Abstract
Since linguistic intuitionistic fuzzy numbers (LIFNs) can more easily express the fuzzy information in a qualitative setting, it is necessary to extend their operational rules and develop the decision making method based on the LIFNs. In this paper, we propose some new operational laws such as the subtraction and division, score function and accuracy function for (LIFNs), and we develop the linguistic intuitionistic fuzzy entropy, which can measure not only the fuzziness but also the intuitionism. Then we propose a method for determining the objective weight of attributes by the proposed entropy. In addition, because the VIKOR (from Serbian: VIseKriterijumska Optimizacija I Kompromisno Resenje) method is an effective decision making tool which can select the optimal alternative by the maximizing “group utility” and minimizing “individual regret”, we extended the VIKOR method to process the linguistic intuitionistic fuzzy information (LIFI) based on the new operational laws, and proposed an extended VIKOR method for the multiple attribute decision making (MADM) problems with LIFI. An illustrative example shows the effectiveness and advantages of the proposed approach.
Keywords
Introduction
Multi-attribute decision making (MADM) and multi-attribute group decision making (MAGDM) problems have play an important role in real life, more and more researches on them have been done and many contributions have been achieved. In real decision making, the decision information is often incomplete and fuzzy, in order to express this information, Zadeh [45] proposed the fuzzy sets (FSs). However, it is difficult to express the some complex fuzzy information, because it only has a membership function, for example, the voting problem in which there exists some of the opposition and abstain. Then Atanassov [1, 2] presented intuitionistic fuzzy sets (IFSs) to express some complex situations. The IFSs are with a membership function and a non-membership function, and we can easily know that IFS is better than FS to describe the fuzzy information. So many experts have studied the intuitionistic fuzzy information and the MADM methods based on them [36–38]. However, in a qualitative setting, it is difficult to express the fuzzy information by the exact numerical value, and it is more feasible by linguistic terms rather than numerical values [18, 39]. For example, when the performance of a car is assessed, terms like “very poor”, “poor”, “good”, “very good” are usually used to express the experts’ preference. Since linguistic set is very useful tool to deal with the MADM problems in in a qualitative setting, it has become a research hotspot, and many achievements have been made [9, 31–35]. In order to easily express the membership degrees and non-membership degree of IFNs in a qualitative setting, one concept called linguistic intuitionistic fuzzy numbers (LIFNs) is proposed by Chen and Liu [5], in which the membership degree and non-membership degree are expressed by linguistic variables based on the given linguistic term set. LIFNs combine the advantages of both linguistic term sets and IFNs, so, they can more rationally and effectively express the fuzzy information because IFNs are more in line with people’s way of thinking and conscious activities.
In order to effectively deal with these incomplete information, the entropy is a hotspot in many research fields, and it is originated from the Thermodynamics and is used to indicate the degree of confusion. Zadeh [45] firstly used the entropy to measure the fuzziness of a fuzzy set, after that, many experts have studied entropy in in the decision making field, De and Termini [10] proposed the concept of axioms which should be complied by the fuzzy entropy, also defined a non-probabilistic entropy for IFSs. Burillo and Bustince [3] introduced the definition of entropy for IFSs and inteval-valued fuzzy sets (IVFSs), in order to measure the intuitionism degree of an IFS or IVFS. Subsequently, Szmidt and Kacprzyk [29] extended the definition of fuzzy entropy provided by De and Termini and gave an axiomatic definition of entropy for IFSs. Based on the geometrical interpretation of IFSs, they also proposed a new entropy measure. Vlachos and Sergiadis [43] proposed an entropy formula from a cross entropy of IFSs. They pointed out that entropy of IFSs could measure both fuzziness and intuitionism of an IFS. Zhao and Xu [49] proposed the entropy measures for interval-valued intuitionistic fuzzy information from a comparative perspective. Guo [13] and Liu et al. [24] presented the axiomatic definition of interval-valued intuitionistic fuzzy entropy. Wang and Wei [30] extended entropy of IFSs to IVIFSs. Subsequently, Gao and Wei [15] defined a new entropy formula based on the improved Hamming distance for IVIFSs. Liang [20] proposed a new entropy measure by the geometrical interpretation of IFSs. The new formula can measure not only the fuzziness but also the intuitionism of an IFS.
