1. INTRODUCTION
In order to deal with fuzzy information in practical applications, Zadeh established the fuzzy set theory [1]. Information aggregation is a common activity in real life, and many fuzzy aggregation operators have been proposed, among which, the weighted averaging (WA) operator [2], the ordered weighted averaging (OWA) operator [3], the weighted geometric averaging (WGA) operator [3], and the ordered weighted geometric averaging (OWGA) operator [3] are the most familiar ones for aggregating information. Recently, adhibition of bipolar scales has been widely used in decision-making [4,5]. There are psychological evidence that in many cases, scores given by humans lie on a bipolar scales, that is to say a scale with a neutral value serving as frontier between good scores and bad scores. In general, our behavior with the good scores are different from the bad sores, so it becomes important to define aggregation operators, which can reflect the diversity of aggregation behavior on bipolar scales. This situation is already considered for uninorm [6], with identity element e serving as frontier. Based on uninorm, a class of very flexible aggregation operators are constructed [5]. This type of information aggregation allows us to command the aggregation process based on the allocation of parameters. But, until now information of bipolar scales is based on fuzzy set introduced by Zadeh [1], which only provides the membership degree of each element in the universe, and the nonmembership degree is 1 minus the membership degree, it is not enough to depict the ambiguity and simulate the life scene.
Although fuzzy set has been widely used, it still cannot handle uncertain information well. As a generalization of fuzzy set, intuitionistic fuzzy set was introduced [7], which is composed of membership degree and nonmembership degree. Intuitionistic fuzzy set can describe not only the fuzziness of objects, but also the uncertainty. It has been shown that intuitionistic fuzzy set is L∗-fuzzy set w.r.t. the lattice (L∗,⩽L∗) [8]. Then, some intuitionsitic fuzzy aggregation operators were developed. The intuitionsitic fuzzy weighted averaging (IFWA) operator, the intuitionsitic fuzzy ordered weighted averaging (IFOWA) operator, and some intuitionsitic fuzzy aggregation hybrid averaging (IFHA) operator were proposed [9]. Based on the algebraic sum, algebraic product, and operational laws on intuitionistic fuzzy sets, the intuitionistic fuzzy weight geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid (IFGH) operator were presented by Xu and Yager [10]. Algebraic product and algebraic sum, as the basic operations of intuitionistic fuzzy sets, are not the only operations to establish the intersection and union model of intuitionistic fuzzy sets. Some algebraic operators such as the Einstein sum operator, the Einstein product operator and the Einstein scalar multiplication operator were introduced by Wang and Liu [11]. In line with these algebraic operators, intuitionistic fuzzy Einstein weighted geometric operator (IFWGε) and the intuitionistic fuzzy Einstein ordered weighted geometric (IFOWGε) operator were proposed. As those aggregation operators [10,11] have the drawback that if there is only one membership degree of intuitionistic fuzzy sets equal to zero, the membership degree of the aggregation result of n intuitionistic fuzzy sets is zero even if the membership degree of n−1 intuitionistic fuzzy sets are not zero. To conquer the defects of aggregation operations, the intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator, the intuitionistic fuzzy ordered weighted geometric interaction averaging (IFOWGIA) operator, and the intuitionistic fuzzy hybrid geometric interaction averaging (IFHGIA) operator were proposed by He et al. [12]. These aggregation operations may result in unconscionable decision results in some cases. Chen and Chang [13] proposed the intuitionistic fuzzy weighted geometric averaging (IFWGA) operator, the intuitionistic fuzzy ordered weighted geometric averaging (IFOWGA) operator, and the intuitionistic fuzzy hybrid geometric averaging (IFHGA) operator. In practical applications, some existing methods aggregate information directly rather than classifying the information into different classes, which results in the incorrect order of preference for alternatives in some cases. For example, A and B be two sets of intuitionistic fuzzy values, and ω1=0.12, ω2=0.13, ω3=0.18, ω4=0.10, ω5=0.17, ω6=0.18, ω7=0.12. Information are shown in the following table.
