Definition 2.1.1 [65] Let U be a universe of discourse

A : U → [ 0,1 ]

u | → A (x)

then A is called fuzzy set on U. A ( x ) is called the membership function of A.

The family of all fuzzy sets on U is denoted by F ( U ) . Let A , B ∈ F ( U ) . Related operations of fuzzy sets.

1) ∀ x ∈ U , B ( x ) ≤ A ( x ) ⇒ B ⊆ A .

2) ( A ∪ B ) ( x ) = A ( x ) ∨ B ( x ) = max ( A ( x ) , B ( x ) ) ; ( A ∩ B ) ( x ) = A ( x ) ∧ B ( x ) = min ( A ( x ) , B ( x ) ) .

3) ( A B ) ( x ) = A ( x ) B ( x ) , A c ( x ) = 1 − A ( x ) .

Definition 2.1.1 [66] Let R is a fuzzy relation, we say that

1) R is referred to as a reflexive relation if for any x ∈ U , R ( x , x ) = 1 .

2) R is referred to as a symmetric relation if for any x , y ∈ U , R ( x , y ) = R ( y , x ) .

3) R is referred to as a transitive relation if for any x , y , z ∈ U , R ( x , y ) ≥ ∨ z ∈ U ( R ( x , z ) ∧ R ( z , y ) ) .

If R is reflexive, symmetric and transitive on U, then we say that R is a fuzzy equivalence relation on U.

Definition 2.1.2 [67] Let ( U , A , P ) be a probability space. Where A is the family of all fuzzy sets that is denoted by F ( U ) . Then A ∈ A is a fuzzy event on U. The probability of A is

P ( A ) ≜ ∫ U   A ( x ) d P .

If U is a finite set, U = { x i | i = 1 , 2 , ⋯ , n } , P ( x i ) = p i , then

P ( A ) ≜ ∑ i = 1 n   A ( x i ) p i

Proposition 2.1.1 Let ( U , A , P ) be a probability space A , B ∈ A . The property of the establishment.

1) P ( U ) = 1 ,that is P ( U ) = ∫ U     d P = 1 ;

2) 0 ≤ P ( A ) ≤ 1 ;

3) A ⊆ B , P ( A ) ≤ P ( B ) ;

4) P ( A c ) = 1 − P ( A ) ;

5) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) ;

6) P ( A ∪ B ) = P ( A ) + P ( B ) , A ∩ B = ∅ .

Definition 2.1.3 [67] Let ( U , A , P ) be a probability space and A , B be two fuzzy events on U. If P ( B ) ≠ 0 then

P ( A | B ) = P ( A B ) P (B)

is called the conditional probability of A given B.

Proposition 2.1.2 Let ( U , A , P ) be a probability space and A be a classical event on X. Then, for each fuzzy event B on X, it holds that

P ( A | B ) + P ( A c | B ) = 1

Proof.

P ( A | B ) + P ( A c | B ) = P ( A B ) P ( B ) + P ( A c B ) P ( B ) = ∫ U A ( x ) B ( x ) d P ∫ U   B ( x ) d P + ∫ U A c ( x ) B ( x ) d P ∫ U   B ( x ) d P = ∫ A   B ( x ) d P + ∫ A c   B ( x ) d P ∫ U   B ( x ) d P = ∫ A ∪ A c B ( x ) d P ∫ U   B ( x ) d P = 1

Definition 2.2.1 [29] Let X be a non empty classic set. The three reorganization in X like A = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ X } meets the following three points.

1) μ A → [ 0,1 ] indicates that the element of X belongs to the A membership degree.

2) ν A → [ 0,1 ] indicates that the non membership degree.

3) 0 ≤ A ( x ) + ν A ( x ) ≤ 1 .

A is called an intuitionistic fuzzy set on the X.

Related operations of IF sets. Suppose

A = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ X } ∈ I F ( X ) ,

B = { 〈 x , μ B ( x ) , ν B ( x ) 〉 | x ∈ X } ∈ I F ( X ) .

A ⊆ B ⇔ μ A ( x ) ≤ μ B ( x )     and     ν A ( x ) ≥ ν B ( x ) , ∀ x ∈ X ;

A ∩ B = { 〈 x , min { μ A ( x ) , μ B ( x ) } , max { ν A ( x ) , ν B ( x ) } 〉 | x ∈ X } ;

A ∪ B = { 〈 x , max { μ A ( x ) , μ B ( x ) } , min { ν A ( x ) , ν B ( x ) } 〉 | x ∈ X } ;

A c = { 〈 x , ν A ( x ) , μ A ( x ) 〉 | x ∈ X } .