Since Atanassov [1, 2] developed the IFS, it has received great attention and has been applied to various fields of real decision making [4, 41]. Atanassov [1, 2] and De et al. [6] introduced some basic operations on IFSs, including “intersection”, “union”, “supplement”, “power” and so on, which ensure that the operational results are also IFSs, Xu and Yager [42] defined the intuitionistic fuzzy number (IFN), also some basic operational laws were defined, such as “addition”, “multiply” “power” about IFNs, Lei and Xu [22] developed the subtraction and division operations for IFNs, obviously, the subtraction and division operations are necessary and important for the study of the IFS theory, and they have received great attention and have been applied to various fields of actual decision making [16, 40]. Because LIFs can combine the advantages of both linguistic term sets and IFNs, now the researches on the LIFNs are rare, Chen and Liu [5] propose the conception of LIFNs, and developed some related operational laws, which include the “intersection”, “union”, “supplement” “addition”, “power”, “multiply”. Then, Liu and Qin [25] defined the Euclidean distance of LIFNs, which is a useful tool to process the fuzzy MADM problems in which the information is expressed by LIFNs. However, until now, there are no the subtraction and division operations for LIFNs, and the operational result of the Euclidean distance is not a LIFN. Obviously, it is very necessary to solve these problems.
In addition, the VIKOR method is an important decision tool to process the fuzzy MADM problems because it can consider the maximum “group utility” and minimum of “individual regret” and can consider two kinds of particular measures of “closeness” to the virtual ideal solution and the virtual negative ideal solution, simultaneously. Comparing with the other decision making methods, such as TOPSIS, ELECTRE, TODIM etc., the advantage of VIKOR can give one compromise optimal choice or a group of choices with no differences based on the maximum “group utility” and minimum of “individual regret”, however, the other methods just can provide an optimal choice. Because the traditional VIKOR method can only deal with the crisp numbers, some new extensions of VIKOR for the different fuzzy information have been studied. Liu and Wang [26] extended VIKOR to generalized interval-valued trapezoidal fuzzy numbers. Wu et al. [44] extended VIKOR to linguistic information. Liao et al. [23] extended VIKOR to Hesitant fuzzy linguistic information. Ghorabaee et al. [14] extended VIKOR to interval type-2 fuzzy sets. Zhang and Wei [48] extended VIKOR to deal with HFS. Liu and Wu [27] extended VIKOR to process the multi-granularity linguistic variables. Zhang and Liu [47] extended VIKOR to process the hybrid information, including linguistic variables, crisp numbers, interval numbers, triangular fuzzy numbers, trapezoid fuzzy numbers, and so on. Du and Liu [8] extended VIKOR to deal with intuitionistic trapezoidal fuzzy numbers. However, all above-mentioned VIKOR methods used the distance formulas to deal with the MADM information, and then the results are a crisp number, and changed its original form. In addition, in some situations, the VIKOR method based on the distance formula cannot give the ranking results because there are the same distances among some different LIFNs. In this paper, we will extend VIKOR method to process LIFNs, in which distance formula will be replaced by the subtraction and division operations for LIFNs.
Since LIFNs combine the advantages of both linguistic term sets and IFNs, it can be better deal with the more complex and fuzzy information in the real world, and the VIKOR method can consider the maximum “group utility” and minimum of “individual regret” and can take into account two kinds of particular measures of “closeness” to the virtual ideal solution and the virtual negative ideal solution, simultaneously. So the motivation and goal of this paper are (1) to propose new subtraction and division operations for LIFNs; (2) to develop a new linguistic intuitionistic entropy measure which can measure not only the fuzziness but also the intuitionism of a LIFN; (3) to propose a method for determining the objective weight of attributes by the proposed entropy; (4) to develop a new VIKOR method by using the subtraction and division operations for LIFNs, which can overcome some defects.
In order to do so, the remainder of this paper is show as follows. In Section 2, we briefly review some basic concepts of LIFNs. In Section 3, we proposed the subtraction and division operations for LIFNs and the entropy of LIFNs. In Section 4, we extend the traditional VIKOR method to the LIFI, and a MADM approach is proposed. In Section 5, we give a numerical example to elaborate the effectiveness and feasibility of our approach, the comparison with other methods is conducted. Concluding remark is made in Section 6.
