Abstract
Linguistic intuitionistic fuzzy number (LIFN) is a special intuitionistic fuzzy number which can more easily describe the uncertainty and the vagueness information existing in the real world, and the power average (PA) operator can relieve some influences of unreasonable attribute values given by biased decision makers. In this paper, we will extend the PA operator to the LIFNs and propose some new operators, and develop some new decision making methods. Firstly, we introduce the definition, properties, score function, and operational rules of the LIFNs. Then, we propose some linguistic intuitionistic fuzzy power operators, such as linguistic intuitionistic fuzzy power averaging (LIFPA) operator, linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator, linguistic intuitionistic fuzzy power geometric (LIFPG) operator, linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator, linguistic intuitionistic fuzzy generalized power averaging (LIFGPA) operator, linguistic intuitionistic fuzzy generalized weighted power averaging (LIFGWPA) operator. At the same time, we study some effective properties of these operators. Then, three methods based on the LIFWPA operator, LIFWPG operator and LIFGWPA operator for multi-attribute decision making are proposed. Finally, we use an illustrative example to demonstrate the practicality and effectiveness of the proposed methods.
Keywords
Introduction
Multiple attribute decision making (MADM) has an extensive use in real decision making. Because of the complexity of decision making problems, it is difficult to describe the attributes by crisp numbers. In order to easily character the attributes, Zadeh [39] defined of the fuzzy set (FS), Atanassov [1] defined the intuitionistic fuzzy set (IFS) which is with membership degree and non-membership degree. Due to its capability of accommodating hesitation in human decision processes, IFSs have been widely applied to decision making process. Zhang and Xu [43] proposed a new ranking method for intuitionistic fuzzy numbers (IFNs). Nguyen [25] developed new similarity degree of IFNs. Guo and Song [6] proposed the information entropy of IFNs based on the amount of knowledge. At the same time, some famous decision making methods have also been extended to process the IFNs, such as TOPSIS method [10], GRA method [24], VIKOR method [26], TODIM method [11], DEMATEL method [5] and so on. Yu [38] discussed the research status of IFS by using a citation network analysis. In real decision making, there exists a great deal of qualitative information, which is expressed by the linguistic information [14, 23]. Sine Zadeh [40–42] firstly defined the linguistic variable (LV), there are many researches on the LVs. Herrera and Verdegay [8] developed the operational rules for the LVs. Liu and Jin [15] defined the intuitionistic uncertain linguistic variable.
As above analysis, an IFN is characterized by real-valued membership and non-membership degree defined on interval [0, 1], and the hesitancy degree can be deduced based on the membership and non-membership degrees. However, due to the vagueness and complexity of the decision making environment, the crisp values are insufficient or inadequate to express the membership and non-membership functions in IFN, a feasible solution is to represent such membership degree and non-membership degree by linguistic variables, So, Chen and Liu [3] defined the concept called linguistic intuitionistic fuzzy numbers (LIFNs). The linguistic intuitionistic fuzzy number (LIFN) is in the form of γ = (S α , S β ) which follows the membership degree and non-membership degree by linguistic variables based on the given linguistic term set. The advantages of both linguistic term sets and IFNs are combined by LIFNs. Because the membership and non-membership degrees are expressed by linguistic variables, LIFNs can easier express the fuzzy information than IFNs.
The information aggregation operators have gotten more and more attentions and have also become an important research topic [12–16, 37]. Some aggregation operators were developed, such as arithmetic and geometric weighted aggregation operators for IFNs [31, 34], generalized aggregation operators of IFNs [44], neutral averaging operators of IFNs [7]. However, these operators cannot consider the relationship between the attributes. Further, Yager [36] presented a power-average (PA) operator by power weighting vectors which depend on the input data. Then some new extended PA operators were developed, such as UPOWG (uncertain power ordered weighted geometric) operator [35], 2-tuple linguistic PA (2TLPA) operator [28], intuitionistic fuzzy PA (IFPA) operator [33], generalized IFPA (GIFPA) operator [45], linguistic weighted PA (LWPA) operator [29], and so on.
Because PA operator can consider the relationship between the attributes by the power weighting, which can relieve the some influences of unreasonable data given by biased decision makers, and linguistic intuitionistic fuzzy numbers can be flexible for decision-makers to exactly quantify their viewpoints with linguistic variables. However, the existing PA operators cannot process the linguistic intuitionistic fuzzy numbers, so there is an important significance to study the new PA operators for the linguistic intuitionistic fuzzy numbers. In this paper, we will propose the power averaging operator under linguistic intuitionistic fuzzy information environment, and then propose some new methods for multi-attribute decision making problems and discuss some cases and properties of them.
