2017 3rd International Conference on Electronic Information Technology and Intellectualization (ICEITI 2017) ISBN: 978-1-60595-512-4
Improved Weighted Synthesis of Two Criteria
for Comparing Intuitionistic Fuzzy Values
Xuan Huang, Binhui Liu and Qi Peng
ABSTRACT
This paper presents a new approach to comparing intuitionistic fuzzy values. Drawbacks of the previously proposed methods, especially the two-criteria method, are first analyzed. Then the score function and the accuracy function are primarily taken into account to construct the synthesis function with a variable weight coefficient which depends on the risk sensitivity of a decision maker. Not only can maintain some advantages of the two-criteria method, this new approach can provide a more reasonable and accurate estimation of the degree that how much an intuitionistic fuzzy value is greater or less than another one. The choice of risk sensitivity is analyzed to give a suggestion for an inexperienced decision maker. Using some convictive examples, it is shown that the proposed scheme is free of the drawbacks of known methods.
INTRODUCTION
The intuitionistic fuzzy set (IFS) proposed in 1986 by Atanassov [1] is an extension of Zadeh’s fuzzy set. The non-membership of an element to a set is used as an addition to the membership to construct the IFS as follows:
Let X be a nonempty set then A X,A,A is called an intuitionistic fuzzy
set of X , where A:X[0,1] and A:X [0,1] are the membership and
non-membership such that 0 1. ________________________
The theories and applications of IFS such as intuitionistic fuzzy logic [2,3], intuitionistic fuzzy numbers [4], intuitionistic fuzzy algebra [5,6], pattern recognition based on IFS [7,8], threat assessment based on IFS [9,10], etc., have been proposed and developed by researchers.
A very important application of IFS is the decision making problem [11-13] where the values of alternatives are presented by intuitionistic fuzzy values (IFVs). As the attributes of alternatives, evaluations and the decision results are all represented by IFVs, the problem of comparing alternatives is transferred to the problem of IFVs comparison.
The score function S proposed by Chen and Tan [14] was applied to compare IFVs in such a way that the greater the score function is, the better the IFV is. But in the case that two score functions are equal, this approach cannot provide a
conclusion. Therefore the accuracy function H was proposed by Hong and Choi [15], and then Xu [16] used the accuracy function as an addition to the score function to solve the case of equality. In the case of equality of two score functions, Xu compared two IFVs in such a way that the greater the accuracy function is, the better the IFV is.
Xu’s method can solve the majority of IFVs comparison problems well [17-18], whereas Dymova [19] pointed out that a decision maker who uses Xu’s method sometimes may obtain an unreasonable result and cannot provide a technique for estimating an extent to which an IFV is greater/lesser than another. In order to avoid the drawbacks of Xu’s method, Dymova proposed the two-criteria method for comparing IFVs. Dymova combined the “net profit” criterion based on the score function and the “risk” criterion based on the accuracy function taking into account the weight of the risk aversion of a decision maker to construct the possibilities of
B
A and B A.
Using the difference between the possibilities P(AB) and P(B A) as the strength of the inequality between A and B , two-criteria method can provide intuitionistic clear results and make it possible to estimate the strength of relationship between IFVs. However, this method does not provide an intrinsic function similar to the score function and the accuracy function to compare IFVs. As a result, the estimation of the strength according to the difference between the possibilities is not precise. On the other hand, the two-criteria method was proposed aiming at the comparison of a pair of IFVs. Therefore for comparing multiple IFVs, it is complicated to calculate many times to traverse every pairs of IFVs.
analyzed emphatically to give a suggestion for an inexperienced decision maker. Concluded remarks are given finally.
