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  [1] , [2]  and [3] . Problems with criterion values taking the form of grey and fuzziness in the meantime are discussed in  [4] , [5]  and [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  [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  [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  [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  [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  [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  2 gives the definition of a grey number, grey stochastic variable and expected probability degree. Section  3 proposes the grey stochastic multi-criteria decision-making method, based on expected probability degree, and illustrates the procedures in detail. Section  4 applies the proposed method to a practical example to explain its rationality and effectiveness. Section  5 is the conclusion.

Definition 1 [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 ${\textstyle \otimes }$ .

An interval grey number can be defined as the grey number with the lower limit, ${\textstyle a^{L}}$ , and the upper limit, ${\textstyle a^{U}}$ . The interval grey number can be denoted as ${\textstyle \otimes \in [a^{L},a^{U}]}$ .

Assume interval grey numbers, ${\textstyle \otimes _{1}\in [a^{L},a^{U}],a^{L}<a^{U},\otimes _{2}\in [b^{L},b^{U}],b^{L}<b^{U}}$ . The interval grey numbers’ operational rules can be defined as follows  [12] :

• ${\textstyle \otimes _{1}+\otimes _{2}\in [a^{L}+b^{L},a^{U}+b^{U}]}$ ,
• ${\textstyle -\otimes _{1}\in [-a^{U},-a^{L}]}$ ,
• ${\textstyle \otimes _{1}-\otimes _{2}=\otimes _{1}+(-\otimes _{2})\in [a^{L}-b^{U},a^{U}-b^{L}]}$ ,
• ${\textstyle k\times \otimes \in [ka^{L},ka^{U}]}$ , where ${\textstyle k}$ 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 ${\textstyle \zeta (\otimes )}$ , and its ${\textstyle i}$ th value can be presented by ${\textstyle \otimes _{i}}$ . Table 1 shows the probability distribution of ${\textstyle \zeta (\otimes )}$ .

Table 1. The probability distribution of ${\textstyle \zeta (\otimes )}$ .

ζ(⊗)	⊗1	⊗2	…	⊗i	…	⊗n
P	P1	P2	…	Pi	…	Pn

In Table 1 , ${\textstyle \zeta (\otimes )}$ is a grey stochastic variable. ${\textstyle \otimes _{i}}$ is the ${\textstyle i}$ th possible value that would be taken by ${\textstyle \zeta (\otimes ),P_{i}}$ is the probability with respect to ${\textstyle \otimes _{i}}$ , while ${\textstyle n}$ is the number of values that a grey stochastic variable can have.

The probability density function can be denoted as

$f(\zeta (\otimes )=\otimes _{i})=p_{i}{\mbox{.}}$

For better understanding of the above definitions, an example is given. Assume that an investment alternative, ${\textstyle a_{i}}$ , 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, ${\textstyle \zeta (\otimes )}$ , in Table 2 .

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:

${\begin{array}{l}\displaystyle f(\zeta (\otimes )=[2.7,2.8])=0.4{\mbox{,}}\quad f(\zeta (\otimes )=[2.9\\\displaystyle 3.0])=0.2{\mbox{,}}f(\zeta (\otimes )=[3.1,3.4])=0.4{\mbox{.}}\end{array}}$

Definition 3.

Assume that ${\textstyle \zeta (\otimes )}$ is a grey stochastic variable, then, the expected value of the grey stochastic variable is defined by ${\textstyle \sum _{i=1}^{n}P_{i}\times \otimes _{i}}$ , if the value of the formula ${\textstyle \sum _{i=1}^{n}P_{i}\times \otimes _{i}}$ can be attained. And the expected value can be denoted as ${\textstyle E(\zeta (\otimes ))}$ , which satisfies:

$E(\zeta (\otimes ))=\sum _{i=1}^{n}P_{i}\times \otimes _{i}{\mbox{.}}$

According to the grey stochastic variable’s operational rules (1)  and (4) , it can be concluded that ${\textstyle E(\zeta (\otimes ))}$ 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.

