.

The grey stochastic variable is denoted as <math display="inline">\zeta (\otimes )</math> , and its <math display="inline">i</math> th value can be presented by <math display="inline">\otimes _i</math> . [[#t000005|Table 1]]  shows the probability distribution of <math display="inline">\zeta (\otimes )</math> .

<span id='t000005'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 1.

The probability distribution of <math display="inline">\zeta (\otimes )</math> .                  

|-

| ζ(⊗)
| ⊗1
| ⊗2
| …
| ⊗i
| …
| ⊗n
|-

| P
| P1
| P2
| …
| Pi
| …
| Pn
|}

In [[#t000005|Table 1]] , <math display="inline">\zeta (\otimes )</math>  is a grey stochastic variable. <math display="inline">\otimes _i</math>  is the <math display="inline">i</math> th possible value that would be taken by <math display="inline">\zeta (\otimes ),P_i</math>  is the probability with respect to <math display="inline">\otimes _i</math> , while <math display="inline">n</math>  is the number of values that a grey stochastic variable can have.      

The probability density function can be denoted as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>f(\zeta (\otimes )=\otimes _i)=p_i\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

For better understanding of the above definitions, an example is given. Assume that an investment alternative, <math display="inline">a_i</math> , may gain an annual revenue of 270–280 million RMB with the probability of 0.4, 290–300 million with 0.2 possibility and 310–340 million with 0.4. This information can be presented by the grey stochastic variable, <math display="inline">\zeta (\otimes )</math> , in [[#t000010|Table 2]] .

<span id='t000010'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 2.

Example of grey stochastic variable.

|-

| ζ(⊗)
| [2.7, 2.8]
| [2.9, 3.0]
| [3.1, 3.4]
|-

| P
| 0.4
| 0.2
| 0.4
|}

Meanwhile, the probability density function can be denoted as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle f(\zeta (\otimes )=[2.7,2.8])=0.4\mbox{,}\quad f(\zeta (\otimes )=[2.9\\\displaystyle 3.0])=0.2\mbox{,}f(\zeta (\otimes )=[3.1,3.4])=0.4\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

==Definition 3.                     ==

Assume that <math display="inline">\zeta (\otimes )</math>  is a grey stochastic variable, then, the expected value of the grey stochastic variable is defined by <math display="inline">\sum _{i=1}^nP_i\times \otimes _i</math> , if the value of the formula <math display="inline">\sum _{i=1}^nP_i\times \otimes _i</math>  can be attained. And the expected value can be denoted as <math display="inline">E(\zeta (\otimes ))</math> , which satisfies:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>E(\zeta (\otimes ))=\sum_{i=1}^nP_i\times \otimes _i\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

According to the grey stochastic variable’s operational rules [[#fd000075|(1)]]  and [[#fd000090|(4)]] , it can be concluded that <math display="inline">E(\zeta (\otimes ))</math>  is an interval grey number.      

The concept of the expected value of the grey stochastic variable can be also illustrated with the above-mentioned example.

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle E(\zeta (\otimes ))=[2.7,2.8]\times 0.4+[2.9,3.0]\times 0.2+[3.1\\\displaystyle 3.4]\times 0.4=[2.9,3.08]\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

==Definition 4 [[#br000065|[13]]] .                                                         ==

Let <math display="inline">L(\otimes _1)=a^U-a^L</math>  be the length of the grey number, <math display="inline">\otimes _1\in [a^L,a^U]</math> , and <math display="inline">L(\otimes _2)=b^U-b^L</math>  be the length of the grey number, <math display="inline">\otimes _2\in [b^L,b^U]</math> , then,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>P(\otimes _1\geq \otimes _2)=\frac{max\lbrace 0,L(\otimes _1)+L(\otimes _2)-max(b^U-a^L,0)\rbrace }{L(\otimes _1)+L(\otimes _2)}</math>
|}
| style="text-align: right;" | 
|}

is called the expected probability degree of <math display="inline">\otimes _1</math>  against <math display="inline">\otimes _2</math> . Therefore, the relationship between <math display="inline">\otimes _1</math>  and <math display="inline">\otimes _2</math>  can be determined as follows:
* If <math display="inline">P(\otimes _1\geq \otimes _2)<0.5</math> , we say that <math display="inline">\otimes _1</math>  is less than <math display="inline">\otimes _2</math> , and denote it as <math display="inline">\otimes _1<\otimes _2</math> ;                                  
* <math display="inline">P(\otimes _1\geq \otimes _2)=0.5</math> , we say that <math display="inline">\otimes _1</math>  is equal to <math display="inline">\otimes _2</math> , and denote it as <math display="inline">\otimes _1=\otimes _2</math> ;                                  
* If <math display="inline">P(\otimes _1\geq \otimes _2)>0.5</math> , we say that <math display="inline">\otimes _1</math>  is larger than <math display="inline">\otimes _2</math> , and denote it as <math display="inline">\otimes _1>\otimes _2</math> .