Definition 2.2.2 An intuitionistic fuzzy relation R on a non-empty set X is a mapping R : X × X ⇒ L defined as R ( x , y ) = 〈 μ R ( x , y ) , ν R ( x , y ) 〉 ∈ L For x , y ∈ X .The family of all IF relations on X is denoted by R . An IF relation R ∈ R is:

1) Reflexive, if R ( x , x ) = 1 for each x ∈ X ;

2) Symmetric, if R ( x , y ) = R ( y , x ) for each x , y ∈ X ;

3) Transitive, if ∨ y ∈ X ( R ( x , y ) ∧ R ( y , z ) ) ≤ L R ( x , z ) for each x , y , z ∈ X .

We write the IF relation R ( x , y ) = ( μ R ( x , y ) , ν R ( x , y ) ) for simplicity, where μ R ( x , y ) , ν R ( x , y ) : X × X → I = [ 0,1 ] and satisfy μ R ( x , y ) + ν R ( x , y ) ≤ 1, ∀ x , y ∈ X .

If X = { x 1 , x 2 , ⋯ , x n } is a finite set, then an IF relation R : X × X → L can be represented by an IF matrix form R = ( R ( x i , x j ) ) n × n , i.e. Then

R = ( ( μ R ( x 1 , x 1 ) , ν R ( x 1 , x 1 ) ) ( μ R ( x 1 , x 2 ) , ν R ( x 1 , x 2 ) ) ⋯ ( μ R ( x 1 , x n ) , ν R ( x 1 , x n ) ) ( μ R ( x 2 , x 1 ) , ν R ( x 2 , x 1 ) ) ( μ R ( x 2 , x 2 ) , ν R ( x 2 , x 2 ) ) ⋯ ( μ R ( x 2 , x n ) , ν R ( x 2 , x n ) ) ⋮ ⋮ ⋱ ⋮ ( μ R ( x n , x 1 ) , ν R ( x n , x 1 ) ) ( μ R ( x n , x 2 ) , ν R ( x n , x 2 ) ) ⋯ ( μ R ( x n , x n ) , ν R ( x n , x n ) ) ) .

V ( R ) is the collection of IFVs R ( x i , x j ) for i , j = 1,2, ⋯ , n , i.e. V ( R ) = { α | α = R ( x i , x j )   forsome   i , j = 1,2, ⋯ , n }

Definition 2.2.3 [40] An IF information system is an ordered quadruple I = ( U , A T , V , f ) .

U = { x 1 , x 2 , ⋯ , x n } is a non-empty finite set of objects;

A T = { a 1 , a 2 , ⋯ , a p } is a non-empty finite set of attributes;

V = ∪ a ∈ A T V a and V a is a domain of attribute a;

f : U × A T ⇒ V is a function such that f ( x , a ) ∈ V a , for each a ∈ A T , x ∈ U , called an information function, where V a is an IF set of universe U. That is f ( x , a ) = 〈 μ a ( x ) , ν a ( x ) 〉 , for all a ∈ A T .

Definition 2.2.4 Let ( U , A ˜ , P ) be a IF probability space. Where A ˜ is the family of all IF sets that is denoted by F ( U ) . Then A ∈ A ˜ is a IF event on U. The probability of A is

P ( A ) = ∫ U   A ( x ) d P = 〈 ∫ U   μ A ( x ) d P , ∫ U   ν A ( x ) d P 〉 = 〈 P ( μ A ) , P ( ν A ) 〉 .

Among P ( μ A ) is probability of membership, P ( ν A ) is probability of nonmembership.

Proposition 2.2.1 Each IF event A is associated with an IF probability P ( A ) . The P is called an IF probability measure on U which is generated by P. If A degenerates into a classical event or a fuzzy event A ′ it follows that P ( A ) = P ( A ′ ) .

Proposition 2.2.2 Also, if U = { x 1 , x 2 , ⋯ , x n } is a finite set and p i = P ( x i ) , then

P ( A ) = ∑ i = 1 n     A ( x i ) p i = 〈 ∑ i = 1 n     μ A ( x i ) p i , ∑ i = 1 n     ν A ( x i ) p i 〉 .

Definition 2.2.4 Let ( U , A , P ) be a probability space and A , B be two IF events on U. If P ( ν B ) ≠ 0 and P ( μ B ) ≠ 0 then

P ( A | B ) = ( P ( μ A | μ B ) , P ( ν A | ν B ) ) .

is called the IF conditional probability of A given B.