SYNTHESIS FUNCTION FOR COMPARING IFVS
Two-criteria Method for Comparing IFVs
Let A A, A and B B, B be IFVs. Then the score and accuracy
functions for A and B are as follows:
A A A
S ,HA A A ,SB B B ,HB B B. (1)
According to the two-criteria method, the expressions of the possibilities
( )
P AB and P B( A) are as follows:
( ) ( ) (1 ) ( )
( ) ( ) (1 ) ( )
A B A B
B A B A
P A B f S S f H H
P B A f S S f H H
(2)
where [0,1] is the weight which depends on the risk aversion of a decision maker and the function
2 ( )
4 f
(3)
is defined on the interval 2 2.
Expression (3) transforms the interval from [ 2, 2] to a non-negative interval [0,1], and f S( ASB)
and f S( BSA) represent the net profit criterion of the
possibilities of AB and B A, f(HAHB) and f(HBHA) represent the
risk criterion of the possibilities of AB andBA, respectively. Then the two
criteria are aggregated by the risk aversion represented by coefficient to construct the possibilities as expression (2). Based on the possibilities, the comparison of the
two IFVs is built up in such a way that if P A( B)P B( A) then AB and
( ) ( ) ( )
ST AB P AB P BA (4)
is the strength of AB.
show that this approach is free of the limitations of Xu’s method and provides an assessment that how much an IFV is greater or less than another one.
Expression (1)-(4) provides a technique to estimate the possibilities P A( B) and P B( A) and uses the difference between the possibilities as the strength of the difference between the two IFVs. However, expression (3) makes the estimation inaccurate. Let us consider the following example with the assumption 0.98.
For two IFVs A 0.6, 0.3 and B 0.5, 0.2 , using (1)-(4) we obtain ( ) 0.501
P AB and P B( A)0.499, then from two-criteria method we have
AB with the strength equal to 0.002 (Example 1 in article [19]).
We can notice from the two-criteria method that the difference between two IFVs is cause by the score and accuracy function. Therefore we can rewrite expression (2) as follows:
( ) (1 ) ( )
A B A B A B
SH SH S S H H (5)
which implies that the interval transformation as (3) is not introduced in the IFVs comparison. We directly aggregate the difference between score functions and the difference between accuracy functions. Similar to expression (2), the aggregation is also based on the net profit criterion and the risk criterion.
From (5) we obtain that SHASHB 0.004, then we can conclude that A is
greater than B with the strength equal to 0.004, which is not equal to the value based on two-criteria method. It can be inferred that this inequality is caused by the interval transformation in two-criteria method. The interval transformation makes
the possibilities P A( B) and P B( A) within the range [0,1], therefore the comparison result ST A( B) is within the range [-1,1], while the difference between two score functions and the difference between two accuracy functions are all within the range [-2,2]. Therefore the interval transformation changes the domain of the fuzzy values. Since the IFVs comparison is based on the score and accuracy functions, we should make sure that the domain of the degree that one IFV is greater than another is the same with the domain of the differences between score and accuracy functions. Hence the value of difference between two possibilities does not reflect the difference between two IFVs directly and accurately. That is to say that the estimation of difference between two IFVs according to the possibilities is unreasonable. The possibilities estimation cannot be treated as the difference between the two IFVs.
of possibilities (P A( B) and P B( A), P A( C) and P C( A), P B( C) and
( )
P CB ) to acquire the ranking for three IFVs. For comparing four IFVs, we need
to calculate 6 pairs of possibilities ( P A( B) and P B( A) , P A( C) and
( )
P CA , P A( D) and P D( A) , P B( C) and P C( B) , P B( D) and
( )
P DB , P C( D) and P D( C)) to acquire the ranking for three IFVs. And more IFVs need to calculate the possibilities more times. Numerous calculations are not convenient to realize in computer program.
Therefore, in order to avoid these drawbacks analysed above, a new way for comparing IFVs should be proposed. From the two-criteria method we can notice that in a decision making problem, the score function can be treated as the net profit criterion of an IFV, and the accuracy function is the risk cost criterion. So we construct a new function which combines the score function and the accuracy function appropriately so that we can obtain integrated information for IFVs comparison according to the new function.
