Management Science and Research December 2015, Volume 4, Issue 4, PP.50-55
A Note about Multi-criteria Fuzzy Decision Making Method Based on a Novel Accuracy Function Defined on Two Opposite Aspects Xinshang You†, Tong Chen College of Management and Economics, Tianjin University, Tianjin 300072, China †
Email: xinshangyou@163.com
Abstract This article proposes a method on multi-criteria decision making problem depending on a novel accuracy function under the interval-valued intuitionistic fuzzy (IVIF) environment. The novel accuracy function is denoted in two opposite aspects, leading to the decision making problem considered more reasonably. In this paper, the IVIF weighted arithmetic operator and the IVIF weighted geometric average operator are both utilized. Based on the developed method, we rank the alternatives and choose the desirable one. Finally, an illustrative example is given to demonstrate the developed approach. Keywords: Interval-valued Intuitionistic Fuzzy Set; Aggregation Operator; Multi-criteria Fuzzy Decision Making; Accuracy Function
1 INTRODUCTION Fuzzy set theory system has been developed successfully, which is attracting the attentions of scholars from different fields. Half of a century ago, Zadeh (1965) proposed the theory of fuzzy set (FS) firstly, which describes the membership degree by an exact number belonging to the interval [0, 1]. In 1975, Zadeh introduced the concept of interval-valued fuzzy set (IVFS), whose membership degree is given by a subinterval of [0, 1]. Atanassov (1986) developed the fuzzy set theory by defining intuitionistic fuzzy set (IFS) which is characterized by a membership degree and a non-membership degree whose values are both numbers between [0, 1]. Later, Atanassov and Gargov (1989) extended IFS to the interval-valued intuitionistic fuzzy set (IVIFS), which is denoted also by a membership degree and a non-membership degree, but the values are subintervals of [0, 1] rather than the real numbers of IFS. Depending on these extensions of above researchers, significant theoretical developments (Z. Xu, 2010; Z. Xu, R. Yager, 2006; Z. Xu, 2007; Y. Jiang, Z. Xu, X. Yu, 2015) and various applications (S. De, R. Biswas, A.Roy, 2001; F. Ye, 2010; M. Medineckene, E. Zavadskas, F. Bjork, Z. Turskis, 2015; J.Wu, Chiclana, 2014) have been published in numerical influential journals. Multi-criteria fuzzy decision making problem (Z. Xu, M. Xia, 2012; N. Chen, Z. Xu, M. Xia, 2013) is an important branch of applications derived from fuzzy sets theory. And this paper mainly discusses this kind of problem. In our daily life, people always need to deal with decision-making problems with respect to choosing the optimal alternative according to the objective criteria. Xu (2007) proposed the score function and accuracy function for IVIFS to rank the elements named interval-valued intuitionistic fuzzy numbers (IVIFNs) of it. However, his definitions loose valid in some cases. To overcome this drawback, Z. Wang et al. (2009) studied the comparisons between any interval-valued intuitionistic fuzzy numbers (IVIFNs) in much more full aspects, considering the uncertainty degree. J. Ye (2009) and V. Lakshmana (2011) proposed two different novel accuracy functions, respectively. However, as known well, the magic charm of fuzzy set theory is that, it describes objective things with the consideration of uncertainty, leading it closer to the realities. The researchers mentioned above, ranked the IVIFSs by combining the membership degree and non-membership degree together, obtaining an exact value for comparison. In this paper, we discuss the ranking problems of IVIFSs in two points of view: the positive aspect and - 50 www.ivypub.org/msr
the negative one, contributing to the problems considered more reasonably for decision maker (DM). The rest of the paper is organized as follows. Section 2 introduces some basic definitions about IVIFSs which are proposed by other researchers. In section 3, we will propose a novel accuracy function by considering it in two aspects: the positive and the negative. And in Section 4, we introduce a multi-criteria fuzzy decision making method based on the new function to judge the alternatives and rank them in decreasing order. Section 5 shows an illustrative example for demonstrating the developed method. This paper concludes in section 6, summarizing our developments.
2 BASIC DEFINITIONS In this section, we introduce basic concepts of fuzzy set theory which will be used in the following sections. Depending on the development of fuzzy set theory, Atanassov and Gargov (1989) firstly introduced the concept of the IVIFS.