|
A |
B |
|
(0.3000,0.6000) |
(0.8000,0.2000) |
|
(1.0000,0.0000) |
(0.7000,0.2000) |
|
(0.2000,0.6000) |
(0.7000,0.2000) |
|
(0.1000,0.7000) |
(0.8000,0.1000) |
|
(0.3000,0.7000) |
(0.9000,0.0000) |
|
(0.1000,0.9000) |
(0.1000,0.8000) |
|
(0.0000,0.8000) |
(0.6000,0.3000) |
| IFWGA |
(1.0000,0.0000) |
(0.7009,0.2991) |
| IFOWGA |
(1.0000,0.0000) |
(0.7889,0.2111) |
| IFHGA |
(1.0000,0.0000) |
(0.7175,0.2825) |
It can be seen from the table that the intuitionistic fuzzy values in A are smaller than those in B, the preference order should be B≻A, but in fact the preference order is A≻B according to three aggregation operations, this is counterintuitive. Most of the time, our behavior with the good items are not the same with the bad items, categorizing information first, and then aggregating them. As uninorms are introduced to model bipolar behavior, in this paper, we will construct aggregation operator using uninorms in L∗-fuzzy set theory.
The rest of this paper consists of the following. Part 2 reviews some basic definitions. Some aggregation operators are constructed based on the uninorm in L∗-fuzzy set theory in part 3. In part 4, we develop the decision-making approach based on the proposed aggregation operator, and apply it to decision experiments. Conclusions are stated in part 5.
2. PRELIMINARIES
In this part, we retrospect some concepts of intuitionistic fuzzy set, L∗-fuzzy set, score function, uninorms, and aggregation operator.
Definition 1.
[7] An object of the form A={(x,μA(x),νA(x))|x∈X} on a universe X is called intuitionistic fuzzy set, where μA(x)(∈[0,1]) is called the degree of membership of x in A, νA(x)(∈[0,1]) is called the degree of nonmembership of x in A, and μA(x),νA(x)∈[0,1] satisfy the following condition: 0⩽μA(x)+νA(x)⩽1(∀x∈X).
Consider the set L∗={(x1,x2)|(x1,x2)∈[0,1]2 and x1+x2⩽1}, (x1,x2)⩽L∗(y1,y2)⇔x1⩽y1 and x2⩾y2, ∀(x1,x2),(y1,y2)∈L∗. (L∗,⩽L∗) is a complete lattice, and 0L∗=(0,1), 1L∗=(1,0). Note that, for x=(x1,x2), y=(y1,y2)∈L∗, if x1<y1 and x2<y2, or x1>y1 and x2>y2, then x and y are incomparable w.r.t. ⩽L∗, denote as x||L∗y [8].
Deschrijiver and Kerre introduced uninorm in L∗-fuzzy set theory [14], which are generalizations of t-norm and t-conorm in L∗-fuzzy set theory.
Definition 2.
[14] (L∗,⩽L∗) is called a complete lattice, where L∗={(x1,x2)|(x1,x2)∈[0,1]2 and x1+x2⩽1}. A uninorm U on L∗ is an increasing, associative and communicative (L∗)2→L∗ mapping that satisfies (∃(e1,e2)∈L∗)(∀(x,y)∈L∗)(U((e1,e2),(x,y))=(x,y)).
The element (e1,e2) corresponding to a uninorm U is clearly unique and is called the identity element of U. If (e1,e2)=0L∗, then we obtain a t-conorm U on L∗, while in case (e1,e2)=1L∗, we obtain a t-norm T on L∗.
The set of all positive integers is denoted by ℕ∗. In fuzzy set theory, aggregation operators are defined as follows.
Definition 3.
[15] A mapping A:⋃n∈ℕ∗[0,1]n→[0,1] is called an aggregation operator A on [0,1] if the following properties are met:
A(x)=x, for all x∈[0,1];
If xi⩽yi for all i∈{1,2,⋯,n}, then A(x1,x2,⋯,xn)⩽A(y1,y2,⋯,yn), for all n∈ℕ∗ and for all (x1,x2,⋯,xn),(y1,y2,⋯,yn)∈[0,1]n;
A(0,0,⋯,0︸ntimes)=0 for all n∈ℕ∗;
A(1,1,⋯,1︸ntimes)=1 for all n∈ℕ∗.
In [16], this definition was extended to L∗-fuzzy set theory. Let (L∗,⩽L∗) be a complete lattice, where L∗={(x1,x2)|(x1,x2)∈[0,1]2 and x1+x2⩽1}.