==Definition 5.                     ==

Assume that <math display="inline">x_i</math>  and <math display="inline">x_j</math>  are grey stochastic variables, and their probability density functions can be denoted as <math display="inline">f_i</math>  and <math display="inline">f_j</math> , respectively. The expected probability degree of grey stochastic variable, <math display="inline">x_i</math> , against <math display="inline">x_j</math> , can be denoted as <math display="inline">E(p(x_i>x_j))</math> :

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle E(p(x_i>x_j))=\\\displaystyle =\sum _{d=1}^t\sum _{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)\times p(\otimes _i^d\geq \otimes _j^g)\mbox{,}\end{array}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">\otimes _i^d</math>  and <math display="inline">\otimes _j^g</math>  are the values that the variables <math display="inline">x_i</math>  and <math display="inline">x_j</math>  may take separately. <math display="inline">d=1,\ldots ,t</math> , and <math display="inline">g=1,\ldots ,t</math> . <math display="inline">p(\otimes _i^d\geq \otimes _j^g)</math>  are the possibility degrees of the interval grey number, <math display="inline">\otimes _i^d</math> , which is more than or equal to <math display="inline">\otimes _j^g</math> .

The expected probability degree is the measurement of the average probability degree in which the grey stochastic variable, <math display="inline">x_i</math> , is better than <math display="inline">x_j</math>    [[#br000055|[11]]] . The properties of expected probability degree can be listed as follows:
* <math display="inline">0\leq E(p(x_i>x_j))\leq 1</math> ,                          
* <math display="inline">E(p(x_i>x_j))+E(p(x_j>x_i))=1</math> ,                          
* if <math display="inline">E(p(x_i>x_j))=E(p(x_j>x_i))</math> , then <math display="inline">E(p(x_i>x_j))=0.5</math> .                  

The proof procedure for these properties will be shown as follows.

==Proof.                     ==

As <math display="inline">\begin{array}{l}\displaystyle E(p(x_i>x_j))=\\\displaystyle =\sum _{d=1}^t\sum _{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)\times p(\otimes _i^d\geq \otimes _j^g)\end{array}</math> ,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>E(p(x_i>x_j))\geq \underset{1\leq d,g\leq t}{min}(p(\underset{i}{\overset{d}{\otimes }}\geq \otimes _j^g))\times \sum_{d=1}^t\sum_{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

and <math display="inline">E(p(x_i>x_j))\leq max_{1\leq d,g\leq t}(p(\otimes _i^d\geq \otimes _j^g))\times \sum _{d=1}^t\sum _{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)</math> .              

Furthermore, considering any value that <math display="inline">d</math>  or <math display="inline">g</math>  may take, there always exist <math display="inline">max_{1\leq d,g\leq t}(p(\otimes _i^d\geq \otimes _j^g))\leq 1</math>  and <math display="inline">min_{1\leq d,g\leq t}(p(\otimes _i^d\geq \otimes _j^g))\geq 0</math> , so, one can reach the conclusion that <math display="inline">0\leq E(p(x_i>x_j))\leq 1</math> .              

Due to <math display="inline">\begin{array}{l}\displaystyle E(p(x_i>x_j))=\\\displaystyle =\sum _{d=1}^t\sum _{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)\times p(\otimes _i^d\geq \otimes _j^g)\end{array}</math> , as well as <math display="inline">\begin{array}{l}\displaystyle E(p(x_j>x_i))=\\\displaystyle =\sum _{g=1}^t\sum _{d=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)\times p(\otimes _j^d\geq \otimes _i^g)\end{array}</math> , we have:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle E(p(x_i>x_j))+E(p(x_j>x_i))=\\\displaystyle =\sum _{d=1}^t\sum _{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)\times [p(\otimes _i^d\geq \otimes _j^g)+\\\displaystyle +p(\otimes _j^d\geq \otimes _i^g)]\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

Since <math display="inline">p(\otimes _i^d\geq \otimes _j^g)+p(\otimes _j^d\geq \otimes _i^g)=1</math>  (see Ref.  [[#br000045|[9]]] ), then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle E(p(x_i>x_j))+E(p(x_j>x_i))=\\\displaystyle =\sum _{d=1}^t\sum _{g=1}^tf_i(\otimes _i^d)\times f_j(\otimes _j^g)=1\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

According to the property [[#fd000080|(2)]] , if <math display="inline">E(p(x_i>x_j))=E(p(x_j>x_i))</math> , then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>E(p(x_i>x_j))+E(p(x_j>x_i))=2\times E(p(x_i>x_j))=1\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Therefore, <math display="inline">E(p(x_i>x_j))=0.5</math> .