${\begin{array}{l}\displaystyle E(\zeta (\otimes ))=[2.7,2.8]\times 0.4+[2.9,3.0]\times 0.2+[3.1\\\displaystyle 3.4]\times 0.4=[2.9,3.08]{\mbox{.}}\end{array}}$

Definition 4 [13] .

Let ${\textstyle L(\otimes _{1})=a^{U}-a^{L}}$ be the length of the grey number, ${\textstyle \otimes _{1}\in [a^{L},a^{U}]}$ , and ${\textstyle L(\otimes _{2})=b^{U}-b^{L}}$ be the length of the grey number, ${\textstyle \otimes _{2}\in [b^{L},b^{U}]}$ , then,

$P(\otimes _{1}\geq \otimes _{2})={\frac {max\lbrace 0,L(\otimes _{1})+L(\otimes _{2})-max(b^{U}-a^{L},0)\rbrace }{L(\otimes _{1})+L(\otimes _{2})}}$

is called the expected probability degree of ${\textstyle \otimes _{1}}$ against ${\textstyle \otimes _{2}}$ . Therefore, the relationship between ${\textstyle \otimes _{1}}$ and ${\textstyle \otimes _{2}}$ can be determined as follows:

• If ${\textstyle P(\otimes _{1}\geq \otimes _{2})<0.5}$ , we say that ${\textstyle \otimes _{1}}$ is less than ${\textstyle \otimes _{2}}$ , and denote it as ${\textstyle \otimes _{1}<\otimes _{2}}$ ;
• ${\textstyle P(\otimes _{1}\geq \otimes _{2})=0.5}$ , we say that ${\textstyle \otimes _{1}}$ is equal to ${\textstyle \otimes _{2}}$ , and denote it as ${\textstyle \otimes _{1}=\otimes _{2}}$ ;
• If ${\textstyle P(\otimes _{1}\geq \otimes _{2})>0.5}$ , we say that ${\textstyle \otimes _{1}}$ is larger than ${\textstyle \otimes _{2}}$ , and denote it as ${\textstyle \otimes _{1}>\otimes _{2}}$ .

Definition 5.

Assume that ${\textstyle x_{i}}$ and ${\textstyle x_{j}}$ are grey stochastic variables, and their probability density functions can be denoted as ${\textstyle f_{i}}$ and ${\textstyle f_{j}}$ , respectively. The expected probability degree of grey stochastic variable, ${\textstyle x_{i}}$ , against ${\textstyle x_{j}}$ , can be denoted as ${\textstyle E(p(x_{i}>x_{j}))}$ :

${\begin{array}{l}\displaystyle E(p(x_{i}>x_{j}))=\\\displaystyle =\sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})\times p(\otimes _{i}^{d}\geq \otimes _{j}^{g}){\mbox{,}}\end{array}}$

where ${\textstyle \otimes _{i}^{d}}$ and ${\textstyle \otimes _{j}^{g}}$ are the values that the variables ${\textstyle x_{i}}$ and ${\textstyle x_{j}}$ may take separately. ${\textstyle d=1,\ldots ,t}$ , and ${\textstyle g=1,\ldots ,t}$ . ${\textstyle p(\otimes _{i}^{d}\geq \otimes _{j}^{g})}$ are the possibility degrees of the interval grey number, ${\textstyle \otimes _{i}^{d}}$ , which is more than or equal to ${\textstyle \otimes _{j}^{g}}$ .

The expected probability degree is the measurement of the average probability degree in which the grey stochastic variable, ${\textstyle x_{i}}$ , is better than ${\textstyle x_{j}}$   [11] . The properties of expected probability degree can be listed as follows:

• ${\textstyle 0\leq E(p(x_{i}>x_{j}))\leq 1}$ ,
• ${\textstyle E(p(x_{i}>x_{j}))+E(p(x_{j}>x_{i}))=1}$ ,
• if ${\textstyle E(p(x_{i}>x_{j}))=E(p(x_{j}>x_{i}))}$ , then ${\textstyle E(p(x_{i}>x_{j}))=0.5}$ .