Definition 4.
[16] A mapping A:⋃n∈ℕ∗(L∗)n→L∗ is called an aggregation operator A on L∗ the following properties are met:
A((x,y))=(x,y), for all (x,y)∈L∗;
If (xi,yi)⩽L∗(xi′,yi′) for all i∈{1,2,⋯,n}, then A((x1,y1),(x2,y2),⋯,(xn,yn))⩽L∗A((x1′,y1′),(x2′,y2′),⋯,(xn′,yn′)), for all n∈ℕ∗ and for all (x1,y1),⋯,(xn,yn),(x1′,y1′),⋯,(xn′,yn′)∈L∗;
A((0,1),(0,1),⋯,(0,1)︸ntimes)=(0,1) for all n∈ℕ∗;
A((1,0),(1,0),⋯,(1,0)︸ntimes)=(1,0) for all n∈ℕ∗.
Definition 5.
[17] Let α̃=(a,b) be an intuitionistic fuzzy number, the score function is defined by:
S(α̃)=a−b2(1)
Next, we will recall some intuitionistic fuzzy aggregation operators.
Definition 6.
(IFWG operator) [10] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. The IFWG operator is defined as follow:
IFWG(ã1,ã2,⋯,ãn)=∏j=1nμãjωj,∏j=1n(1−νãj)ωi(2)
where
ωj is the weight of
ãj,
ωj∈[0,1],
1⩽j⩽n and
∑j=1nωj=1.
Definition 7.
(IFOWG operator) [10] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the ordered weighted geometric operator, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. The IFOWG operator is defined as follow:
IFOWG(ã1,ã2,⋯,ãn)=∏j=1nμãσ(j)ωj,∏j=1n(1−νãσ(j))ωj(3)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
ãσ(j) is the
jth largest intuitionistic fuzzy value among
ã1,ã2,⋯, and
ãn, where
1⩽j⩽n.
Definition 8.
(IFHG operator) [10] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj∈[0,1] and 1⩽j⩽n. Let ω1,ω2,⋯, and ωn be the weights of the intuitionistic fuzzy values ã1,ã2,⋯, and ãn, respectively, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the hybrid geometric, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. Let āj=(ã)nwj, where n is the balancing coefficient, the intuitionistic fuzzy hybrid geometric (IFHG) operator is defined as follow:
IFHG(ã1,ã2,⋯,ãn)=∏j=1nμāσ(j)ωj,∏j=1n(1−νāσ(j))ωj(4)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
āσ(j) is the
jth largest intuitionistic fuzzy value among
ā1,ā2,⋯, and
ān, where
1⩽j⩽n.
Definition 9.
(IFWGε operator) [11] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. The IFWGε operator is defined as follow:
IFWGε(ã1,ã2,⋯,ãn)=2∏j=1nμãjωj∏j=1n(2−μãj)ωj+∏j=1nμãjωj,∏j=1n(1+νãj)ωi−∏j=1n(1−νãj)ωj∏j=1n(1+νãj)ωj+∏j=1n(1−νãj)ωj(5)
where
ωj is the weight of
ãj,
ωj∈[0,1],
1⩽j⩽n and
∑j=1nωj=1.
Definition 10.
(IFOWGε operator) [11] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the Einstein ordered weighted geometric operator, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. The IFOWGε operator is defined as follow:
IFOWGε(ã1,ã2,⋯,ãn)=2∏j=1nμãσ(j)ωj∏j=1n(2−μãσ(j))ωj+∏j=1nμãσ(j)ωj,∏j=1n(1+νãσ(j))ωj−∏j=1n(1−νãσ(j))ωj∏j=1n(1+νãσ(j))ωj+∏j=1n(1−νãσ(j))ωj(6)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
ãσ(j) is the
jth largest intuitionistic fuzzy value among
ã1,ã2,⋯, and
ãn, where
1⩽j⩽n.
Definition 11.
(IFWGIA operator) [12] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. The IFWGIA operator is defined as follow:
IFWGIA(ã1,ã2,⋯,ãn)=∏j=1n(1−νãj)ωj−∏j=1n(1−(μãj+νãj))ωj,1−∏j=1n(1−νãj)ωj(7)
where
ωj is the weight of
ãj,
ωj∈[0,1],
1⩽j⩽n and
∑j=1nωj=1.