==Definition 6.                     ==

Assume that <math display="inline">GRS=\lbrace x_1,x_2,\ldots ,x_m\rbrace </math>  is a discrete set of <math display="inline">m</math>  grey stochastic variables, then the expected probability degree judgment matrix can be defined as a matrix that consists of all the expected probability degrees in which each grey stochastic variable is better than other variables separately. It can be denoted as <math display="inline">MATRIXP=(E(p(x_i>x_j)))_{m\times m}</math> , where: <math display="inline">i=1,\ldots ,m;j=1,\ldots ,m</math> .

<span id='e000040'></span>

==Theorem 1.                     ==

''The expected probability degree judgment matrix is a complementary judgment matrix.''

==Proof.                     ==

According to the definition for the expected probability degree, it is easy to reach the conclusion that <math display="inline">E(p(x_i>x_i))=0.5</math>  in the expected probability degree judgment matrix. Furthermore, there always exists the equality <math display="inline">E(p(x_i>x_j))+E(p(x_j>x_i))=1</math> , so, the judgment matrix of the expected probability degree is a complementary judgment matrix.

<span id='s000015'></span>
==3. Grey stochastic multi-criteria decision-making method based on expected probability degree==

There is a grey stochastic multi-criteria decision-making problem. Assume that <math display="inline">A=\lbrace a_1,\ldots ,a_i,\ldots ,a_m\rbrace </math>  is a discrete alternative set of <math display="inline">m</math>  possible alternatives, and <math display="inline">C=\lbrace c_1,\ldots ,c_k,\ldots ,c_n\rbrace </math>  is a set of <math display="inline">n</math>  criteria. <math display="inline">W=\lbrace w_1,\ldots ,w_k,\ldots ,w_n\rbrace </math>  is the vector of criteria weights, and the weights are subject to the constraints as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\sum_{k=1}^nw_k=1\mbox{,}\quad w_k\geq 0,k=1,2,\ldots ,n\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

The criterion value of alternative <math display="inline">a_i</math> , with respect to criterion <math display="inline">c_k</math> , is denoted as <math display="inline">x_{ik}</math> , and <math display="inline">x_{ik}</math>  is a grey stochastic variable. The incompletely certain information of the criteria weights can be represented by symbol <math display="inline">\Omega </math>  and it can be also shown in the form of linear equalities and inequalities, as follows  [[#br000070|[14]]] :
* <math display="inline">\lbrace W:A_1W\geq b,W>0,b\geq 0\rbrace </math> ,                          
* <math display="inline">\lbrace W:A_1W\leq b,W>0,b\geq 0\rbrace </math> ,                          
* <math display="inline">\lbrace W:A_1W=b,W>0,b\geq 0\rbrace </math> ,                                      where <math display="inline">A_1</math>  is a matrix with one line and <math display="inline">n</math>  columns, and <math display="inline">W=\lbrace w_1,\ldots ,w_k,\ldots ,w_n\rbrace </math> .      

The order of the alternatives is needed to be listed under the above conditions.

For the mentioned grey stochastic multi-criteria decision-making problems, the solving procedure can be summarized as follows:

''Step''  1. Establish the expected probability degree judgment matrix.      

Calculate the expected probability degree of every alternative, with respect to each criterion, and form the expected probability degree judgment matrix. As far as criterion <math display="inline">c_k</math>  is concerned, the expected probability degree judgment matrix for the alternative set can be denoted as <math display="inline">MATRIXP_k</math> :

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>MATRIXP_k=(E(p(x_{ik}>x_{jk})))_{m\times m}\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">i=1,\ldots ,m;j=1,\ldots ,m;k=1,2,\ldots ,n</math> .      

According to [[#e000040|Theorem 1]] , <math display="inline">MATRIXP_k</math>  is a complementary judgment matrix.      

''Step''  2. Form the comprehensive judgment matrix of expected probability degree.      