Definition 2.1 (Atanassov, Gargov, 1989) Let X = { x1 , x2 ,..., xn } be a non-empty finite set and I be the set of all closed subintervals of the interval [ 0,1] . An interval-valued intuitionistic fuzzy set (IVIFS) S ⊂ X is defined in the following form:
S =< { xi , µ S ( xi ) ,ν S ( xi ) >| xi ∈ X } where the functions
(1)
µ S ( xi ) : X I and ν S ( xi ) : X I stand for the degree of membership and degree of non-
membership of the element xi ∈ X to the set S , respectively, satisfying 0 ≤ sup
{µ ( x )} + S
i
i = 1, 2, ..., n . As µ S ( xi ) and ν S ( xi ) are closed intervals, we rewrite Eq. (1) as follows:
sup {ν S ( xi )} ≤ 1 ,
{ xi , [µ Sl ( xi ) , µ Su ( xi )], [ν Sl ( xi ) ,ν Su ( xi )] >| xi ∈ X } S =<
(2)
Denote D = ([ a, b],[c, d ]) for simple and named interval-valued intuitionistic fuzzy number (IVIFN).
Definition 2.2 (Z. Xu, 2007) Let A j ∈ IVIFS ( X ) , ( j = 1, 2,..., n) . The weighted arithmetic average operator is defined by n
Fw ( A1 , A2 ,..., An ) = ∑ w j Aj
(3)
j =1
n n n n wj wj wj w l u l 1 (1 ( )) ,1 (1 ( )) , ( ( )) , (ν Au j ( x)) j w A x x x µ µ ν = − − − − ∑ ∏ ∏ ∏ ∏ j j A A A j j j j =1 = j 1 =j 1 = j 1 =j 1 n
where w j is the weight of A j , ( j = 1, 2,..., n) , w j ∈ [ 0,1] and
∑
n j =1
(4)
wj = 1 .
Definition 2.3 (Z. Xu, 2007) Let A j ∈ IVIFS ( X ) , ( j = 1, 2,..., n) . The weighted geometric average operator is defined by n
Gw ( A1 , A2 ,..., An ) = ∑ Aj
wj
(5)
j =1
n n n n n w w w w w Aj j ∏ ( µ Al j ( x)) j , ∏ ( µ Au j ( x)) j , 1 − ∏ (1 −ν Al j ( x)) j ,1 − ∏ (1 −ν Au j ( x)) j = ∑ j 1 =j 1 j= =1 = j 1 =j 1
where w j is the weight of A j , ( j = 1, 2,..., n) , w j ∈ [ 0,1] and
∑
n j =1
(6)
wj = 1 .
It is obvious that Eq. (4) and Eq. (6) have different highlights. The weighted arithmetic average operator stresses influence of the group. And the weighted geometric average operator emphasizes the individual influence. Xu gave - 51 www.ivypub.org/msr
the next famous accuracy function in 2007.
Definition 2.4(Z. Xu, 2007) Let A = ([a, b], [c, d]) be an interval-valued intuitionistic fuzzy number, an accuracy function H of an interval-valued intuitionistic fuzzy value can be represented as follows:
H ( A= )
1 (a + b + c + d ) 2
(7)
where the accuracy number is H ( A ) ∈ [ 0, 1] .