Definition 12.
(IFOWGIA operator) [12] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the ordered weighted geometric interaction averaging operator, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. The IFOWGIA operator is defined as follow:
IFOWGIA(ã1,ã2,⋯,ãn)=∏j=1n(1−νãσ(j))ωj−∏j=1n(1−(μãσ(j)+νãσ(j)))ωj,1−∏j=1n(1−νãσ(j))ωj(8)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
ãσ(j) is the
jth largest intuitionistic fuzzy value among
ã1,ã2,⋯, and
ãn, where
1⩽j⩽n.
Definition 13.
(IFHGIA operator) [12] Let ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω1,ω2,⋯, and ωn be the weights of the intuitionistic fuzzy values ã1,ã2,⋯, and ãn, respectively, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the hybrid geometric interaction averaging operator, where ωj∈[0,1], 1⩽j⩽n and ∑j=1nωj=1. Let āj=(ã)nwj, where n is the balancing coefficient, the IFHGIA operator is defined as follow:
IFHGIA(ã1,ã2,⋯,ãn)=∏j=1n(1−νāσ(j))ωj−∏j=1n(1−(μāσ(j)+νāσ(j)))ωj,1−∏j=1n(1−νāσ(j))ωj(9)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
āσ(j) is the
jth largest intuitionistic fuzzy value among
ā1,ā2,⋯, and
ān, where
1⩽j⩽n.
Definition 14.
(IFWGA operator) [13] Let A={ã1,ã2,⋯,ãn}, ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω1,ω2,⋯, and ωn be the weights of the intuitionistic fuzzy values ã1,ã2,⋯, and ãn, respectively, where ωj∈[0,1], 1⩽j⩽n and Σj=1nωj=1. The IFWGA operator is as follow:
IFWGA(ã1,ã2,⋯,ãn)=1−∏j=1n(1−μãiwj, ∏j=1n(1−μãj)wj−∏ji=1n(1−μãj−νãj)wj)(10)
Definition 15.
(IFOWGA operator) [13] Let A={ã1,ã2,⋯,ãn}, ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the OWGA operator, where ωj∈[0,1], 1⩽j⩽n and Σj=1nωj=1. The IFOWGA operator is as follow:
IFOWGA(ã1,ã2,⋯,ãn)=1−∏j=1n(1−μãσ(j)ωj, ∏j=1n(1−μãσ(j))ωj−∏j=1n(1−μãσ(j)−νãσ(j))ωj)(11)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
ãσ(j) is the
jth largest intuitionistic fuzzy value among
ã1,ã2,⋯, and
ãn and
S(ãσ(j−1))>S(ãσ(j)), where
1⩽j⩽n.
Definition 16.
(IFHGA operator) [13] Let A={ã1,ã2,⋯,ãn}, ãj=(μãj,νãj), μãj∈[0,1], νãj∈[0,1], 0⩽μãj+νãj⩽1 and 1⩽j⩽n. Let ω1,ω2,⋯, and ωn be the weights ã1,ã2,⋯, and ãn, respectively, where ωj∈[0,1], 1⩽j⩽n and Σj=1nωj=1. Let ω={ω1,ω2,⋯,ωn}T be the weighting vector of the hybrid geometric averaging operator, where ωj∈[0,1], 1⩽j⩽n and Σj=1nωj=1. Let āj=(ãj)nwj, where n is the balancing coefficient. The IFHGA operator is as follow:
IFHGA(ã1,ã2,⋯,ãn)=1−∏j=1n(1−μāσ(j)ωj, ∏j=1n(1−μāσ(j))ωj−∏j=1n(1−μāσ(j)−νāσ(j))ωj)(12)
where “
σ(1),σ(2),⋯, and
σ(n)” is a permutation of “
1,2,⋯, and
n,” such that
āσ(j) is the
jth largest intuitionistic fuzzy value among
ā1,ā2,⋯, and
ān and
S(āσ(j−1))>S(āσ(j)), where
1⩽j⩽n.