Preliminaries
The intuitionistic fuzzy numbers and the linguistic term set
Atanassov [1] defined intuitionistic fuzzy set (IFS), which is shown as follows:
For an IFS A and a given x, Xu and Yager [34] called the pair (μA(x), vA(x)) an intuitionistic fuzzy number (IFN).
In real decision making, there are great deals of qualitative information which can be easily expressed by linguistic variables [4, 40]. In order to describe them, it is necessary to set the appropriate linguistic term set. Suppose that S = {s i |i = 0, 1, …, t} is a linguistic term set with odd cardinality, where t is a positive integer.
For example, a set of seven linguistic terms S could be given as follows [36]:
In general, in order to minimize the loss of information, the above linguistic term set S is extended to a continuous linguistic set which is explained by Xu [30]. It is omitted here.
Linguistic intuitionistic fuzzy numbers
For convenience, suppose Γ[0,t] is the set of all LIFNs.
In the following, we will introduce the comparison method of two LIFNs [3].
Then we call Ls (γ) and Lh (γ) the score function and the accuracy function of the LIFN γ, respectively.
If Ls (γ1) < Ls (γ2), then γ1 is smaller than γ2, denoted by γ1 ≺ γ2; If Ls (γ1) = Ls (γ2), then if Lh (γ1) < Lh (γ2), then γ1 is smaller than γ2, denoted by γ1 ≺ γ2; if Lh (γ1) = Lh (γ2), then γ1 and γ2 have the same information, denoted by γ1 = γ2;
Obviously, we have (s0, s t ) ≤ (s α , s β ) ≤ (s t , s0) for any (s α , s β ) ∈ Γ[0,t]. For any two LIFNs γ1 = (s α 1 , s β 1 ), γ2 = (s α 2 , s β 2 ) ∈ Γ[0,t]. If α1 ≥ α2 and β1 ≤ β2, then γ1 ≥ γ2.
In Section 2, we have introduced the concept of linguistic intuitionistic fuzzy numbers where membership and non-membership are represented as linguistic terms. In addition, the PA operator can take the relationship between the attributes. So it is necessary to extend the PA to the LIFNs. In this section, we will develop some kinds of linguistic intuitionistic fuzzy power aggregation operators.
Linguistic intuitionistic fuzzy power averaging operator
Sup (γ i , γ j ) = Sup (γ j , γ i ) , Sup (γ i , γ j ) ∈ [0, 1],and Sup (γ i , γ j ) ≥ Sup (γ m , γ n ), if |γ i - γ j | ≤ |γ m - γ n |.
when n = 1, it’s obvious that the Equation (12) is right. Suppose when n = k, the Equation (12) is right, i.e.,
Then when n = k + 1, we have
So, when n = k + 1, the formula (12) is right.
According to (i) and (ii), we can get the formula (12) holds for any n.
In the following, we will discuss some properties of LIFPA operator as follows:
we can get
Since is any permutation of (γ1 , γ2, …, γ
n
), then we have
In Equations (10) and (12), we think all aggregated attributes are of equivalent importance. However, in real decision making, the weights of the attributes should also be taken into consideration. In the following, we will propose another power averaging operator which is called the linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator.
The LIFWPA operator has the following properties:
The LIFPG operator has some properties as follows:
Similar to the LIFPA operator, the LIFPG operator also neglects their own weights. The next, we will extend the LIFPG operator to the linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator which can consider the weights of the aggregated arguments.
The LIFWPG operator has the idempotency, commutativity and boundedness.
The LIFGPA operator has the following properties.
In the following, we will discuss some cases of the LIFGPA operator. when λ → 0,
when λ = 1,
Since the LIFGPA operator neglects the importance of their own weights. Here, we will introduce another power averaging operator which is called the linguistic intuitionistic fuzzy generalized weighted power averaging (LIFGWPA) operator to overcome the shortcoming.
In the following, we will discuss some properties of the LIFGWPA operator. It can be proved that the LIFGWPA operator has the following properties.
In the following, we will discuss some cases of the LIFGWPA operator. when λ → 0,
when λ = 1,
In this section, we will use the LIFWPA, LIFWPG and LIFGWPA operators to solve the MADM problems in which the attribute values take the form of LIFNs.