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

Proof.

As ${\textstyle {\begin{array}{l}\displaystyle E(p(x_{i}>x_{j}))=\\\displaystyle =\sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})\times p(\otimes _{i}^{d}\geq \otimes _{j}^{g})\end{array}}}$ ,

$E(p(x_{i}>x_{j}))\geq {\underset {1\leq d,g\leq t}{min}}(p({\underset {i}{\overset {d}{\otimes }}}\geq \otimes _{j}^{g}))\times \sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g}){\mbox{,}}$

and ${\textstyle E(p(x_{i}>x_{j}))\leq max_{1\leq d,g\leq t}(p(\otimes _{i}^{d}\geq \otimes _{j}^{g}))\times \sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})}$ .

Furthermore, considering any value that ${\textstyle d}$ or ${\textstyle g}$ may take, there always exist ${\textstyle max_{1\leq d,g\leq t}(p(\otimes _{i}^{d}\geq \otimes _{j}^{g}))\leq 1}$ and ${\textstyle min_{1\leq d,g\leq t}(p(\otimes _{i}^{d}\geq \otimes _{j}^{g}))\geq 0}$ , so, one can reach the conclusion that ${\textstyle 0\leq E(p(x_{i}>x_{j}))\leq 1}$ .

Due to ${\textstyle {\begin{array}{l}\displaystyle E(p(x_{i}>x_{j}))=\\\displaystyle =\sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})\times p(\otimes _{i}^{d}\geq \otimes _{j}^{g})\end{array}}}$ , as well as ${\textstyle {\begin{array}{l}\displaystyle E(p(x_{j}>x_{i}))=\\\displaystyle =\sum _{g=1}^{t}\sum _{d=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})\times p(\otimes _{j}^{d}\geq \otimes _{i}^{g})\end{array}}}$ , we have:

${\begin{array}{l}\displaystyle E(p(x_{i}>x_{j}))+E(p(x_{j}>x_{i}))=\\\displaystyle =\sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})\times [p(\otimes _{i}^{d}\geq \otimes _{j}^{g})+\\\displaystyle +p(\otimes _{j}^{d}\geq \otimes _{i}^{g})]{\mbox{.}}\end{array}}$

Since ${\textstyle p(\otimes _{i}^{d}\geq \otimes _{j}^{g})+p(\otimes _{j}^{d}\geq \otimes _{i}^{g})=1}$ (see Ref.  [9] ), then:

${\begin{array}{l}\displaystyle E(p(x_{i}>x_{j}))+E(p(x_{j}>x_{i}))=\\\displaystyle =\sum _{d=1}^{t}\sum _{g=1}^{t}f_{i}(\otimes _{i}^{d})\times f_{j}(\otimes _{j}^{g})=1{\mbox{.}}\end{array}}$

According to the property (2) , if ${\textstyle E(p(x_{i}>x_{j}))=E(p(x_{j}>x_{i}))}$ , then:

$E(p(x_{i}>x_{j}))+E(p(x_{j}>x_{i}))=2\times E(p(x_{i}>x_{j}))=1{\mbox{.}}$

Therefore, ${\textstyle E(p(x_{i}>x_{j}))=0.5}$ .

Definition 6.

Assume that ${\textstyle GRS=\lbrace x_{1},x_{2},\ldots ,x_{m}\rbrace }$ is a discrete set of ${\textstyle m}$ 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 ${\textstyle MATRIXP=(E(p(x_{i}>x_{j})))_{m\times m}}$ , where: ${\textstyle i=1,\ldots ,m;j=1,\ldots ,m}$ .