Synthesis Function
In expression (5), we directly aggregate the difference between score functions and the difference between accuracy functions with a weight coefficient. At the same time, the interval transformation as (3) is neglected.
Then the weight coefficient can be introduced into the estimation of an IFV to form a new function named synthesis function combining the two criterions as follows:
(1 )
SFS H (6)
where [0,1] is the weight which depends on the risk sensitivity of a decision maker, just as the coefficient in Dymova’s two-criteria method.
Therefore the comparison of IFVs is transformed to the comparison of the synthesis function in such a way that an alternative associated with a greater
synthesis function is a greater one, i.e., if SFA SFB then AB and
A B A B
SF SF SF
(7)
is treated as the degree of AB with a given .
Let’s consider some examples to verify if this modification of criterion is useful and correct to break the limitations of the old method.
Example 1. Consider A 0.4,0.2 and B 0.3, 0.5 . Since SA 0.2 ,
0.6
A
H
of SA SB. But from (6)-(7) we can obtain the opposite results presented in Figure
1 if we choose parameter properly.
0 0 -0.2 -0.4 0.2 0.4 0.6 0.8 1 SF
0.2 0.4 0.6 0.8 1 SFA
SFB
1/3
[image:6.612.223.371.150.282.2]
Figure 1. The synthesis functions for A 0.4,0.2 and B 0.3, 0.5 .
Just let equal to 0.95 to compare A and B as an example. From (1) and (6)
we have: SFA 0.95SA0.05HA 0.22 , SFB 0.95SB0.05HB 0.15 , and
0.37
A B A B
SF SF SF
is the degree of AB at 0.95. Then we have AB as the conclusion which agrees with Xu’s method.
While 0.05 , we have: SFA 0.05SA0.95HA 0.58 ,
0.05 0.95 0.75
B B B
SF S H
, and SFB A SFBSFA0.17 is the degree of
B A at 0.05. Then we have BA as the conclusion against to Xu’s method.
Therefore we can get different conclusion by different risk sensitivity, just as we can get according to the two-criteria method. Now let’s consider the degree of difference between the two IFVs.
According to the two-criteria method, at 0.95 we have:
( ) ( ) ( ) 0.555 0.4625 0.0925
ST AB P AB P BA . And at 0.05 we
have: ST B( A)P B( A)P A( B)0.6875 0.645 0.0425.
We can notice that 0.09250.1 1 and 0.04250.1 1, which may imply us that the difference of this two IFVs is tiny, then we can conclude that A is close to
B . However this conclusion obviously is incorrect in fact. This unreasonable
conclusion is cause by the interval transformation in two-criteria method.
We can notice that SASB=0.4 and HAHB= 0.2 , hence we can conclude
consider to compare the two IFVs, then we can determine that AB with the
degree of 0.4 based on the score function. And the same as the degree of -0.2 if we don’t consider the effect of the score function.
According to the synthesis function, the score function representing the net profit criterion and the accuracy function representing the risk cost criterion are aggregated directly without the interval transformation. This aggregation does not lose any
important information. It is clear that SFASFB 0.4 at 1 and
0.2
A B
SF SF
at 0. Furthermore, from Figure 1 we can see that the range of
A B
SF SF
is [ 0.2, 0.4] , which is in accordance with the analysis above. The value
0.37 at 0.95 is in the range of SFA SFB. Not the same as the value 0.00925
based on the two-criteria method, the value based on the synthesis function clearly indicates that AB at 0.95. In addition, the value is close to 0.4, reflecting the
fact that we place more emphasis on the score function at 0.95. And the value -0.17 at 0.05 is close to -0.2, which reflects the fact that the accuracy function is the more important criterion at 0.05. Therefore the degree of AB based on
the synthesis function is more reasonable and accurate than the degree based on the two-critera method.
By the effective and reasonable aggregation of the score function and the accuracy function, a synthesis function reflects the intrinsic characteristic of an IFV at a given risk sensitivity of a decision maker. Therefore the synthesis function of an IFV can be regarded as an intrinsic function of the IFV, just as the effect of the score function and the accuracy function of the IFV.