After gaining the expected probability degree judgment matrix, with respect to each criterion, the comprehensive judgment matrix of expected probability degree can be denoted by <math display="inline">CP=(\sum _{k=1}^nw_k\times E(p(x_{ik}>x_{jk})))_{m\times m}</math> , where <math display="inline">i=1,\ldots ,m;j=1,\ldots ,m;k=1,2,\ldots ,n</math> .      

It can be proved that the comprehensive judgment matrix of the expected probability degree is also a complementary judgment matrix, and the process proof can be shown as follows.

According to property [[#fd000085|(3)]]  of the expected probability degree, there always exists <math display="inline">E(p(x_{ik}>x_{ik}))=0.5</math> , so does the expression <math display="inline">\sum _{k=1}^nw_k\times E(p(x_{ik}>x_{ik}))=0.5\times \sum _{k=1}^nw_k=0.5</math> . Besides,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \sum _{k=1}^nw_k\times E(p(x_{ik}>x_{jk}))+\sum _{k=1}^nw_k\times E(p(x_{jk}>x_{ik}))=\\\displaystyle =\sum _{k=1}^nw_k\times (E(p(x_{ik}>x_{jk}))+E(p(x_{jk}>x_{ik})))=\\\displaystyle =\sum _{k=1}^nw_k=1\mbox{,}\end{array}</math>
|}
| style="text-align: right;" | 
|}

thus, matrix <math display="inline">CP</math>  is a complementary judgment matrix.      

''Step''  3. Calculate the criteria weights.      

According to the sorting vector, <math display="inline">\omega =(\omega _1,\omega _2,\ldots ,\omega _i,\ldots ,\omega _m)^T</math>  was introduced in  [[#br000075|[15]]]  for solving the ranking problem of the complementary judgment matrix. <math display="inline">\omega _i</math>  can be calculated by the method as follows:

<span id='fd000075'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\omega _i=\frac{(\sum_{l=1}^m\sum_{k=1}^nw_k\times E(p(x_{ik}>x_{lk})))+\frac{m}{2}-1}{m\times (m-1)}\mbox{,}</math>
|}
| style="text-align: right;" | ( 1)
|}

where <math display="inline">i=1,2,\ldots ,m</math> .      

If the criteria weights are already known, the order of the alternative set can be attained by comparing <math display="inline">\omega _i</math> . However, the situation of criteria weights, which are incompletely certain, is common in real multi-criteria decision-making. Consequently, by drawing lessons from Refs.  [[#br000080|[16]]] , [[#br000085|[17]]]  and [[#br000090|[18]]] , establishing an optimization model based on the closeness degree of <math display="inline">\omega _i</math>  is a major way of obtaining the criteria weights in this paper.      

For Formula [[#fd000075|(1)]] , it can be transformed into Formula [[#fd000080|(2)]] , as follows:

<span id='fd000080'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\omega _i=\frac{(\sum_{k=1}^nw_k\times \sum_{l=1}^mE(p(x_{ik}>x_{lk})))+\frac{m}{2}-1}{m\times (m-1)}\mbox{,}</math>
|}
| style="text-align: right;" | ( 2)
|}

where <math display="inline">i=1,2,\ldots ,m</math> .      

Therefore, the comprehensive sorting value of alternative <math display="inline">a_i</math>  can be represented by <math display="inline">\omega _i=f_i(W)</math> . Furthermore, the comprehensive sorting vector of the alternative set can be denoted by <math display="inline">F(W)=(f_1(W),f_2(W),\ldots ,f_m(W))</math> . As for given <math display="inline">\Omega </math> , obviously, if the value of <math display="inline">f_i(W)</math>  is bigger, the corresponding alternative, <math display="inline">a_i</math> , would be better. Consequently, the multi-objective decision-making model can be built as follows:

<span id='fd000085'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{c}
maxF(W)=(f_1(W),f_2(W),\ldots ,f_m(W))\\
\mbox{s.t.}W\in \Omega \mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | ( 3)
|}

Solve the programming model as follows:

<span id='fd000090'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{c}
maxf_i(W)\\
\mbox{s.t.}W\in \Omega \mbox{,}
\end{array}\quad i=1,2,\ldots ,m\mbox{.}</math>
|}
| style="text-align: right;" | ( 4)
|}

Since model [[#fd000090|(4)]]  is a linear programming model, it can be solved to obtain the optimal weight vector <math display="inline">W_i^+=(w_1^+,w_2^+,\ldots ,w_n^+)^T</math> , with respect to alternative <math display="inline">a_i</math> , using the simplex method. Then, the positive comprehensive sorting value of alternative <math display="inline">a_i</math>  can be attained in terms of the <math display="inline">W_i^+</math> , namely, <math display="inline">\omega _i^+=f_i(W_i^+)</math> .      

