Definition 1. (Atanassov 1986) ⁴

Let U be a nonempty objects set. An IFS A has the form A = {〈µ_A(x), ν_A(x)〉 : x ∈ U}, where the mappings µ_A : U → [0,1] and ν_A : U → [0,1] denote the degree of membership and nonmembership of each element x ∈ U to the set A (namely µ_A(x) and ν_A(x)) respectively, and 0 ≤ µ_A(x) + ν_A(x) ≤ 1 for each x ∈ U. 1 − µ_A(x) − ν_A(x) can be interpreted as the hesitancy degree of x to A.

The family of all IFS subsets of U is denoted by IFS(U). We call A(x) = 〈µ_A(x), ν_A(x)〉 an intuitionistic fuzzy value(IFV). Especially, for any A ∈ P(U), if x ∈ A, then A(x) = 〈1,0〉, if x ∉ A, then A(x) = 〈0,1〉. Let Bel_A(x) = µ_A(x), Pl_A(x) = 1 − ν_A(x), then the intuitionistic fuzzy set A can also be written as A = {BI_A(x) : x ∈ U}, in which BI_A(x) = [Bel_A(x),Pl_A(x)] expresses the belief interval of x to A.

Distance measure is an important uncertainty measurements in IFS theory.

Definition 2.

Let U be a finite and nonempty set, A,B ∈ IFS(U), the distance between A and B is defined as D(A,B)=1|U|∑x∈U[|μA(x)−μB(x)|+|νA(x)−νB(x)|4+max{|μA(x)−μB(x)|,|νA(x)−νB(x)|}2].

The intuitionistic fuzzy binary relation is an intuitionistic subset on U × U. IR is defined as IR = {〈µ_IR(x,y), ν_IR(x,y)〉 : (x,y) ∈ U × U}, where µ_IR : U × U → [0,1], ν_IR : U × U → [0,1], and satisfied the condition 0 ≤ µ_IR(x,y) + ν_IR(x,y) ≤ 1, ∀(x,y) ∈ U × U. We denote all intuitionistic fuzzy binary relation sets on U as IFR(U × U). For IR ∈ IFR(U × U), if IR(x,x) = 〈1,0〉 is satisfied for all x ∈ U, then IR is reflexive; if IR(x,y) = IR(x,y) is satisfied for all (x,y) ∈ U × U, then IR is symmetric.

Definition 3. (Wu & Zhou 2008) ²⁸

Let U be a finite and nonempty set, IR is an intuitionistic fuzzy binary relation on U × U. The ordered pair (U,IR) is called intuitionistic fuzzy approximation space.

Definition 4. (Shafer 1976) ²⁹

A set function m : 2^Θ → [0,1] is referred to as a basic probability assignment or mass distribution, if it satisfies the following axioms:

1. (1)

   m(∅) = 0

2. (2)

   ∑_A⊆Θm(A) = 1.

If m(A) ≠ 0, then subset A is called the focal element of m. The value of m(A) represents the degree of belief that a specific element of Θ belongs to A. All focal elements of m is denoted as M, the pair (M,m) is called a belief structure on Θ.

Definition 5. (Shafer 1976) ²⁹

Let m : 2^Θ → [0,1] be a basic belief assignment function. ∀X ⊆ Θ, the belief function (Bel : m → [0,1]) and plausibility function (Pl : m → [0,1]) of X are respectively defined as:

The belief interval of X is defined as Bl(X) = [Bel(X),Pl(X)].

∀X,Y ⊆ Θ, The belief intervals of X,Y are Bl(X) = [Bel(X),Pl(X)] and Bl(Y) = [Bel(Y),Pl(Y)] respectively. If Bel(X) > Bel(Y), then Bl(X) > Bl(Y), denoted as X > Y. If Bel(X) = Bel(Y) and Pl(X) > Pl(Y), then Bl(X) > Bl(Y), denoted as X > Y. If Bel(X) = Bel(Y) and Pl(X) = Pl(Y), then Bl(X) = Bl(Y), denoted as X = Y.