3. AGGREGATION OPERATORS IN L*-FUZZY SETS
Let (L∗,⩽L∗) be a complete lattice, where L∗={(x1,x2)|(x1,x2)∈[0,1]2,x1+x2⩽1}, (x1,x2)⩽L∗(y1,y2) iff x1⩽y1 and x2⩾y2, for (x1,x2),(y1,y2)∈L∗. We know that uninorms where all arguments are below or above identity behave like t-norms and t-conorms. The proposed aggregation operation has neutral value hold the post of frontier between these four types of scores. As information be classified into different classes before aggregating is more intuitive, a neutral value serving as frontier used to classify information is needed.
Theorem 1.
Let (e,f)∈L∗ be a neutral value serving as frontier used to classify information. Arguments (x11,y11),(x12,y12),⋯,(x1k,y1k), (x21,y21),(x22,y21),⋯,(x2l,y2l),(x31,y31),(x32,y32),⋯,(x3m,y3m), (x41,y41),(x42,y42),⋯,(x4n,y4n)∈L∗ are ordered:
(x11,y11),(x12,y12),⋯,(x1k,y1k)⩽L∗(e,f);
(e,f)⩽L∗(x21,y21),(x22,y21),⋯,(x21,y21);
e<x31,x32,⋯,x3m, f⩽y31,y32,⋯,y3m;
x41,x42,⋯,x4n<e, y41,y42,⋯,y4n⩽f.
ℛG∗((x11,y11),⋯,(x1k,y1k),(x21,y21),⋯, (x2l,y2l),(x31,y31),⋯,(x3m,y3m), (x41,y41),⋯,(x4n,y4n)) =A(U1((x11,y11),⋯,(x1k,y1k)), U2((x21,y21),⋯,(x2l,y2l)), U3((x31,y31),⋯,(x3m,y3m)), U4((x41,y41),⋯,(x4n,y4n)) =A((X1,Y1),(X2,Y2),(X3,Y3),(X4,Y4)) =(A(X1,X2,X3,X4),A(Y1,Y2,Y3,Y4)),
where A(X1,X2,X3,X4)=kX1+lX2+mX3+nX4k+l+m+n,
A(Y1,Y2,Y3,Y4)=kY1+lY2+mY3+nY4k+l+m+n,
(X1,Y1)=U1((x11,y11),⋯,(x1k,y1k)),
U1 is an uninorm in L∗ with identity element (e1,f1),
and (e,f)⩽L∗(e1,f1).
(X2,Y2)=U2((x21,y21),⋯,(x2l,y2l)),
U2 is an uninorm in L∗ with identity element (e2,f2),
and (e2,f2)⩽L∗(e,f).
(X3,Y3)=U3((x31,y31),⋯,(x3m,y3m)),
U3 is an uninorm in L∗ with identity element (e3,f3),
and e3⩽e,f3⩽f.
(X4,Y4)=U4((x41,y41),⋯,(x4n,y4n)),
U4 is an uninorm in L∗ with identity element (e4,f4),
and e⩽e4,f⩽f4.
Then ℛ∗ is an aggregation operator.
Proof.
(1) It is obvious that for all (x,y)∈L∗, ℛ∗((x,y))=(x,y).
(2) Monotonicity. Suppose that (x11,y11)⩽L∗(x11′,y11′), ⋯, (x1k,y1k)⩽L∗(x1k′,y1k′), (x21,y21)⩽L∗(x21′,y21′), ⋯, (x2l,y2l)⩽L∗(x2l′,y2l′), (x31,y31)⩽L∗(x31′,y31′), ⋯, (x3m,y3m)⩽L∗(x3m′,y3m′), (x41,y41)⩽L∗(x41′,y41′), ⋯, (x4n,y4n)⩽L∗(x4n′,y4n′), according to the monotonicity of uninorms, we have U1((x11,y11),⋯,(x1k,y1k))⩽L∗U1((x11′,y11′),⋯,(x1k′,y1k′)), U2((x21,y21),⋯,(x2l,y2l))⩽L∗U2((x21′,y21′),⋯,(x2l′,y2l′)), U3((x31,y31),⋯,(x3m,y3m))⩽L∗U3((x31′,y31′),⋯,(x3m′,y3m′)), U4((x41,y41),⋯,(x4n,y4n))⩽L∗U4((x41′,y41′),⋯,(x4n′,y4n′)) then
kXk+lXl+mXm+nXnk+l+m+n⩽kXk′+lXl′+mXm′+nXn′k+l+m+n
kYk+lYl+mYm+nYnk+l+m+n⩾kYk′+lYl′+mYm′+nYn′k+l+m+n
it is obvious that
ℛ∗((x11,y11),⋯,(x1k,y1k),(x21,y21), ⋯,(x2l,y2l),(x31,y31),⋯,(x3m,y3m), (x41,y41),⋯,(x4n,y4n)) ⩽ℛ∗((x11′,y11′),⋯,(x1k′,y1k′),(x21′,y21′), ⋯,(x2l′,y2l′),(x31′,y31′),⋯,(x3m′,y3m′), (x41′,y41′),⋯,(x4n′,y4n′))
(3) ℛ∗((0,1),(0,1),⋯,(0,1)︸ntimes)=U((0,1),(0,1),⋯,(0,1)︸ntimes)=(0,1).