Then, we solve the programming model as follows:

<span id='fd000095'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{c}
minf_i(W)\\
\mbox{s.t.}W\in \Omega \mbox{,}
\end{array}\quad i=1,2,\ldots ,m\mbox{.}</math>
|}
| style="text-align: right;" | ( 5)
|}

Since model [[#fd000095|(5)]]  is also a linear programming model, it can be solved to obtain the optimal weight vector, <math display="inline">W_i^{-}=(w_1^{-},w_2^{-},\ldots ,w_n^{-})^T</math> , with respect to alternative <math display="inline">a_i</math> , using the simplex method. Then, the negative comprehensive sorting value of alternative <math display="inline">a_i</math>  can be attained in terms of <math display="inline">W_i^{-}</math> , namely, <math display="inline">\omega _i^{-}=f_i(W_i^{-})</math> .      

Vectors <math display="inline">\omega ^+=(\omega _1^+,\omega _2^+,\ldots ,\omega _m^+)^T</math>  and <math display="inline">\omega ^{-}=(\omega _1^{-},\omega _2^{-},\ldots ,\omega _m^{-})^T</math>  are called the positive and negative ideal points, respectively, in this multi-criteria decision-making problem.      

Consequently, the closeness degree function for the alternative is defined by the following expression, as:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>C(W)=\frac{\sqrt{\sum_{i=1}^m(\omega _i^{-})^2}\sum_{i=1}^m\omega _i^+f_i(W)}{\sqrt{\sum_{i=1}^m(\omega _i^{-})^2}\sum_{i=1}^m\omega _i^+f_i(W)+\sqrt{\sum_{i=1}^m(\omega _i^+)^2}\sum_{i=1}^m\omega _i^{-}f_i(W)}\mbox{.}</math>
|}
| style="text-align: right;" | ( 6)
|}

For any weight vector, <math display="inline">W</math> , which belongs to <math display="inline">\Omega </math> , if the value that <math display="inline">C(W)</math>  can have is bigger, then <math display="inline">F(W)</math>  is closer to <math display="inline">\omega ^+</math> , which means the alternatives are closer to the optimal status from the whole view, and vice versa. Therefore, the programming model can be built as follows:

<span id='fd000105'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{c}
maxC(W)\\
\mbox{s.t.}W\in \Omega \mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | ( 7)
|}

As model [[#fd000105|(7)]]  is a nonlinear programming model, and the objective function is complicated, it is difficult to solve it by common ways. In this paper, the genetic algorithm  [[#br000095|[19]]]  is introduced to deal with the mentioned model, consequently, the optimal criteria weight vector can be calculated and denoted as <math display="inline">W^{{_\ast}}=(w_1^{{_\ast}},w_2^{{_\ast}},\ldots ,w_n^{{_\ast}})^T</math> .      

''Step''  4. Rank the alternatives.      

According to <math display="inline">W^{{_\ast}}</math>  and Formula [[#fd000080|(2)]] , the order of the alternatives can be obtained.      

<span id='s000020'></span>
==4. Examples illustration==

An investment bank is planning to invest in three listed companies, which are denoted as <math display="inline">a_1,a_2</math>  and <math display="inline">a_3</math> , accordingly. There are three criteria taken into account, namely, annual product income, <math display="inline">c_1</math> , social benefit, <math display="inline">c_2</math> , and environmental pollution degree, <math display="inline">c_3</math> . Among these three criteria, <math display="inline">c_1</math>  and <math display="inline">c_2</math>  belong to the benefit type of criterion, while <math display="inline">c_3</math>  is a cost criterion. All three companies would have three possible values, which are in the form of a grey interval number under each criterion, and the corresponding probabilities are known. The vector of criteria weights are denoted with <math display="inline">\begin{array}{l}\displaystyle \Omega =\lbrace 0.1\leq w_1\leq 0.3,0.2\leq w_2\leq 0.4,0.5\leq w_2\leq 0.7\\\displaystyle w_1+w_2+w_3=1\rbrace \end{array}</math> . The ranking of the alternatives need to be given under the above-mentioned conditions. The data for the alternatives are shown in [[#t000015|Table 3]] . In [[#t000015|Table 3]] , “<math display="inline">A</math> ” is the abbreviation of “alternative”, “<math display="inline">V</math> ” is “possible value”, “<math display="inline">P</math> ” is “Probability”, and “<math display="inline">C</math> ” is “Criterion”.