Decision-theoretic rough sets were first proposed by Yao [42] for the Bayesian decision process. Based on the thoughts of three-way decisions, DTRS adopt two state sets and three action sets to depict the decision-making process. The state set is denoted by Ω = { X , X c } showing that an object belongs to X and is outside X, respectively. The action sets with respect to a state are given by A = { a P , a B , a N } , where a P , a B and a N represent three actions about deciding x ∈ P O S ( X ) , x ∈ B N D ( X ) , and x ∈ N E G ( X ) , namely an object x belongs to X, is uncertain and not in X, respectively. The loss function concerning the loss of expected by taking various actions in the different states is given by the 3 × 2 matrix in Table 1.

In Table 1, λ P P , λ B P and λ N P express the losses happened for taking actions of a P , a B and a N , respectively, when an object belongs to X. Similarly, λ P N , λ B N and λ N N indicate the losses incurred for taking the same actions when the object does not belong to X. For an object x, the expected loss on taking the actions could be expressed as:

R ( a P | [ x ] R ) = λ P P P ( X | [ x ] R ) + λ P N P ( X c | [ x ] R ) ; (1)

R ( a B | [ x ] R ) = λ B P P ( X | [ x ] R ) + λ B N P ( X c | [ x ] R ) ; (2)

R ( a N | [ x ] R ) = λ N P P ( X | [ x ] R ) + λ N N P ( X c | [ x ] R ) . (3)

By the Bayesian decision process, we can get the following minimum-risk decision rules:

(P) If R ( a P | [ x ] R ) ≤ R ( a B | [ x ] R ) and R ( a P | [ x ] R ) ≤ R ( a N | [ x ] R ) , then decide x ∈ P O S ( X ) ;

(B) If R ( a B | [ x ] R ) ≤ R ( a P | [ x ] R ) and R ( a B | [ x ] R ) ≤ R ( a N | [ x ] R ) , then decide x ∈ B N D ( X ) ;

(N) If R ( a N | [ x ] R ) ≤ R ( a P | [ x ] R ) and R ( a N | [ x ] R ) ≤ R ( a B | [ x ] R ) , then decide x ∈ N E G ( X ) .

In addition, By taking into account the loss of receiving the right things is not greater than the latency, and both of them are less than the loss of refusing the accurate things; at the same time, the loss of rejecting improper things is less than or equal to the delation in accepting the correct things, and both shall be smaller than the loss of receiving the invalidate things. Hence, a reasonable assumption is that 0 ≤ λ P P ≤ λ B P < λ N P and 0 ≤ λ N N ≤ λ B N < λ P N .

Accordingly, the conditions of the three decision rules (P)-(N) are reducible to the following form.

(P) If P ( X | [ x ] R ) ≥ α and P ( X | [ x ] R ) ≥ γ , then decide x ∈ P O S ( X ) ;

(B) If P ( X | [ x ] R ) ≤ α and P ( X | [ x ] R ) ≥ β , then decide x ∈ B N D ( X ) ;

(N) If P ( X | [ x ] R ) ≥ β and P ( X | [ x ] R ) ≤ γ , then decide.

where the thresholds values are given by:

Definition 3.1.1 Let be an IF information system, , , , , then

is called the relative similarity degree of and; or

is regarded as the relative similarity degree of and.

From the above two formulas, the relative similarity degree of and, that is the similarity of objects and under the attribute. In addition, the greater the value of, the greater the similarity degree. Particularly, when, then IF number is completely similar to. In other words, the property value of objects and are identical.

Proposition 3.1.1 For any three IF numbers, , , the following properties can be obtained:

1) is bounded,;

2) is reflexive,;

3) is symmetric,;

4) is transitive, if and, then;

5) is contiguous, if is closer to than, then; if is closer to than, then.

Definition 3.1.2 Let be an IF information system, , , , . The similarity degree of objects and under attribute set AT is as follows:

Through establishing analogical relations, we could turn IF information system into a fuzzy approximation space in accordance with definition 3.1 and 3.2. The subscript AT will be omitted in the rear. It holds:

1) Firstly, U is a non-empty classical set, a binary relation R from U to U indicates a fuzzy set. So R is a fuzzy relation on the universe U.

2) Furthermore, R is a fuzzy equivalence relation on U. The reasons are as follows:

• , , R is reflexive;

• , , R is symmetric;

• , , R is transitive

These three conditions are very obvious. Therefore, ordered pair is a fuzzy approximation space.