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 ${\textstyle E(p(x_{i}>x_{i}))=0.5}$ in the expected probability degree judgment matrix. Furthermore, there always exists the equality ${\textstyle E(p(x_{i}>x_{j}))+E(p(x_{j}>x_{i}))=1}$ , so, the judgment matrix of the expected probability degree is a complementary judgment matrix.

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 ${\textstyle A=\lbrace a_{1},\ldots ,a_{i},\ldots ,a_{m}\rbrace }$ is a discrete alternative set of ${\textstyle m}$ possible alternatives, and ${\textstyle C=\lbrace c_{1},\ldots ,c_{k},\ldots ,c_{n}\rbrace }$ is a set of ${\textstyle n}$ criteria. ${\textstyle W=\lbrace w_{1},\ldots ,w_{k},\ldots ,w_{n}\rbrace }$ is the vector of criteria weights, and the weights are subject to the constraints as follows:

$\sum _{k=1}^{n}w_{k}=1{\mbox{,}}\quad w_{k}\geq 0,k=1,2,\ldots ,n{\mbox{.}}$

The criterion value of alternative ${\textstyle a_{i}}$ , with respect to criterion ${\textstyle c_{k}}$ , is denoted as ${\textstyle x_{ik}}$ , and ${\textstyle x_{ik}}$ is a grey stochastic variable. The incompletely certain information of the criteria weights can be represented by symbol ${\textstyle \Omega }$ and it can be also shown in the form of linear equalities and inequalities, as follows  [14] :

• ${\textstyle \lbrace W:A_{1}W\geq b,W>0,b\geq 0\rbrace }$ ,
• ${\textstyle \lbrace W:A_{1}W\leq b,W>0,b\geq 0\rbrace }$ ,
• ${\textstyle \lbrace W:A_{1}W=b,W>0,b\geq 0\rbrace }$ , where ${\textstyle A_{1}}$ is a matrix with one line and ${\textstyle n}$ columns, and ${\textstyle W=\lbrace w_{1},\ldots ,w_{k},\ldots ,w_{n}\rbrace }$ .

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 ${\textstyle c_{k}}$ is concerned, the expected probability degree judgment matrix for the alternative set can be denoted as ${\textstyle MATRIXP_{k}}$ :

$MATRIXP_{k}=(E(p(x_{ik}>x_{jk})))_{m\times m}{\mbox{,}}$

where ${\textstyle i=1,\ldots ,m;j=1,\ldots ,m;k=1,2,\ldots ,n}$ .

According to Theorem 1 , ${\textstyle MATRIXP_{k}}$ 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 ${\textstyle CP=(\sum _{k=1}^{n}w_{k}\times E(p(x_{ik}>x_{jk})))_{m\times m}}$ , where ${\textstyle i=1,\ldots ,m;j=1,\ldots ,m;k=1,2,\ldots ,n}$ .

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 (3) of the expected probability degree, there always exists ${\textstyle E(p(x_{ik}>x_{ik}))=0.5}$ , so does the expression ${\textstyle \sum _{k=1}^{n}w_{k}\times E(p(x_{ik}>x_{ik}))=0.5\times \sum _{k=1}^{n}w_{k}=0.5}$ . Besides,

${\begin{array}{l}\displaystyle \sum _{k=1}^{n}w_{k}\times E(p(x_{ik}>x_{jk}))+\sum _{k=1}^{n}w_{k}\times E(p(x_{jk}>x_{ik}))=\\\displaystyle =\sum _{k=1}^{n}w_{k}\times (E(p(x_{ik}>x_{jk}))+E(p(x_{jk}>x_{ik})))=\\\displaystyle =\sum _{k=1}^{n}w_{k}=1{\mbox{,}}\end{array}}$

thus, matrix ${\textstyle CP}$ is a complementary judgment matrix.

Step 3. Calculate the criteria weights.