Definition 6. (Shafer 1976) ²⁹

Let Bel₁,Bel₂, ⋯ ,Bel_n be belief functions of Θ. Then combined evidence can be calculated by orthogonal sum m = m₁ ⊕ m₂ ⊕ ⋯ ⊕ m_n for fusing independent information sources m_i. The orthogonal sum is associative and commutative; it is defined in Dempster’ s rule of combination:

Definition 7. (Eugenia, Medina & Ramirez 2013)³⁰

Let (P₁, ≤₁), (P₂, ≤₂), (P₃, ≤₃) be three posets. The pair (&, ↙,↖) is called an adjoint triple with respect to P₁,P₂,P₃, if the three mappings & : P₁ × P₂ → P₃, ↙: P₃ × P₂ → P₁, ↖: P₃ × P₁ → P₂ satisfy,

• •

  x ≤₁ z ↙ y iff x&y ≤₃ z iff y ≤₂ z ↖ x, for any x ∈ P₁,y ∈ P₂,z ∈ P₃;

• •

  & is order-preserving on both arguments;

• •

  ↙,↖ are order-preserving on the first argumentand order-reserving on the second argument.

There are some special examples of adjoint triples:

• Gödel adjoint triple:

  x&Gy=min{x,y},z↖Gx={1,if x≤zz,otherwise;

• Product adjoint triple:

  x&Py=x·y,z↖Gx=min{1,z/x};

• Łukasiewicz adjoint triple:

  x&Ly=max{0,x+y−1},z↖Lx=min{1,1−x+z}.

The above adjoint pairs formed by a t-norm and its residuated implications, which can be seen as degenerate examples of general adjoint triples, and in this three adjoint pairs ↖_G=↙_G, ↖_P=↙_P, ↖_L=↙_L, since &_G, &_P, &_L are commutative.

Definition 8.

Given an intuitionistic fuzzy approximation space (U,IR), ∀A ∈ IFS(U), the lower and the upper approximations of A are defined respectively as

Definition 9.

For all A ∈ IFS(U), the pair (IR¯(A),IR¯(A)) is called a multi-adjoint intuitionistic fuzzy rough set of A with respect to IR.

An important feature of the presented framework is that the user may represent explicit preferences among the objects and acquire different upper and lower approximations in an intuitionistic fuzzy decision system, by associating different family adjoint triples. In next section, the probability of approximations will be calculated to construct the belief interval which will be used in the decision system.

Definition 10. (Feng, Zhang & Mi) ³¹

Let U be a nonempty finite set, (U,P(U),P) is a probability space. ∀A ∈ IFS(U) the probability measure P^* of A is defined as

It is obviously that P^*(∅) = 0P^*(U) = 1 .

If (U,P(U),P) is a probability space, IR ∈ IFR(U × U) is a reflexive and symmetric intuitionistic fuzzy binary relation, (U,IR) is an intuitionistic fuzzy approximate space mentioned in Definition 3. Let IR(x,y) = IR_S(x)(y), IR_S(x) is an intuitionistic fuzzy set on U.

Definition 11. (Feng, Zhang & Mi) ³¹

Let U be a nonempty finite set, a set function m : IFS(U) → [0,1] is referred to as a basic probability assignment (also called mass function) if it satisfies axioms (M1) and (M2)

• (M1) m(∅) = 0;

• (M2) ∑_A∈IFS(U) m(A) = 1.

For A ∈ IFS(U) with m(A) ≠ 0 is referred to as a focal element of m. We denote all focal elements of m by M. The pair (M,m) is called an IF belief structure.

Theorem 1. (Feng, Zhang & Mi) ³¹

Let U = {x₁,x₂, ⋯ ,x_n} be a nonempty and finite universe of discourse, (U,IR) a reflective and symmetric intuitionistic fuzzy approximation space. ∀A ∈ IFS(U)we define

Then m_IR is a mass function.