<span id='t000015'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 3.

The alternatives’ evaluations under the criteria.

|-

! <math display="inline">C</math>
! <math display="inline">P</math>
! <math display="inline">a_1</math>
! <math display="inline">a_2</math>
! <math display="inline">a_3</math>
|-

| rowspan="3" | C1
| 0.4
| [2.7, 2.7]
| [2.5, 2.5]
| [3.1, 3.1]
|-

| 0.2
| [3.0, 3.0]
| [2.1, 2.1]
| [3.5, 3.5]
|-

| 0.4
| [2.8, 2.8]
| [2.7, 2.7]
| [2.9, 2.9]
|-

| rowspan="3" | C2
| 0.4
| [3.5, 4.0]
| [3.5, 3.9]
| [3.3, 3.5]
|-

| 0.2
| [3.9, 4.4]
| [4.4, 4.5]
| [2.6, 3.1]
|-

| 0.4
| [3.3, 3.8]
| [3.8, 4.1]
| [3.2, 3.7]
|-

| rowspan="3" | C3
| 0.4
| [0.25, 0.4]
| [0.4, 0.6]
| [0.4, 0.6]
|-

| 0.2
| [0.1, 0.25]
| [0.6, 0.75]
| [0.25, 0.4]
|-

| 0.4
| [0.4, 0.6]
| [0.25, 0.4]
| [0.6, 0.75]
|}

The steps to solve the above problem can be displayed as follows:

''Step''  1. Calculate the expected probability degree of the alternative, with regard to each criterion, and establish the expected probability degree judgment matrix.      

In [[#t000010|Table 2]] , <math display="inline">c_1</math>  and <math display="inline">c_2</math>  are benefit criteria, but <math display="inline">c_3</math>  is a cost criterion. Consequently, <math display="inline">c_3</math>  should be transformed to the benefit criterion for the unification of evaluation. [[#t000020|Table 4]]  shows the results.

<span id='t000020'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 4.

The results for the alternatives under the criteria after unification.

|-

! <math display="inline">C</math>
! <math display="inline">P</math>
! <math display="inline">a_1</math>
! <math display="inline">a_2</math>
! <math display="inline">a_3</math>
|-

| rowspan="3" | C1
| 0.4
| [2.7, 2.7]
| [2.5, 2.5]
| [3.1, 3.1]
|-

| 0.2
| [3.0, 3.0]
| [2.1, 2.1]
| [3.5, 3.5]
|-

| 0.4
| [2.8, 2.8]
| [2.7, 2.7]
| [2.9, 2.9]
|-

| rowspan="3" | C2
| 0.4
| [3.5, 4.0]
| [3.5, 3.9]
| [3.3, 3.5]
|-

| 0.2
| [3.9, 4.4]
| [4.4, 4.5]
| [2.6, 3.1]
|-

| 0.4
| [3.3, 3.8]
| [3.8, 4.1]
| [3.2, 3.7]
|-

| rowspan="3" | C3
| 0.4
| [−0.4, −0.25]
| [−0.6, −0.4]
| [−0.6, −0.4]
|-

| 0.2
| [−0.25, −0.1]
| [−0.75, −0.6]
| [−0.4, −0.25]
|-

| 0.4
| [−0.6, −0.4]
| [−0.4, −0.25]
| [−0.75, −0.6]
|}

The expected probability degree judgment matrix, with regard to each criterion, can be attained conveniently in Matlab 7.0, and the matrices can be denoted as <math display="inline">MATRIXP_1,MATRIXP_2</math> , and <math display="inline">MATRIXP_3</math> . <math display="inline">MATRIXP_1</math>  represents the judgment matrix with regard to <math display="inline">c_1,MATRIXP_2</math>  with regard to <math display="inline">c_2</math> , and <math display="inline">MATRIXP_3</math>  with regard to <math display="inline">c_3</math> .

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>MATRIXP_1=[\begin{array}{ccc}
0.5000 & 0.9200 & 0.0800\\
0.0800 & 0.5000 & 0\\
0.9200 & 1.0000 & 0.5000
\end{array}]\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

''Step''  2. Form the comprehensive judgment matrix of the expected probability degree.      