Preliminaries
Linguistic intuitionistic fuzzy numbers
Chen [5] put forward the conception of linguistic intuitionistic fuzzy number (LIFN), in which the membership degree and the non-membership degree are presented by linguistic variables [7, 46]. The definition is shown as follows:
For convenience, suppose Γ[0,t] is the set of all LIFNs.
Chen [5] also introduced some operational laws of LIFNs as follows:
In addition, the complement operation of LIFNs can be presented as
Then, Liu and Qin [25] introduced the Euclidean distance of LIFNs which is shown as follows:
The VIKOR is one useful method to solve the MADM problems, it can consider both the “group utility” and “individual regret”. The decision making problem can be expressed as follows:
Suppose there are m alternatives which are expressed as (X1, X2, …, X
m
), and there are n attributes which are presented as (A1, A2, …, A
n
), the evaluation value of alternative X
i
with respect to attribute A
j
is expressed by x
ij
, i = 1, 2, …, m . j = 1, 2, … n. Suppose the
In the VIKOR method, the maximum group utility can be presented by min S i , and minimum individual regret can be presented by min R i , where S i = L1,i, and R i = L∞,i.
The steps of the VIKOR method can be described as follows:
Where
If one of above two conditions is not met, then we will get a collection of compromise alternatives and not one compromise solution. If Condition 2 is not met, then we can get that alternatives X(1) and X(2) should be compromise solutions. If Condition 1 is not met, then the maximum M can be got by the formula Q (X(M)) - Q (X(1)) < MQ,
Based on the above analysis, we know that the best solution is the one with the minimum Q value when the conditions 1 and 2 are met, and when one of two conditions is not met, the compromise solutions may be have more than one.
Some new operational laws of linguistic intuitionistic fuzzy numbers
Chen [5] proposed the some operational laws of LIFNs but not including the subtraction and division operations, these two operations of the LIFNs are necessary to solve the MADM problems, so we propose the subtraction and division operations for LIFNs as follows:
(1) γ1 ! γ2 = (s α 1 ! s α 2 , s β 2 ! s β 2 ), where
(2) γ1 % γ2 = (s α 1 % s α 2 , s β 1 % s β 2 ), where
From definition 6, we can see the subtraction and division operations for LIFNs are still LIFNs.
In the following, we give two functions to compare two LIFNs.
If Ls (γ1) < Ls (γ2), then γ1 is smaller than γ2, denoted by γ1 ≺ γ2; If Ls (γ1) = Ls (γ2), and Lh (γ1) < Lh (γ2), then γ1 is smaller than γ2, denoted by γ1 ≺ γ2; and Lh (γ1) = Lh (γ2), then γ1 and γ2 have the same information, denoted by γ1 = γ2;
Next, we introduce the include relationships between two LIFNs:
γ1 ⊆ γ2 if and only if
γ1 = γ2 if and only if
In this section, we propose linguistic intuitionistic fuzzy entropy.
Let M ={ γ i 〈 s α (γ i ) , s β (γ i ) 〉 |γ i ∈ Γ[0, t] }, the mapping E (M) is called as entropy if E satisfies the following conditions:
If E (M) = 0, since 0 ≤ |α (γ
i
) - β (γ
i
) | ≤ t ≤ t + π (γ
i
), so
If E (M) = 1, since 0 ≤ |α (γ i ) - β (γ i ) | ≤ t ≤ t + π (γ i ), so t - |α (γ i ) - β (γ i ) | + π (γ i ) = t + π (γ i ) must be established, we can get |α (γ i ) - β (γ i ) | = 0, so, α (γ i ) = β (γ i ) namely, s α (γ i ) = s β (γ i ).
Thus (t - α
M
) (2t - α
N
- β
N
) ≤ (t - α
N
) (2t - α
M
- β
M
), which implies
Similarity, when M ⊆ N and s α (N) ≤ s β (N) for every γ i ∈ Γ[0,t], we can also prove that E (M) ≤ E (N).