The probability measures of IR¯(A) and IR¯(A) have the following expressions.

Definition 12.

Suppose U be a nonempty and finite universe of discourse, (U,IR) a reflective and symmetric intuitionistic fuzzy approximation space. P is a probability measure on U, ∀A ∈ IFS(U),

Therefore, we have the following results.

Theorem 2.

Suppose U be a nonempty and finite universe of discourse, (U,IR) a reflective and symmetric intuitionistic fuzzy approximation space. P is a probability measure on U, ∀A ∈ IFS(U),

Proof.

 BelIR(A)=P*(IR¯(A))=∑x∈UP(x)((1−νIR¯(A)(x))2−(1−μIR¯(A)(x)−νIR¯(A)(x))2)=∑RS(x)∈G(∑y:RS(y)=RS(x)P(1y))((1−νIR¯(A)(x))2−(1−μIR¯(A)(x)−νIR¯(A)(x))2)=∑X∈GmIR(X)((1−sup{(1−νX(y))&xyνA(y):y∈U})2−(1−inf{μA(y)↖xyμX(y):y∈U}−sup{(1−νX(y))&xyνA(y):y∈U})2)=∑X∈GmIR(X)((1−νIR¯(X⊆A)(x))2−(1−μIR¯(X⊆A)(x)−νIR¯(X⊆A)(x))2)..

Similarly, we get PlIR(A)=P*(IR¯(A))=∑x∈UP(x)((1−νIR¯(A)(x))2−(1−μIR¯(A)(x)−νIR¯(A)(x))2)=∑RS(x)∈G(∑y:RS(y)=RS(x)P(1y))((1−νIR¯(A)(x))2−(1−μIR¯(A)(x)−νIR¯(A)(x))2)=∑X∈GmIR(X)((1−inf{νA(y)↖xy(1−νX(y)):y∈U})2−(1−sup{μX(y)&xyμA(y):y∈U}−inf{νA(y)↖xy(1−νX(y)):y∈U})2)=∑X∈GmIR(X)((1−νIR¯(X∩A)(x))2−(1−μIR¯(X∩A)(x)−νIR¯(X∩A)(x))2)..

Theorem 3.

Suppose U be a nonempty and finite universe of discourse, IR ∈ IFR(U × U) a reflective and symmetric intuitionistic fuzzy binary relation, the following properties hold:

1. 1.

   Bel_IR(∅) = Pl_IR(∅) = 0, Bel_IR(U) = Pl_IR(U) = 1;

2. 2.

   Bel_IR(A) ≤ P^*(A) ≤ Pl_IR(A),∀A ∈ IFS(U);

3. 3.

   Bel_IR(A) ≤ Bel_IR(B),Pl_IR(A) ≤ Pl_IR(B), ∀A,B ∈ IFS(U),A ⊆ B.

Proof.

1. (1)

   According to Definition 8 and 10, we have Bel_IR(∅) = Pl_IR(∅) = P^*(∅) = 0, Bel_IR(U) = Pl_IR(U) = P^*(U) = 0;

2. (2)

   BelIR(A)=P*(IR¯(A))≤P*(A)≤P*(IR¯(A))=PlIR(A)

3. (3)

   ∀A,B ∈ IFS(U), if A ⊆ B, then BelIR(A)=P*(IR¯(A))≤P*(IR¯(B))=BelIR(B). Similarly, we have Pl_IR(A) ≤ Pl_IR(B).

It is easy to verify that Bel_IR and Pl_IR degenerate into the crisp belief and plausibility functions when the belief structure (M,m) and A are crisp. Thus Bel_IR and Pl_IR are measures of belief and plausibility functions on IFS(U). Convert belief function and plausibility function into Mass functions, we have:

Assume

The sources of information in decision-making system are mostly reliable. That means the intuitionistic fuzzy relation matrices given by most experts are reliable. Otherwise, the system has lapsed and the group decision-making information integration is meaningless.