Example 2. Consider five IFVs: A1 0.4,0.2 , A2 0.6,0.1 , A3 0.5,0.3 ,
4 0.4,0.3
A
, and A5 0.3,0.5 . For 0.95, we have SF10.22,SF2 0.51,
3 0.23, 4 0.13, 5 0.15.
SF SF SF
So we can easily get the ranking of them at 0.95 based on the synthesis
function as follows: A2 A3 A1A4 A5.
Example 2 shows the explicitness and convenience of the synthesis function in solving the multiple IFVs comparison problems. As an intrinsic function of the IFV, the synthesis function can help us acquire the sequence of multiple IFVs explicitly and easily. Without pairwise comparison of the possibilities based on the two-criteria method, the synthesis function can decrease the amount of calculation and simplify the multiple comparison, which is helpful for us to make the comparison of multiple IFVs realization in computer program conveniently.
above, we can conclude that the combination of the score function and the accuracy function is appropriate and effective. In addition, the estimation of the degree of difference between IFVs based on the synthesis function is more reasonable and accurate than the estimation according to the two-criteria method. Therefore the synthesis function of an IFV can be regarded as a new intrinsic function. Therefore the synthesis function can be regarded as a new criterion in the IFVs comparison problem.
THE WEIGHT OF RISK SENSITIVITY FOR COMPARING IFVS
For an experienced decision maker, the weight of risk sensitivity can be regarded as 0.95 if the decision maker thinks that the net profit criterion is much more important (with degree of 0.95) than the risk cost criterion. Similarly, the weight also can be regarded as 0.3 (or other values) if the risk cost criterion is much more important (with degree of 0.3) than the net profit criterion for a decision maker who is sensitive to risk. But for an inexperienced decision maker who has no opinion to balance the contradiction between net profit and risk cost, it’s difficult to determine the weight of risk sensitivity, which makes the synthesis function lose its effect of adjustable to a decision maker’s opinion.
For the two IFVs shown in Example.1, A 0.4,0.2 andB 0.3, 0.5 , we have:
1) If [0,1/ 3), then AB; 2) If [1/ 3,1], then AB;
3) If 1 3
, then AB.
We can notice that the range of for AB is two times long as which for
AB. It can be inferred that the possibility of AB is double of the possibility of AB in an inexperienced decision maker’s opinion. This is a suggestion that we
can acquire from the curves of the two synthesis functions.
For another two IFVs, A 0.7, 0.3 and B 0.5,0.1 , we have SFA SFB no
matter how changes, which implies that A is always greater than B. Particularly,
we have SFA=SFB at 1.
For any A A, A and B B, B , according to expression (10) we have:
0 0
(1 2 ) (1 2 )
A A B B
(8)
Where 0 is the critical point at which SFA SFB. Therefore if A B then we
0
( + ) ( ) =
2( ) 2( )
B B A A B A
B A B A
H
(9)
If 0（ ）0,1 , then the curves of the two synthesis functions have an intersection
within the range 0 1 as Example 1. In this case, if HB A 0, which implies
B A
SF SF at 0, then we can conclude the possibilities of AB and AB as follows:
0
0
( )
( ) 1
P A B P A B
(10)
and in the case of HB A 0 we have the opposite conclusion.
If 0[0,1] , which means that the two synthesis functions do not have
intersection within the range 0 1, it can be inferred that one synthesis function is always greater than another one. Then the conclusion of AB or AB holds
for all .
Particularly, if the critical point 0 is equal to 1 as Example 2, which means that
the two synthesis function intersect at the bound 1, it can be inferred that one synthesis function is always greater than another one besides the bound.
Similar, the conclusion that AB or AB holds for all if 0 is equal to 0,
which implies HA HB, which one is the greater one is based on the difference
between SA and SB.
Moreover, in the case of A B, 0 does not exist so that the curves of the two
synthesis functions have no intersection and are parallel.