Let X ={ x1, x2, …, x m } be a discrete set of alternatives, C ={ c1, c2, …, c n } be a set of attributes, and ω = (ω1, ω2, …, ω n ) T be the weighting vector of the attributes, where ω j ∈ [0, 1] , j = 1, 2, …, n . . For each alternative x i ∈ X, the decision maker gives his/her preference value γ ij with respect to attribute c j ∈ C. where γ ij = (s α ij , s β ij ) takes the form of LIFNs. The goal of the decision making is to give the ranking of all alternatives.
Since there are two types of attributes, i.e. benefit type and cost type, we can transform the cost attribute values to benefit type, and the transformed decision matrix is expressed by , (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, where
In this section, we will give an example to explain the proposed method. Suppose there is a decision making problem of selecting the best global supplier is describe as follows.
A manufacturing company desires to select a best global supplier for its most critical parts used in assembling process. Suppose that X ={ x1, x2, x3, x4 } is a set of four potential global suppliers (i.e., alternatives) under consideration and C ={ c1, c2, c3, c4, c5 } is a set of attributes, where (c1, c2, c3, c4, c5) stand for “overall cost of the production”, “quality of the production”, “service performance of supplier”, “supplier’s profile”, “risk factor”, respectively. And the attribute weight ω = (0.3, 0.3, 0.2, 0.1, 0.1)
T
. The four alternatives x
i
(1, …, 4) are to be evaluated using the LIFNs based on the linguistic term set:
Then we can construct the decision matrix R = (γ ij ) 4×5 shown in Table 1.
Procedure of decision making based on the LIFWPA operator
Then calculate the weights ω
k
(k = 1, 2, …, n), and we get
So the best choice is x4.
Procedure of decision making based on the LIFWPG operator
So the best choice is x4.
Procedure of decision making based on the LIFGWPA operator
Obviously, there are the same ranking results for these three methods.
In addition, in order to demonstrate the influence of the parameter λ on ranking results, we use the different values λ in LIFGWPA operator in step 4 to rank the alternatives. The ranking results are shown in Table 2.
As we can see from Table 2, the ordering of the alternatives may be different for the different value λ in LIFGWPA operator. When 0 < λ ≤ 6, the ordering of the alternatives is x4 ≻ x3 ≻ x2 ≻ x1 and the best alternative is x4. When 7 ≤ λ, the ordering of the alternatives is x4 ≻ x3 ≻ x1 ≻ x2 and the best alternative is x4.
Comparing with the existing methods
In order to verify the validness of the methods in this paper, we use the method proposed by Chen [3] to rank this example, and get
Then we calculate the score function, and get
So we can get the ranking result x4 ≻ x3 ≻ x2 ≻ x1 and the best choice is x4.
Obviously, there are the same ranking result and the same best alternative from these methods. Comparing with the proposed methods in this paper, the method proposed by Chen [3] is based on the LIFHA operator, it is only a special case of the LIFWPA operator and LIFGWPA operator proposed in this paper and it doesn’t consider the relationship between the attributes by the power weighting. Obviously, the linguistic intuitionistic fuzzy power operators in this paper can relieve the some influences of unreasonable data given by biased decision makers, the fundamental aspect of these operators is that the weight of each input argument depends on the other input arguments and allows argument values to support each other. In the LIFGPA operator, when we change the value of λ, the result is different, and it provides the more flexible features as λ is assigned different values.
Conclusion
In this paper, we explore some Power aggregation operators based on LIFNs and develop some methods to deal with the MADM problems in which the attribute values take the form of LIFNs. Firstly, we proposed the linguistic intuitionistic fuzzy power averaging (LIFPA) operator, the linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator, the linguistic intuitionistic fuzzy power geometric (LIFPG) operator, the linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator, the linguistic intuitionistic fuzzy generalized power averaging (LIFGPA) operator and linguistic intuitionistic fuzzy generalized weighted power averaging (LIFGWPA) operator. Then, we propose some methods for multi-criteria decision making based on the LIFWPA operator, LIFWPG operator and LIFGWPA operator, and describe the operational processes in detail. Finally, an application example is given to describe the developed approach and to verify its practicality and effectiveness. The advantages of these methods are that they can consider the relationship between the attributes so as to relieve the some influences of unreasonable data given by biased decision makers. In the further research, it is necessary and meaningful to study some new aggregation operators based on the linguistic intuitionistic fuzzy numbers because the linguistic intuitionistic fuzzy numbers are widely applied to each domain in the real world, or extend the proposed methods to interval valued intuitionistic hesitant fuzzy set [2] and interval neutrosophic hesitant set [19] and so on.
Footnotes
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province, National Soft Science Project of China (2014GXQ4D192), Shandong Provincial Social Science Planning Project (No. 15BGLJ06), the Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (2015Z057).