Let yes express supportive of the expert M_k to a decision object; No indicate appositive of expert M_k to a decision objects; (yes, no) mean cannot judge (or hesitate).

• Step 1 From Definition 8, the lower and upper approximations according to expert are obtained. They are Dk¯(ol) and Dk¯(ol), which indicate the degrees of certainty and probability of the expert M_k with regard to decision object o_l on each attribute. The calculation is as follows

  μ¯ik(ol)=inf{μj(ol)↖ijμijk:j≤m};ν¯ik(ol)=sup{(1−νijk)&ijνj(ol):j≤m};μ¯ik(ol)=sup{μijk&ijμj(ol):j≤m};ν¯ik(ol)=inf{vj(ol)↖ij(1−νijk):j≤m}.

• Step 2 From Definition 12, the belief function BelDk(ol)=P*(Dk¯(ol)) and plausibility function PlDk(ol)=P*(Dk¯(ol)) of the expert M_k with regard to decision object o_l are calculated. The weight of a_i in D^k can be used as the probability of the attribute a_i under expert M_k. The calculation is as follows

  Let D^k be the intuitionistic fuzzy relation matrix given by expert M_k. μijk, νijk denote the degree to which the expert k considers the attribute i to be relevant and irrelevant with the attribute j. Then the function sijk=μijk−νijk is denoted as the relevant score of the attribute i to the attribute j by expert k.

  The weight of a_i under expert M_k is

  (6)Pk(ai)=∑j=1msij∑i=1m(∑j=1msij).

  P^k(a_i) reflects the degree to which attribute a_i is recognized by expert M_k. The higher the correlation with other attributes, the more important of attribute a_i, and the bigger weight of a_i. Therefore, the weights of attributes based on correlation are reasonable.

• Step 3 Convert the belief function and the plausibility function to the Mass function. Let

  mlk(yes)=BelDk(ol),mlk(no)=1−PlDk(ol),mlk(yes,no)=PlDk(ol)−BelDk(ol).

• Step 4 Due to the interference of different factors, some experts give evidence that may be larger in conflict with other experts, and the rule of evidence combination does not recognize strong evidence nor the weak evidence. To get a reasonable and reliable evidence combination results, it is necessary to analyze the conflict of evidence before combination. We use distance measure to modify the evidence, the bigger the distance, the smaller the degree of support and the bigger the discount factor. According to definition 2, the distance between any two evidences can be calculated. Let Δ = (ϑ_ij)_{K × K} be the distance matrix of objects, discount factor of evidences is defined as:

  (7)ωk=1−∑k=1Kϑij1−min{∑k=1Kϑij:i=1,2⋯K}.

  The revised evidence indicates that:

  ml*k(yes)=ωkmlk(yes),ml*k(no)=ωkmlk(no),ml*k(yes,no)=1−ml*k(yes)−ml*k(no).

• Step 5 Combine all experts evidence for each decision object. The calculation is as follow

  (8)ml(A)=∑A1∩A2∩⋯∩AK≠∅ml*1(A1)ml*2(A2)⋯ml*K(AK)1−N.
  in which N=∑A1∩A2∩⋯∩AK≠∅ml*1(A1)ml*2(A2)⋯ml*K(AK), and A,A₁,A₂, ⋯ ,A_K ∈ {yes,no,(yes,no)}.

• Step 6 Compares the belief interval of each decision object and makes the optimal decision.

Example 1

Suppose O = {o₁,o₂,o₃,o₄,o₅,o₆} represent six alternative houses for sale, AT = {a₁,a₂,a₃,a₄} be a set of attributes such as the location of the house, the quality of the property service, the building structure and the average price. The intuitionistic fuzzy sets of the six houses according to AT are shown in Table 2.

P = {M₁,M₂,M₃,M₄} is a advisory group consists of four property consultants. The four members evaluate the attributes with their personal experience and information. Their evaluate intuitionistic fuzzy relation matrix about attribute set AT are shown in Table 3. Decision maker should rank houses based on individual preferences and specialists’ opinions. And the MD process by means of the proposed method is described as follows.