CONCLUSIONS
accurate than the estimation according to the two-criteria method. Synthesis function can decrease the amount of calculation in multiple comparison problems so that it can help us to simplify the multiple comparison problems. Some convictive examples are used to show that synthesis function method is free of the limitations of known methods, and the combination of the score function and the accuracy function is appropriate and effective. Therefore the synthesis function method is reasonable and effective and can be treated as a new technique for comparing IFVs.
REFERENCES
1. K.T. Atanassov. 1986. “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, 20(1):87-96.
2. K.T. Atanassov. 1998. “Remark on the intuitionistic fuzzy logics,” Fuzzy Sets and Systems,
95(1):127-129.
3. K.T. Atanassov, G. Gargov. 1998. “Elements of intuitionistic fuzzy logic, Part I,” Fuzzy Sets and
Systems, 95(1):39-52.
4. H.B. Mitchell. 2004. “Ranking intuitionistic fuzzy numbers,” Knowledge-Based Systems,
12(3):368-377.
5. R. Biswas. 1989. “Intuitionistic fuzzy subgroups,” Mathematical. Forum X, 2(2):233-243.
6. W.A. Dudek, B. Davvaz, Y.B. Jun. 2005. “On intuitiontistic fuzzy sub-hyperquasigroups of
hyperquasigroups,” Information Sciences, 170(2):251-262.
7. S.M. Chen, C.H. Chang. 2015. “A novel similarity measure between Atanassov’s intuitionistic
fuzzy sets based on transformation techniques with applications to pattern recognition,”
Information Sciences, 291(C):96-114.
8. C.H. Chu, K.C. Hung, P. Julian. 2014. “A complete pattern recognition approach under
Atanassov’s intuitionistic fuzzy sets,” Knowledge-Based Systems, 66(4):36-45.
9. Y.J. Lei, B.S. Wang, Y. Wang. 2009. “Technique for threat assessment based on intuitionistic
fuzzy reasoning,” Journal of Electronics and Information Technology, 29(9): 2077-2081.
10. D.F. Chen, Y. Feng, Y.X. Liu. 2012. “Threat assessment for air defense operations based on
intuitionistic fuzzy logic,” Procedia Engineering, 29(4):3302-3306.
11. H.W. Liu, G.J. Wang. 2007. “Multicriteria decision-making methods based on intuitionistic fuzzy
sets,” European Journal of Operational Research, 179(1):220-233.
12. L.H. Chen, C.C. Hung, C.C. Tu. 2012. “Considering the decision maker’s attitudinal character to
solve multi-criteria decision-making problems in an intuitionistic fuzzy environment,”
Knowledge-Based Systems, 36(36):129-138.
13. X. Zhao, G. Wei. 2013. “Some intuitionistic fuzzy Einstein hybrid aggregation operators and
their application to multiple attribute decision making,” Knowledge-Based Systems,
37(2):472-479.
14. S.M. Chen, J.M. Tan. 1994. “Handling multicriteria fuzzy decision-making problems based on
vague set theory,” Fuzzy Sets and Systems, 67(2):163-172.
15. D.H. Hong, C.H. Choi. 2000. “Multicriteria fuzzy decision-making problems based on vague set
theory,” Fuzzy Sets and Systems, 114(1):103-113.
16. Z.S. Xu. 2007. “Intuitionistic preference relations and their application in group decision
making,” Information Sciences, 177(11):2363-2379.
17. Z.S. Xu. 2011. “Approaches to multiple attribute group decision making based on intuitionistic
fuzzy power aggregation operators,” Knowledge-Based Systems, 24(6):749-760.
18. S. Zeng, W. Su. 2011. “Intuitionistic fuzzy ordered weighted distance operator,”
19. L. Dymova, P. Sevastjanov, A. Tikhonenko. 2013. “Two-criteria method for comparing
real-valued and interval-real-valued intuitionistic fuzzy values,” Knowledge-Based Systems,