(4) ℛ∗((1,0),(1,0),⋯,(1,0)︸ntimes)=U((1,0),(1,0),⋯,(1,0)︸ntimes)=(1,0).
By Theorem 1, we have two special aggregation operators:
Corollary 2.
Take U1=U2=U3=U4=U with identity element (e,f), which is neutral value serving as frontier used to classify information, we have
ℛG∗(((x11,y11),⋯,(x1k,y1k),(x21,y21),⋯, (x2l,y2l),(x31,y31),⋯,(x3m,y3m), (x41,y41),⋯,(x4n,y4n)) =A(U((x11,y11),⋯,(x1k,y1k)),U((x21,y21), ⋯,(x2l,y2l))U((x31,y31),⋯,(x3m,y3m)), U((x41,y41),⋯,(x4n,y4n))), =A((X1,Y1),(X2,Y2),(X3,Y3),(X4,Y4)) =(A(X1,X2,X3,X4),A(Y1,Y2,Y3,Y4))
Corollary 3.
Obviously, t-norms and t-conorms are uninorms with identity elements are (1,0) and (0,1), respectively. Then take U1=T, U2=S, U3=U4=U with identity element (e,f), which is neutral value serving as frontier used to classify information, we have
ℛG∗((x11,y11),⋯,(x1k,y1k),(x21,y21),⋯, (x2l,y2l)(x31,y31),⋯,(x3m,y3m),(x41,y41), ⋯,(x4n,y4n)) =A(T((x11,y11),⋯,(x1k,y1k)),S((x21,y21), ⋯,(x2l,y2l)),U((x31,y31),⋯,(x3m,y3m)), U((x41,y41),⋯,(x4n,y4n)) =A((X1,Y1),(X2,Y2),(X3,Y3),(X4,Y4)) =(A(X1,X2,X3,X4),A(Y1,Y2,Y3,Y4))
4. APPLICATION FOR MULTI-ATTRIBUTES DECISION-MAKING
4.1. Multi-attributes Decision Method
Mathematically speaking, the multi-attributes decision-making problem of n alternatives with m attributes can be expressed as
c1c2⋯cm A1A2⋮An(x11,y11)(x21,y21)⋮(xn1,yn1)(x12,y12)(x22,y22)⋮(xn2,yn2)⋯⋯⋱⋯(x1m,y1m)(x2m,y2m)⋮(xnm,ynm)A1A2⋮An
where
U={A1,A2,⋯,An} is the set of alternatives,
C={c1,c2,⋯,
cm} is the set of attributes,
(xij,yij)(i=1,2,⋯,n,j=1,2,⋯,m) is the evaluation information of alternatives
Ai under
cj provided by experts.
The decision procedure is summarized as follow:
Step 1: For every Ai, 1⩽i⩽n, according to the neutral value serving as frontier used to classify information, (xi1,yi1),⋯,(xim,yim) are classified into four classes.
Step 2: For the four classes, aggregating them using different uninorms, respectively, then we can obtain (X1,Y1),(X2,Y2),(X3,Y3),(X4,Y4).
Step 3: By Theorem 1, aggregating (X1,Y1),(X2,Y2),(X3,Y3),(X4,Y4), then we can obtain the aggregation result ℛ∗(Ai)(1⩽i⩽n), respectively.