Given the probability P with its description R, a fuzzy probability approximation space is constructed in which U is a domain of discourse, R is a fuzzy equivalence relation on U, and P is fuzzy probability of U.

Assume be a fuzzy probability approximation space, for each, is denoted by for any. In light of (1)~(3), thus, the expected costs of adopting various actions in different states for x are expressed as follows:

Proposition 3.2.1 The condition probability and are calculated by:

In the above equations,. The computing method of condition probability is not the same as we know before. Since for every. Thus, (4) - (6) is further expressed as:

Loss function to meet the conditions: and. According to Bayesian decision process, the decision rules can be characterized by the following form:

(P₁) If and, then decide;

(B₁) If and, then decide;

(N₁) If and, then decide.

The decision rules (P₁)-(N₁) are the three-way decisions, which have three regions:, and. These rules mainly relies on the comparisons among, and which are essentially computing the fuzzy probabilities. Decision rules (P₁)-(N₁) of three-way decisions can be simplified as:

(P₂) If and, then decide;

(B₂) If and, then decide;

(N₂) If and, then decide.

Proposition 3.2.2 In this case, we have the following simplified fuzzy probability region:

In the fuzzy relation R, the fuzzy probability upper approximation and the fuzzy probability of X are respectively:

Under the discussions in Proposition3.2.2, the additional conditions of decision rule (B₂) suggest that, namely, it follows that, the rules are:

(P₃) If, then decide;

(B₃) If, then decide;

(N₃) If, then decide.

Proposition 3.2.3 In this case, we have the following simplified fuzzy probability regions:

In the fuzzy relation R, the fuzzy probability lower approximation and the fuzzy probability upper approximation of X are respectively:

According to decision-theoretic rough set, suppose the loss function satisfies, and , then we can get . Meanwhile, this paper also discusses the relationship between the value of and 1.

Case 1: When, the loss function must satisfies ;

Case 2: When, the loss function must satisfies ;

Case 3: When, the loss function must satisfies .

Set 10 investment objects, from the perspective of risk factors for their assessment, risk factors for 5 categories: market risk, technical risk, management risk, environmental risk and production risk. Table 2 is the risk assessment form of investment, among, A = {market risk, technology risk, management risk, environment risk, production risk}. For simplicity and without loss of generality, using said the market risk, technology risk, management risk, environment risk, production risk.

Any one of the IF numbers in Table 2 is among. indicates the degree of risk under the attribute. indicates the degree of insurance under the attribute.

On the basis of Table 2, the hypothesis is a fuzzy probability approximation space, including, R is a fuzzy relation, and the fuzzy relation on U as shown in Table 3. Now assume that the preference probability distribution on U is, , , , , , , , ,. Let denotes a decision class in which the classes are excellent. In the Bayesian decision process, some experts will provide values of the loss function for X, i.e.. It exhibits three cases in Table 4. Consider the loss function of Table 4, there are .

And the fuzzy conditional probabilities for every are computed as follows (by Equations. (7)):

, , ,

, , ,

, ,

,.

Case 1: When, namely, , it follows that,.

And

, ,

.

Based on these achievements, we can get the corresponding decision rules as follows:

(P₁) The investors most probably choose this scheme with a possibility not less than 0.54;

(B₁) The investors are less likely to invest in terms of the current conditions;

(N₁) We are not sure for who need further investigation.

Case 2: When, namely, , it follows that

,.

And

, ,

.

According to the calculation results, the decision rules in case 2 can present as follows:

(P₂) The investors most probably choose this scheme with a possibility not less than 0.5;

(B₂) The investors are less likely to invest with respect the given conditions and loss function;

(N₂) We are not sure for who need further investigation.

Case 3: When, namely, , it follows that

,.

And

, ,

.

Analogously, we can get the rest of the decision rules associate with these rough regions, as follows:

(P₃) The investors most probably choose this scheme with a possibility not less than 0.59;

(B₃) The investors are less likely to invest in terms of the given conditions and loss function;

(N₃) We are not sure for who need further investigation under the current conditions.

Definition 4.1.1 Let be an IF information system, , , , , then

are called the degree of membership similarity and the degree of nonmembership similarity of and.

The similarity degree of objects and under attribute set AT is as follows:

Through establishing analogical relations, we could turn IF information system into a IF approximation space in accordance with Definition 4.1. The subscript AT will be omitted in the rear. It holds:

1) Firstly, U is a non-empty classical set, a IF relation from U to U indicates a IF set. So is a IF relation on the universe U.