According to the sorting vector, ${\textstyle \omega =(\omega _{1},\omega _{2},\ldots ,\omega _{i},\ldots ,\omega _{m})^{T}}$ was introduced in  [15] for solving the ranking problem of the complementary judgment matrix. ${\textstyle \omega _{i}}$ can be calculated by the method as follows:

$\omega _{i}={\frac {(\sum _{l=1}^{m}\sum _{k=1}^{n}w_{k}\times E(p(x_{ik}>x_{lk})))+{\frac {m}{2}}-1}{m\times (m-1)}}{\mbox{,}}$

( 1)

where ${\textstyle i=1,2,\ldots ,m}$ .

If the criteria weights are already known, the order of the alternative set can be attained by comparing ${\textstyle \omega _{i}}$ . However, the situation of criteria weights, which are incompletely certain, is common in real multi-criteria decision-making. Consequently, by drawing lessons from Refs.  [16] , [17]  and [18] , establishing an optimization model based on the closeness degree of ${\textstyle \omega _{i}}$ is a major way of obtaining the criteria weights in this paper.

For Formula (1) , it can be transformed into Formula (2) , as follows:

$\omega _{i}={\frac {(\sum _{k=1}^{n}w_{k}\times \sum _{l=1}^{m}E(p(x_{ik}>x_{lk})))+{\frac {m}{2}}-1}{m\times (m-1)}}{\mbox{,}}$

( 2)

where ${\textstyle i=1,2,\ldots ,m}$ .

Therefore, the comprehensive sorting value of alternative ${\textstyle a_{i}}$ can be represented by ${\textstyle \omega _{i}=f_{i}(W)}$ . Furthermore, the comprehensive sorting vector of the alternative set can be denoted by ${\textstyle F(W)=(f_{1}(W),f_{2}(W),\ldots ,f_{m}(W))}$ . As for given ${\textstyle \Omega }$ , obviously, if the value of ${\textstyle f_{i}(W)}$ is bigger, the corresponding alternative, ${\textstyle a_{i}}$ , would be better. Consequently, the multi-objective decision-making model can be built as follows:

${\begin{array}{c}maxF(W)=(f_{1}(W),f_{2}(W),\ldots ,f_{m}(W))\\{\mbox{s.t.}}W\in \Omega {\mbox{.}}\end{array}}$

( 3)

Solve the programming model as follows:

${\begin{array}{c}maxf_{i}(W)\\{\mbox{s.t.}}W\in \Omega {\mbox{,}}\end{array}}\quad i=1,2,\ldots ,m{\mbox{.}}$

( 4)

Since model (4) is a linear programming model, it can be solved to obtain the optimal weight vector ${\textstyle W_{i}^{+}=(w_{1}^{+},w_{2}^{+},\ldots ,w_{n}^{+})^{T}}$ , with respect to alternative ${\textstyle a_{i}}$ , using the simplex method. Then, the positive comprehensive sorting value of alternative ${\textstyle a_{i}}$ can be attained in terms of the ${\textstyle W_{i}^{+}}$ , namely, ${\textstyle \omega _{i}^{+}=f_{i}(W_{i}^{+})}$ .

Then, we solve the programming model as follows:

${\begin{array}{c}minf_{i}(W)\\{\mbox{s.t.}}W\in \Omega {\mbox{,}}\end{array}}\quad i=1,2,\ldots ,m{\mbox{.}}$

( 5)

Since model (5) is also a linear programming model, it can be solved to obtain the optimal weight vector, ${\textstyle W_{i}^{-}=(w_{1}^{-},w_{2}^{-},\ldots ,w_{n}^{-})^{T}}$ , with respect to alternative ${\textstyle a_{i}}$ , using the simplex method. Then, the negative comprehensive sorting value of alternative ${\textstyle a_{i}}$ can be attained in terms of ${\textstyle W_{i}^{-}}$ , namely, ${\textstyle \omega _{i}^{-}=f_{i}(W_{i}^{-})}$ .