Step 4: Compute the score functions S(ℛ∗(Ai))(1⩽i⩽n) by Eq. (1), respectively.
Step 5: Rank these score functions S(ℛ∗(Ai))(1⩽i⩽n) and choose the best one.
4.2. Application for Multi-attributes Decision-Making
In this part, we compare the proposed method with other existing methods by some examples. Assume the neutral value serving as frontier used to classify information is (0.5,0.5), U1((x1,y1),(x2,y2))=(x1∧x2,y1∨y2), U2((x1,y1),(x2,y2))=(x1∨x2,y1∧y2), U3((x1,y1),(x2,y2))=x1x21−x1−x2+2x1x2,if{x1,x2}≠{0,1}0otherwise.,1∧(y1+y2), U4((x1,y1),(x2,y2))=(x1x2,0∨(y1+y2−1)), U5((x1,y1),(x2,y2))=(x1x2,y1∨y2), U6((x1,y1),(x2,y2))=(x1∨x2,y1y2).
Example 1.
[12] Let's say an investment company wants to invest some money in the most promising firm. Through market analysis, three possible choices A, B, and C will be considered, as shown below: A is a car firm, B is a food firm, and C is a computer firm. The five attributes c1, c2, c3, c4, and c5 are used to evaluate the three alternatives, as shown below: c1 denotes the risk analysis, c2 denotes the growth analysis, c3 denotes the social-political impact analysis, c4 denotes the environment impact analysis, and c5 denotes the development of the money. The intuitionistic fuzzy decision matrix is shown as Table 1.
|
c1 |
c2 |
c3 |
c4 |
c5 |
| A |
(0.2000,0.5000) |
(0.4000,0.2000) |
(0.5000,0.4000) |
(0.3000,0.3000) |
(0.7000,0.1000) |
| B |
(0.2000,0.7000) |
(0.6000,0.3000) |
(0.4000,0.3000) |
(0.4000,0.4000) |
(0.6000,0.1000) |
| C |
(0.2000,0.7000) |
(0.5000,0.3000) |
(0.4000,0.5000) |
(0.3000,0.4000) |
(0.6000,0.2000) |
Table 1The decision matrix represented by intuitionistic fuzzy values.
Assume that the weight w1, w2, w3, w4, and w5 of the attributes c1, c2, c3, c4, and c5 are 0.112, 0.236, 0.304, 0.236, and 0.112, respectively, that is, w1=0.112, w2=0.236, w3=0.304, w4=0.236, and w5=0.112. Assume that IFOWGA operator has the weighting vector ω={0.25,0.20,0.15,0.18,0.22}T, and IFHGA operator has the weighting vector ω={0.25,0.20,0.15,0.18,0.22}T. Table 2 compares the preference order of the different methods.
| Methods |
Aggregation Operators |
Preference order |
Results |
| Method [10] |
IFWG |
A≻B≻C |
A |
|
IFOWG |
A≻B≻C |
A |
|
IFHG |
A≻B≻C |
A |
| Method [11] |
IFWGε |
A≻B≻C |
A |
|
IFOWGε |
A≻B≻C |
A |
| Method [12] |
IFWGIA |
A≻B≻C |
A |
|
IFOWGIA |
A≻B≻C |
A |
|
IFHGIA |
A≻B≻C |
A |
| Method [13] |
IFWGA |
A≻B≻C |
A |
|
IFOWGA |
A≻B≻C |
A |
|
IFHGA |
A≻B≻C |
A |
| The proposed method |
|
|
|
| (U1,U2,U3,U4) |
ℛ∗ |
A≻B≻C |
A |
| The proposed method |
|
|
|
| (U5,U6,U3,U4) |
ℛ∗ |
A≻B≻C |
A |
Table 2A comparison of the preference orders of the alternatives for the different methods.
From Table 2, we can see that the method of Xu and Yager [10], the method of Wang and Liu [11], the method of He et al. [12], the method of Chen and Chang [13], and the proposed method get the same preference order of the alternatives A≻B≻C, although different uninorms are chosen. Therefore, there is no direct relationship between the result of decision and the choice of uninorms.