Since the criteria weights are incompletely certain, the comprehensive judgment matrix of the expected probability degree can be denoted by matrix <math display="inline">CP</math>  as given in [[#tb000005|Box I]] :      

<span id='tb000005'></span>

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>CP=[\begin{array}{ccc}
0.5w_1+0.5w_2+0.5w_3 & 0.92w_1+0.3222w_2+0.68w_3 & 0.08w_1+0.8583w_2+0.8w_3\\
0.08w_1+0.6778w_2+0.32w_3 & 0.5w_1+0.5w_2+0.5w_3 & 0.9644w_2+0.64w_3\\
0.92w_1+0.1417w_2+0.2w_3 & w_1+0.0356w_2+0.36w_3 & 0.5w_1+0.5w_2+0.5w_3
\end{array}]</math>
|}
| style="text-align: right;" | 
|}

''Step''  3. Calculate the criteria weights.      

According to Formula [[#fd000080|(2)]] , the comprehensive sorting value of alternative <math display="inline">a_i</math>  can be shown as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \omega _1=f_1(W)=\frac{1.5w_1+1.6805w_2+1.98w_3+1.5-1}{3\times 2}=0.25w_1+\\\displaystyle +0.2801w_2+0.33w_3+0.0833\mbox{;}\end{array}</math>
|}
| style="text-align: right;" | 
|}

Consequently, the programming model for alternative <math display="inline">a_i</math> , which maximizes the value of <math display="inline">f_i(W)</math> , can be established as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>maxf_i(W)</math>
|}
| style="text-align: right;" | 
|}

Using the simplex method to solve the above models, the results can be shown as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\omega _1^+=0.3953\mbox{,}\quad \omega _2^+=0.3572\mbox{,}\quad \omega _3^+=0.3152\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

And the programming model for alternative <math display="inline">a_i</math> , which minimizes the value of <math display="inline">f_i(W)</math> , can be established as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>minf_i(W)</math>
|}
| style="text-align: right;" | 
|}

Consequently, <math display="inline">\omega _1^{-}=0.3793,\omega _2^{-}=0.3053,\omega _3^{-}=0.2571</math> .      

Therefore, the nonlinear programming model can be built as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>maxC(W)=\frac{0.1434w_1+0.1508w_2+0.1503w_3+0.049}{0.2846w_1+0.302w_2+0.3019w_3+0.0975}</math>
|}
| style="text-align: right;" | 
|}

The optimal criteria weights can be easily obtained using the genetic algorithm toolbox in Matlab 7.0, and they are <math display="inline">W^{{_\ast}}=(0.3,0.2,0.5)^T</math> .      

''Step''  4: Rank the alternatives.      

According to <math display="inline">W^{{_\ast}}</math>  and Formula [[#fd000080|(2)]] , we calculate the sorting vector, which is <math display="inline">\omega =(0.3793,0.3054,0.3152)^T</math> . Thus, the order of alternatives is <math display="inline">a_1,a_3,a_2</math> , which means <math display="inline">a_1</math>  should be the priority for the investment bank, while <math display="inline">a_3</math>  is the second choice, and <math display="inline">a_2</math>  is the last one. If we use the approach proposed in  [[#br000100|[20]]] , the same ranking order of the alternatives is obtained. Thus, the proposed approach is valid.      

<span id='s000025'></span>
==5. Conclusion==

An approach is proposed in this paper for grey stochastic multi-criteria decision-making problems with incompletely uncertain criteria weights. Firstly, the definition of the expected probability degree of a grey stochastic variable is given by introducing the concept of probability preference, and its properties are discussed in this paper. Then, the evaluations of the alternatives, with respect to the criteria, can be transformed into the expected probability degree judgment matrices. The comprehensive judgment matrix was consequently formed. As the criteria weights are incompletely certain, an optimal programming model, which is based on the closeness degree of the sorting vector, is built, and solved by the genetic algorithm to obtain optimal criteria weights. Therefore, the order of the alternative is listed. This approach is verified by an example.

==Acknowledgement==

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China  (Nos. [[#gs000005|71271218]]  and [[#gs000005|71221061]] ) and the Layout Foundation of Humanities and Social Sciences of PR.C Ministry of Education  (No. [[#gs000010|11YJCZH227]] ).