Therefore, E (M) ≤ E (N).
From above four conditions, we can know that the linguistic intuitionistic fuzzy entropy can consider the uncertainty both from fuzziness and from the lack of information based on LIFNs, when LIFNs are (s t , s0) or (s0, s t ), the entropy value is smallest; When LIFNs meet s α (γ i ) = s β (γ i ) for each γ i ∈ Γ(0,t), the information of LIFNs is most fuzzy and when γ i = (s0, s0) γ i ∈ Γ(0,t), the information of LIFNs lack all the information, the entropy value is largest; For given a LIFN, the more fuzzy of the information, and the more information missing, the greater of the entropy value. So the defined linguistic intuitionistic fuzzy entropy can measure not only the fuzziness but also the intuitionism of LIFNs.
In this paper, we will extend the VIKOR method to solve MADM problem with the linguistic intuitionistic fuzzy information (LIFI), the subtraction and division operations for LIFNs will be used to this new method and the linguistic intuitionistic fuzzy entropy will be used to determine the each attribute’s weight.
In order to do so, we describe the decision making problem firstly.
For a multiple attribute decision making problem, let X ={ x1, x2, …, x m } be a group of alternatives, C ={ c1, c2, …, c n } be a group of attributes, and the attribute weights are unknown. Suppose that γ ij = (sα(γ ij ), sβ(γ ij )) is the evaluation value of the alternative x i with respect to the attributes c j which is expressed by the LIFI, where sα(γ ij ), sβ(γ ij ) represent the membership degree and non-membership degree of LIFNs, and sα(γ ij ), sβ(γ ij ) ∈ S(0,t). The decision matrix denoted by LIFNs is listed in Table 1, and the goal of this MADM problem is to rank the alternatives and get the best one.
Decision making matrix with the linguistic intuitionistic fuzzy information
Decision making matrix with the linguistic intuitionistic fuzzy information
Because the weight information is unknown, we can use the linguistic intuitionistic fuzzy entropy to calculate the attribute weights.
The procedures of the proposed method are shown as follows:
Since there are different types of attributes, we should convert different type to the same type. In general, we can transform the cost attribute values to benefit type, and the transformed decision matrix is expressed by
According to the partial order relation, we have the virtual positive ideal solution (PIS):
Where
In this step, We utilize the subtraction and division operations for LIFNs to get the results, which are defined by Equations (12–15). It can make sure the results are also LIFNs.
Description of the example
In this section, we will give an example to show the proposed method. Suppose there is a decision making problem of selecting the best global supplier which is described 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 unknown. The four alternatives x
i
(1, …, 4) are to be evaluated by using the LIFNs based on the linguistic term set:
Then we can construct the decision matrix R = (γ ij ) 4×5 shown in Table 2.
The evaluation values of four alternatives with respect to the five attributes
The evaluation values of four alternatives with respect to the five attributes
Since “overall cost of the production”, “risk factor” are the cost attribute values, “quality of the production”, “service performance of supplier”, “supplier’s profile” are the benefit attribute values, we transform the cost attribute values to benefit type according to formula (21), and get
We firstly gave the ranking results by the values Ls (S i ), Ls (R i ) and Ls (Q i ) , i = 1, 2, 3, 4, the smaller the values Ls (S i ), Ls (R i ) and Ls (Q i ) , i = 1, 2, 3, 4, the better the alternatives are. The results are listed in Table 3.
The ranking and the compromise solution
As MQ = 1/(m-1) = 1/(4-1) = 0.333333, so
Which is not satisfied the condition 1.
So, x4 is the only compromise solution.
The VIKOR method for linguistic intuitionistic fuzzy numbers based on distance measure formula
Before the subtraction and division operations for LIFNs are not proposed, the distance formula of LIFNs is usually used to solve the MADM problems. In order to prove the valid of our method, we use the extended VIKOR method based on the Euclidean distance of linguistic intuitionistic fuzzy numbers to calculate our example again, the procedures are shown as follows:
The ranking and the compromise solution by distance formula
The ranking and the compromise solution by distance formula
As MQ = 1/(m-1) = 1/(4-1) = 0.333333, so
Which is not satisfied the condition 1.