In the following research, we discovered that methods in Xu and Yager [10], Wang and Liu [11], He et al. [12], Chen and Chang [13] are flawed in that they fail to differentiate the preference order of the alternatives in some situation.
Example 2.
Let's say the investor wants to invest some money in the most promising firm. Through market analysis, three possible choices A, B, and C will be considered, as shown below: A is a car firm, B is a food firm, and C is a computer firm. The five attributes c1, c2, c3, c4, and c5 are used to evaluate the three alternatives, as shown below: c1 denotes the risk analysis, c2 denotes the growth analysis, c3 denotes the social-political impact analysis, c4 denotes the environment impact analysis, and c5 denotes the development of the money. The intuitionistic fuzzy decision matrix is shown as Table 3.
|
c1 |
c2 |
c3 |
c4 |
c5 |
| A |
(0.3416,0.2300) |
(0.6900,0.3100) |
(0.0000,0.8000) |
(0.2000,0.7500) |
(0.0000,0.8700) |
| B |
(0.2000,0.7500) |
(0.7100,0.2300) |
(0.1200,0.3100) |
(0.2000,0.8000) |
(0.0000,0.8700) |
| C |
(0.1000,0.8700) |
(0.2500,0.7500) |
(0.0000,0.8000) |
(0.36000,0.2300) |
(0.0000,0.3100) |
Table 3The decision matrix represented by intuitionistic fuzzy values.
Assume that the weight w1, w2, w3, w4, and w5 of the attributes c1, c2, c3, c4, and c5 are 0.2, 0.2, 0.2, 0.2, and 0.2, respectively, that is, w1=w2=w3=w4=w5=0.2. Assume that IFOWGA operator has the weighting vector ω={0.2,0.2,0.2,0.2,0.2}T, and IFHGA operator has the weighting vector ω={0.2,0.2,0.2,0.2,0.2}T. Table 4 compares the preference order of the different methods.
| Methods |
Aggregation Operators |
Preference Order |
Results |
| Method [10] |
IFWG |
A=B=C |
Don't know |
|
IFOWG |
A=B=C |
Don't know |
|
IFHG |
A=B=C |
Don't know |
| Method [11] |
IFWGε |
A=B=C |
Don't know |
|
IFOWGε |
A=B=C |
Don't know |
| Method [12] |
IFWGIA |
A=B=C |
Don't know |
|
IFOWGIA |
A=B=C |
Don't know |
|
IFHGIA |
A=B=C |
Don't know |
| Method [13] |
IFWGA |
A=B≻C |
Don't know |
|
IFOWGA |
A=B≻C |
Don't know |
|
IFHGA |
A=B≻C |
Don't know |
| The proposed method |
|
|
|
| (U1,U2,U3,U4) |
ℛ∗ |
A≻B≻C |
A |
| The proposed method |
|
|
|
| (U5,U6,U3,U4) |
ℛ∗ |
A≻B≻C |
A |
Table 4A comparison of the preference orders of the alternatives for the different methods.
Table 4 shows a comparison of the preference order of the alternatives A, B, and C for different methods. We can see that the method of Xu and Yager [10], the method of Wang and Liu [11] and the method of He et al. [12] cannot distinguish the preference order among the alternatives A, B, and C, and the method of Chen and Chang [13] cannot distinguish the preference order among the alternatives A and B. On the contrary, although different uninorms are chosen, the proposed method can distinguish the preference order of the alternatives A≻B≻C. Therefore, the method based on the proposed aggregation operator can conquer the defects of Xu and Yager's [10], Wang and Liu's [11], He et al.'s [12] and Chen and Chang's [13] approaches. Moreover, The result shows that different uninorms do not affect the ranking of alternatives.
4.3. Analysis of the Effectiveness of the Proposed Method
In practical application, some existing methods aggregate information directly rather than classifying the information into different classes, which resulted to the defects that they get wrong preference orders of alternatives or cannot be ranked in some situations. Most of the time, our behavior with the good items are not the same with the bad items, therefore these information be classified into different classes before aggregating is reasonable. The proposed aggregation operator based on uninorms in L∗-fuzzy set theory is effective and reasonable, because uninorm can model the data of bipolar behavior. Moreover, we show that there is no direct relationship between the result of decision and the choice of uninorms.
, Yue Zhang