2) Furthermore, is a IF equivalence relation on U. The reasons are as follows:

• , , is reflexive;

• , , is symmetric;

• , , is transitive.

These three conditions are very obvious. Therefore, ordered pair is a IF approximation space.

Given the probability with its description, a IF probability approximation space is constructed in which U is a domain of discourse, is a IF equivalence relation on U, and is fuzzy probability of U.

Let be a IF information system and P be a probability measure on U. The decision-theoretic procedure in this section adopts two states and three actions. The row states is express an element is in X or not, respectively. The set of actions is by a interval-valued matrix shown in Table 5. The subscript X represents this loss function is for X, which is omitted in the following.

The expected losses of each action for object are as follows:

In Table 5, and are lower bound and upper bound of ., , and indicates the costs for taking actions of, , and,respectively, when an element is in X. Equally, , and denotes the losses for taking the same actions when an element belongs to. On the basis of conditions in Table 5, a particular kind of loss function is considered:

Likewise, is the description of x based on the IF relation, i.e. and it is assumed that.

Proposition 4.2.1 Note that and are two IF condition probabilities. X is a classical event,., and. If and, then

is called the conditional probability of X given, is called the conditional probability of X given.

In light of Bayesian decision procedure, the decision rules in Section 2 could be re-expressed as follows:

If and, deciding;

If and, deciding.

If the remainder elements x’s satisfying neither nor, we decide;

Definition 4.2.1 Let be an IF probabilistic approximation space. The loss function is the interval value. The are defined as follows:

In the IF relation, the [λ]-IF probability upper approximation and the [λ]-IF probability upper approximation are respectively:

is called the [λ]-IF probability rough set of X.

The decision rules are the three-way decisions, which have three regions:, and. These rules mainly rely on the comparisons among, and which are essentially computing the IF probabilities. Therefore, the conditions for calculating decision rules are as follows.

For the rule:

For the rule:

Therefore, in light of Bayesian decision procedure, the decision rules could be rewritten as follows:

If and , then decide;

If and , then decide.

If the remainder elements x’s satisfying neither nor, then decide;

Where

For any interval valued we define “” operations and “” relation as follows:

Proposition 4.2.2 For simplicity, it is denoted by, and. In this case, we have the following simplified IF probability region:

In the fuzzy relation R, the fuzzy probability upper approximation and the fuzzy probability of X are respectively:

is called the -IF probability rough set of X.

Under the discussions in Proposition 4.2.2, the additional conditions of decision rule suggest that, namely, it follows that, the rules are:

If, then decide;

If, then decide;

If the remainder elements x’s satisfying neither nor, then decide;

Proposition 4.2.3 In this case, we have the following simplified IF probability region:

In the IF relation, the IF probability lower approximation and the IF probability upper approximation of X are respectively:

is called the -IF probability rough set of X.

According to decision-theoretic rough set, suppose the loss function satisfies, and , then we can get . Meanwhile, this paper also discusses the relationship between the value of and 1.

Case 1: When, the loss function must satisfies ;

Case 2: When, the loss function must satisfies ;

Case 3: When, the loss function must satisfies .

Now continue to use case 3.3 as the research object, and make the rough set theory of decision making under the IF probability approximation space. On the basis of Table 2, the hypothesis is a IF probability approximation space, including, is a IF relation, as shown in Table 6. Now assume that the preference probability distribution on U is, , , , , , , , ,. Let denotes a decision class in which the classes are excellent. In the Bayesian decision process, some experts will provide values of the loss function for X, i.e.. It exhibits three cases in Table 7. Consider the loss function of Table 7, there are;; .

And the IF conditional probabilities for every are computed as follows (by Equations. (12)):

, ,

, ,

, ,

, ,

,.

Case 1: When, namely, , it follows that

,.

and

, ,

.

Based on these achievements, we can get the corresponding decision rules as follows:

The investors most probably choose this scheme.

The investors are less likely to invest.

We are not sure for who need further investigation.

Case 2: When, namely, , it follows that

,.

and

, ,

.

According to the calculation results, the decision rules in case 2 can present as follows:

The investors most probably choose this scheme;

The are less likely to invest.

We are not sure for who need further investigation.

Case 3: When, namely, , it follows that

,.

and

, ,

.

Analogously, we can get the rest of the decision rules associate with these rough regions, as follows:

The investors most probably choose this scheme;

The are less likely to invest;

We are not sure for who need further investigation.