• Step 1 Let τ : a_i × a_j → product. According to Definition 8, the the lower and upper approximations of alternatives w.r.t experts are obtained.

• Step 2 According to formulas (6)we can easily get the weight of a_i under expert M^k, as is shown in Table 4. According to the formulas (3) to (5) the belief function and plausibility function of each house in the intuitionistic fuzzy relation matrix given by the specialists are calculated.

  Experts	Pk(a1)	Pk(a2)	Pk(a3)	Pk(a4)
  M1	0.33	0.23	0.28	0.16
  M2	0.26	0.26	0.25	0.23
  M3	0.21	0.27	0.25	0.28
  M4	0.28	0.29	0.26	0.17

  Table 4.

  The weight of attributes.

• Step 3 Translate them into mass functionsas shown in Table 5.

  Experts	o1	o2	o3	o4	o5	o6
  M1	〈0.65,0.33,0.02〉	〈0.59,0.41,0.00〉	〈0.62,0.28,0.11〉	〈0.61,0.37,0.02〉	〈0.60,0.36,0.04〉	〈0.64,0.27,0.09〉
  M2	〈0.61,0.29,0.11〉	〈0.57,0.36,0.07〉	〈0.58,0.24,0.18〉	〈0.57,0.31,0.11〉	〈0.57,0.34,0.09〉	〈0.58,0.24,0.18〉
  M3	〈0.61,0.28,0.10〉	〈0.57,0.37,0.06〉	〈0.59,0.28,0.12〉	〈0.57,0.33,0.10〉	〈0.61,0.35,0.04〉	〈0.59,0.28,0.13〉
  M4	〈0.63,0.32,0.05〉	〈0.59,0.39,0.02〉	〈0.58,0.25,0.17〉	〈0.59,0.33,0.08〉	〈0.57,0.35,0.08〉	〈0.59,0.25,0.16〉

  Table 5.

  The mass function of each house in Example 1.

• Step 4–5 Revise the evidence above. Combine all the experts’ evidence, we get the belief interval for each house:

  BI(o1)=[0.947,0.948],BI(o2)=[0.846,0.847],BI(o3)=[0.958,0.963],BI(o4)=[0.904,0.905],BI(o5)=[0.889,0.889],BI(o6)=[0.961,0.956].

• Step 6 Compare the belief interval of each house, we know that o₆ > o₃ > o₁ > o₄ > o₅ > o₂. Therefore, the consumer should buy the house o₆.

Example 2

In the example above, we only used Product adjoint pair to calculate the intuitionistic fuzzy upper and lower approximations. Assumes that the main consideration of the consumer is good education resources and community environment when buying a house. The consumer insists that the location of the house, the quality of the property service are more important than other factors. Thus, let

Because the Product adjoint triple implication results in lower values and has more influence on the infimum in the lower approximation. According to the formulas (1) to (6), the belief function and plausibility function of each house in the intuitionistic fuzzy relation matrix given by the specialists are calculated. Translate them into mass functions, as shown in Table 6.

Calculate the discount factor of each specialist and revise the evidence above. Then using the same method to combine all the experts’ evidence, we get the belief interval for each house:

Comparing the belief interval of each house, we know that o₆ > o₁ > o₃ > o₅ > o₄ > o₂. Therefore, the consumer should buy the house o₆.

If the consumer puts the dwelling Environment quality in the first place, the quality of the property service and the building structure will be considered more than other factors. Thus, let

Similarly, we get the following results o₆ > o₃ > o₁ > o₄ > o₂ > o₅. Therefore, the consumer should buy the house o₆.

Example 3

We use five different pairs of level value to rank the objects. For example, the L(0.70,0.30)-level, the L(mid,mid)-level, the L(top,bot)-level, the L(top,top)-level, the L(bot,bot)-level. Compute the level soft set L(s,t), as is shown in Table 7.