So, x4 is the only compromise solution.
Clearly, both two different VIKOR methods has the same compromise solution x4, it can prove the proposed method in this paper is effective. Different from the VIKOR based on distance measure, our method based on subtraction and division operations can make sure the aggregate result are still the form of LIFNs, and use the score function to compare the result. In a sense, the form of result is easier to accept with the linguistic intuitionistic fuzzy information, and it can reflect the real world better.
In addition, the distance measure exist some defects that it can be invalid in some cases, but the subtraction is efficient in these cases to solve MADM problems. For example, suppose there are γ1 = (s7, s1) , γ2 = (s6, s2) γ3 = (s5, s3) , γ4 = (s4, s4) ∈ Γ[0,8].
First, we calculate the distance between γ
i
to γi+1, i = 1, 2, 3 by distance formula of LIFNs and utilize the subtraction to calculate γ
i
- γi+1, i = 1, 2, 3, we get
Next, in order to compare the results by subtraction, we deal with the results by score index, we get
Finally, we give a ranking, The results is shown in Table 5.
The comparison of the ranking between distance measure and subtraction of LIFNs
From Table 5, we can see that the ranking based on distance formula is invalid, simultaneously, the ranking based on subtraction is efficient, and so it can prove that the subtraction of LIFNs is more flexible and more useful to solve MADM problems.
Obviously, the advantage of the new VIKOR method can keep that the form of aggregate result is still a LIFN because the subtraction of LIFNs can solve this problem that the distance measure between two LIFNs is a real number and it is not a LIFN. This is a highlight by replacing distance measure in VIKOR method by the subtraction of LIFNs, so it is more flexible and efficient for the proposed VIKOR method than the traditional one because it can solve the existing problem when there is the same distance between any two LIFNs which aredifferent.
In order to describe the advantage of VIKOR, in this section, we use another method to calculate our example once again. Hu and Xu [19] proposed TOPSIS method with interval-valued intuitionistic fuzzy information, we extend it to LIFNs. the procedures are shown as follow:
the distance between each alternative and the virtual positive ideal solution is shown as follows:
the distance between each alternative and the virtual positive ideal solution is shown as follows:
It is easy to see that the ranking results between our proposed method and the TOPSIS method for LIFNs are same and has the same best alternative x4, it can prove our method is valid again, the advantage of the our method is that it can provide the compromise alternative set by the maximum “group utility” and minimum “individual regret”. Accord the real decision problem, we can adjust the values of the balance parameter v to balance the factors between group utility and individual regret, and get different alternative rankings, but the TOPSIS cannot do this, our method is more feasible and scientific to solve the MADM problems in the real world, the disadvantage is the TOPSIS method concise but the VIKOR method is a little complex to calculate.
Conclusions
The linguistic intuitionistic fuzzy numbers have become more and more useful tools to deal with fuzzy and uncertainty information. In this paper, we introduce some new operational laws of LIFNs, such as the subtraction and division; it can substitute the distance measure formula in some way. We also give some functions to compare two LIFNs. Next, we introduce the linguistic intuitionistic fuzzy entropy, which can measure not only the fuzziness but also the intuitionism of LIFNs, and then we give a method for determining the objective weight of attributes by the proposed entropy. Further, the traditional VIKOR method is extended to process the LIFNs and use it to solve the MADM problems, which is based on the maximum “group utility” and minimum “individual regret”. Compared with the earlier VIKOR which is calculated by the distance formula, the advantages of our proposed method are (1) it can solve the existing problem that the VIKOR method based on the distance formula cannot give the ranking results because there are the same distances among some different LIFNs; (2) the form of the calculating result is still LIFNs; (3) the new score function and accuracy function are used to rank the alternatives; (4) the proposed method can consider the “group utility’ and “individual regret” and can give a compromise best solution or a set of compromise solutions. In the further research, it is necessary and meaningful to study some other operational laws of LIFNs and extend some other methods deal with the LIFNs.
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 (No. ts201511045), and Shandong Provincial Social Science Planning Project (Nos. 15BGLJ06,16CGLJ31 and 16CKJJ27), and Key research and development program of Shandong Province (2016GNC110016).