Vectors ${\textstyle \omega ^{+}=(\omega _{1}^{+},\omega _{2}^{+},\ldots ,\omega _{m}^{+})^{T}}$ and ${\textstyle \omega ^{-}=(\omega _{1}^{-},\omega _{2}^{-},\ldots ,\omega _{m}^{-})^{T}}$ 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:

$C(W)={\frac {{\sqrt {\sum _{i=1}^{m}(\omega _{i}^{-})^{2}}}\sum _{i=1}^{m}\omega _{i}^{+}f_{i}(W)}{{\sqrt {\sum _{i=1}^{m}(\omega _{i}^{-})^{2}}}\sum _{i=1}^{m}\omega _{i}^{+}f_{i}(W)+{\sqrt {\sum _{i=1}^{m}(\omega _{i}^{+})^{2}}}\sum _{i=1}^{m}\omega _{i}^{-}f_{i}(W)}}{\mbox{.}}$

( 6)

For any weight vector, ${\textstyle W}$ , which belongs to ${\textstyle \Omega }$ , if the value that ${\textstyle C(W)}$ can have is bigger, then ${\textstyle F(W)}$ is closer to ${\textstyle \omega ^{+}}$ , 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:

${\begin{array}{c}maxC(W)\\{\mbox{s.t.}}W\in \Omega {\mbox{.}}\end{array}}$

( 7)

As model (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  [19] is introduced to deal with the mentioned model, consequently, the optimal criteria weight vector can be calculated and denoted as ${\textstyle W^{_{\ast }}=(w_{1}^{_{\ast }},w_{2}^{_{\ast }},\ldots ,w_{n}^{_{\ast }})^{T}}$ .

Step 4. Rank the alternatives.

According to ${\textstyle W^{_{\ast }}}$ and Formula (2) , the order of the alternatives can be obtained.

4. Examples illustration

An investment bank is planning to invest in three listed companies, which are denoted as ${\textstyle a_{1},a_{2}}$ and ${\textstyle a_{3}}$ , accordingly. There are three criteria taken into account, namely, annual product income, ${\textstyle c_{1}}$ , social benefit, ${\textstyle c_{2}}$ , and environmental pollution degree, ${\textstyle c_{3}}$ . Among these three criteria, ${\textstyle c_{1}}$ and ${\textstyle c_{2}}$ belong to the benefit type of criterion, while ${\textstyle c_{3}}$ 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 ${\textstyle {\begin{array}{l}\displaystyle \Omega =\lbrace 0.1\leq w_{1}\leq 0.3,0.2\leq w_{2}\leq 0.4,0.5\leq w_{2}\leq 0.7\\\displaystyle w_{1}+w_{2}+w_{3}=1\rbrace \end{array}}}$ . The ranking of the alternatives need to be given under the above-mentioned conditions. The data for the alternatives are shown in Table 3 . In Table 3 , “${\textstyle A}$ ” is the abbreviation of “alternative”, “${\textstyle V}$ ” is “possible value”, “${\textstyle P}$ ” is “Probability”, and “${\textstyle C}$ ” is “Criterion”.

Table 3. The alternatives’ evaluations under the criteria.

${\textstyle C}$	${\textstyle P}$	${\textstyle a_{1}}$	${\textstyle a_{2}}$	${\textstyle a_{3}}$
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]
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]
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 Table 2 , ${\textstyle c_{1}}$ and ${\textstyle c_{2}}$ are benefit criteria, but ${\textstyle c_{3}}$ is a cost criterion. Consequently, ${\textstyle c_{3}}$ should be transformed to the benefit criterion for the unification of evaluation. Table 4 shows the results.

Table 4. The results for the alternatives under the criteria after unification.