==References==

<ol style='list-style-type: none;margin-left: 0px;'><li><span id='br000005'></span>
[[#br000005|[1]]] H. Katagiri; A fuzzy random multiobjective 0–1 programming based on the expectation optimization model using possibility and necessity measures; Mathematical and Computer Modelling, 40 (2004), pp. 411–421</li>
<li><span id='br000010'></span>
[[#br000010|[2]]] H. Katagiri, S. Masatoshi, K. Kosuke, N. Ichiro; Interactive multiobjective fuzzy random linear programming: maximization of possibility and probability; European Journal of Operational Research, 188 (2008), pp. 530–539</li>
<li><span id='br000015'></span>
[[#br000015|[3]]] E.E. Ammar; On solutions of fuzzy random multiobjective quadratic programming with application in portfolio problem; Information Sciences, 178 (2008), pp. 468–484</li>
<li><span id='br000020'></span>
[[#br000020|[4]]] G.Z. Bu, Y.W. Zhang; Grey fuzzy comprehensive evaluation method based on interval numbers of three parameters; Systems Engineering and Electronics, 23 (1) (2001), pp. 43–46</li>
<li><span id='br000025'></span>
[[#br000025|[5]]] G.Z. Bu, Y.W. Zhang; Grey fuzzy comprehensive evaluation based on the theory of grey fuzzy relation; Systems Engineering—Theory & Practice, 22 (4) (2002), pp. 141–144</li>
<li><span id='br000030'></span>
[[#br000030|[6]]] S.Q. Zhu, K. Meng, H.X. Zhang; Interval numbers grey fuzzy comprehensive evaluation and its application; Electronics Optics & Control, 5 (2006), pp. 36–38</li>
<li><span id='br000035'></span>
[[#br000035|[7]]] J.Z. Kou; On the mutual complementarity between random methods and grey methods; Journal of Systemic Dialectics, 4 (1998), pp. 81–83</li>
<li><span id='br000040'></span>
[[#br000040|[8]]] Y.B. Yu, B.D. Wang, P. Liu, W. Li; Risky multiobjective decision-making theory and its application; Chinese Journal of Management Science, 12 (2003), pp. 9–13</li>
<li><span id='br000045'></span>
[[#br000045|[9]]] D. Luo, S.F. Liu; Research on grey multi-criteria risk decision-making method; Systems Engineering and Electronics, 26 (8) (2004), pp. 1057–1060</li>
<li><span id='br000050'></span>
[[#br000050|[10]]] C.J. Rao, X.P. Xiao; Method of grey matrix relative degree for dynamic hybrid multi-attribute decision making under risk; Systems Engineering and Electronics, 28 (9) (2006), pp. 1353–1357</li>
<li><span id='br000055'></span>
[[#br000055|[11]]] S.B. Yao; Research on Risk Multi-Attribute Decision Making: Theory, Methods and Applications; Huazhong University of Science and Technology (2006)</li>
<li><span id='br000060'></span>
[[#br000060|[12]]] S.F. Liu, T.B. Guo, Y.G. Dang; Grey System Theory and its Application; Science Press (1999)</li>
<li><span id='br000065'></span>
[[#br000065|[13]]] G.D. Li, D. Yamaguchia, M. Nagaib; A grey-based decision-making approach to the supplier selection problem; Mathematical and Computer Modelling, 46 (2007), pp. 573–581</li>
<li><span id='br000070'></span>
[[#br000070|[14]]] J.Q. Wang; Overview on fuzzy multi-criteria decision-making approach; Control and Decision, 23 (6) (2008), pp. 601–606</li>
<li><span id='br000075'></span>
[[#br000075|[15]]] Z.S. Xu, Z.D. Sun; Priority method for a kind of multi-attribute decision-making problems; Journal of Management Sciences in China, 6 (2002), pp. 34–38</li>
<li><span id='br000080'></span>
[[#br000080|[16]]] Z.D. Sun, Z.S. Xu, Q.L. Da; A model based on alternative similarity scale for uncertain multi-attribute decision making; Journal of Management Sciences in China, 9 (6) (2001), pp. 58–62</li>
<li><span id='br000085'></span>
[[#br000085|[17]]] Z.S. Xu, Z.D. Sun; A method based on satisfactory degree of alternative for uncertainty multi-attribute decision-making; Systems Engineering, 19 (3) (2001), pp. 76–79</li>
<li><span id='br000090'></span>
[[#br000090|[18]]] Z.H. Mei, W.D. Jing; A method based on satisfaction degree of decision-maker for multi-attribution decision-making; Journal of Chengdu University of Information Technology, 19 (2) (2004), pp. 306–310</li>
<li><span id='br000095'></span>
[[#br000095|[19]]] G. Xuan, R.W. Cheng; Genetic Algorithm and Project Optimization; Tsinghua University Press, Beijing, China (2004)</li>
<li><span id='br000100'></span>
[[#br000100|[20]]] J.Q. Wang, L. Zhou; Grey random multi-criteria decision-making approach based on maximum membership degree; Control and Decision, 25 (4) (2010), pp. 493–496</li>
</ol>