${\textstyle C}$	${\textstyle P}$	${\textstyle a_{1}}$	${\textstyle a_{2}}$	${\textstyle a_{3}}$
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]
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]
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 ${\textstyle MATRIXP_{1},MATRIXP_{2}}$ , and ${\textstyle MATRIXP_{3}}$ . ${\textstyle MATRIXP_{1}}$ represents the judgment matrix with regard to ${\textstyle c_{1},MATRIXP_{2}}$ with regard to ${\textstyle c_{2}}$ , and ${\textstyle MATRIXP_{3}}$ with regard to ${\textstyle c_{3}}$ .

$MATRIXP_{1}=[{\begin{array}{ccc}0.5000&0.9200&0.0800\\0.0800&0.5000&0\\0.9200&1.0000&0.5000\end{array}}]{\mbox{,}}$

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 ${\textstyle CP}$ as given in Box I :

$CP=[{\begin{array}{ccc}0.5w_{1}+0.5w_{2}+0.5w_{3}&0.92w_{1}+0.3222w_{2}+0.68w_{3}&0.08w_{1}+0.8583w_{2}+0.8w_{3}\\0.08w_{1}+0.6778w_{2}+0.32w_{3}&0.5w_{1}+0.5w_{2}+0.5w_{3}&0.9644w_{2}+0.64w_{3}\\0.92w_{1}+0.1417w_{2}+0.2w_{3}&w_{1}+0.0356w_{2}+0.36w_{3}&0.5w_{1}+0.5w_{2}+0.5w_{3}\end{array}}]$

Step 3. Calculate the criteria weights.

According to Formula (2) , the comprehensive sorting value of alternative ${\textstyle a_{i}}$ can be shown as follows:

${\begin{array}{l}\displaystyle \omega _{1}=f_{1}(W)={\frac {1.5w_{1}+1.6805w_{2}+1.98w_{3}+1.5-1}{3\times 2}}=0.25w_{1}+\\\displaystyle +0.2801w_{2}+0.33w_{3}+0.0833{\mbox{;}}\end{array}}$

Consequently, the programming model for alternative ${\textstyle a_{i}}$ , which maximizes the value of ${\textstyle f_{i}(W)}$ , can be established as follows:

$maxf_{i}(W)$

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

$\omega _{1}^{+}=0.3953{\mbox{,}}\quad \omega _{2}^{+}=0.3572{\mbox{,}}\quad \omega _{3}^{+}=0.3152{\mbox{.}}$

And the programming model for alternative ${\textstyle a_{i}}$ , which minimizes the value of ${\textstyle f_{i}(W)}$ , can be established as follows:

$minf_{i}(W)$

Consequently, ${\textstyle \omega _{1}^{-}=0.3793,\omega _{2}^{-}=0.3053,\omega _{3}^{-}=0.2571}$ .

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

$maxC(W)={\frac {0.1434w_{1}+0.1508w_{2}+0.1503w_{3}+0.049}{0.2846w_{1}+0.302w_{2}+0.3019w_{3}+0.0975}}$

The optimal criteria weights can be easily obtained using the genetic algorithm toolbox in Matlab 7.0, and they are ${\textstyle W^{_{\ast }}=(0.3,0.2,0.5)^{T}}$ .

Step 4: Rank the alternatives.

According to ${\textstyle W^{_{\ast }}}$ and Formula (2) , we calculate the sorting vector, which is ${\textstyle \omega =(0.3793,0.3054,0.3152)^{T}}$ . Thus, the order of alternatives is ${\textstyle a_{1},a_{3},a_{2}}$ , which means ${\textstyle a_{1}}$ should be the priority for the investment bank, while ${\textstyle a_{3}}$ is the second choice, and ${\textstyle a_{2}}$ is the last one. If we use the approach proposed in  [20] , the same ranking order of the alternatives is obtained. Thus, the proposed approach is valid.

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. 71271218 and 71221061 ) and the Layout Foundation of Humanities and Social Sciences of PR.C Ministry of Education (No. 11YJCZH227 ).

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