3 NOVEL ACCURACY FUNCTION In interval-valued intuitionistic fuzzy sets, it is well known that we describe the membership degree and nonmembership degree by closed intervals that are the sub-interval of [0, 1]. For [0, 1], we know [0.5, 0.5] = 0.5 is the most hesitant degree no matter for membership or non-membership. Next, we will take the special point into consideration during proposing the novel accuracy function. Let D = ([a, b], [c, d]) be an interval-valued intuitionistic fuzzy number. And P =[a, b], N = [c, d], which stand for the positive and negative attitude to the assessed alternative by decision maker, respectively. The positive accuracy function is defined as follows:
= RD ( P )
(a − 0.5) + (b − 0.5) a + b − 1 = b−a b−a
(8)
The negative accuracy function is defined as follows:
= RD ( N )
(0.5 − c) + (0.5 − d ) 1 − c − d = d −c d −c
(9)
Derived from the above Equations (8) and (9), the higher value of RD(P) the better assurance of the positive attitude; the higher value of RD(N) the better assurance of the negative attitude. Next, we will reconsider the same numerical examples taken from Ye (2009). Example 1. A1= ([0.4, 0.5], [0.3, 0.4]) and A2= ([0.4, 0.6], [0.2, 0.4]) are two IVIFNs for some two alternatives. The desirable alternative is chosen by the accuracy function. By applying Eq. (7), we can obtain H(A1) = H(A2) = 0.8. Then we cannot judge which alternative is better. While by applying Eq. (8), we obtain RA1(P)=-1 and RA2(P)=0. Obviously, in the opinion of positive side, the alternative A2 is better than the alternative A1, whose conclusion is the same with Ye’s (2009). Example 2. A1= ([0.2, 0.3], [0.3, 0.6]) and A2= ([0.1, 0.2], [0.5, 0.6]) are two IVIFNs for some two alternatives. The desirable alternative is selected also by the accuracy function. Obtain H(A1) = H(A2) = 0.7 by applying Eq. (7). We do not know which alternative is better again. By utilizing Eq. (8), we can obtain RA1(P)=-5 and RA2(P)=-7. Obviously, in the opinion of positive, the alternative A1 is better than the alternative A2, whose conclusion is also the same with Ye’s (2009). We solve the above two examples only in the opinion of positive, for the reason that the values of the function R(P) are significantly different. When it comes to the condition that the values of the function R(P) are similar, we should take the function R(N) into consideration, and we will show this condition in Section 5.
4 METHOD PROCESSES BASED ON THE NOVEL ACCURACY FUNCTION In this section, we introduce a method for multi-criteria fuzzy decision making problems with weights considered, depending on the novel accuracy functions proposed above. Denote A = {A1, A2, ... , Am} be a set of alternatives and C = {C1, C2, ... , Cn} be a set of criteria whose weights given by the decision maker and satisfy
Aij
∑
n j =1
w j = 1 . Alternative Ai is characterized by an IVIFS as follows:
{[µ Al i ( C j ) , µ Aui ( C j )],[ν Al i ( C j ) ,ν Aui ( C j )] | C j ∈ C} - 52 www.ivypub.org/msr
(10)
( )
( )
( )
( )
where 0 ≤ µ Aui C j + ν Aui C j ≤ 1 , µ Al i C j ≥ 0 , ν Al i C j ≥ 0 ,
j = 1, 2,..., n , and i = 1, 2,..., m . For short,
denote Aij = ([µ , µ ],[ν ,ν ]) . Thus, a multi-attribute decision making problem (MADMP) can be expressed in the from of decision matrix M = (Aij)m×n. Then, based on the accuracy function proposed in section 3, we introduce the decision method in the following process: l ij
u ij
l ij
u ij
Step 1. Identify the evaluation criteria and alternatives. Enter the decision maker’s opinions in the form of IVIFNs to the decision matrix M. And make sure the weight set about the criteria set properly. Step 2. Calculate the weighted arithmetic average values by Eq. (4) when we pay much attention to the group’s influence. In other cases, apply Eq. (6) to computing the weighted geometric average for stressing the individual’s influence. Step 3. Apply Eq. (8) to calculating the accuracy number R(P) of Ai and Eq. (9) to calculating the accuracy number R(N) of Ai. Step 4. Compare the alternatives Ai(i = 1, 2, ..., m) according to the values of R(P ) if the differences between them are significant, otherwise take the values of R(N) into consideration and judge these alternatives on the negative aspect. Step 5. Rank all the alternatives in non-increasing order and determine the best one.
5 A REAL EXAMPLE ANALYSIS In this section, we choose the numerical example adapted from Herrera and Herrera-Viedma (2000) about a multicriteria decision making problem, which is also used in (J.Ye, 2009), (V. Lakshmana Gomathi Nayagam, S. Muralikrishnan, Geetha Sivaraman,2011), for the convenience of showing the rationality and meaningfulness of the method introduced above. Step 1. There is a panel with the following four possible alternatives to invest the money: A1 (car company), A2 (food company), A3 (computer company), A4 (arms company). The investment company must make a decision according to the following three attributes: C1 (the risk analysis), C2 (the growth analysis), C3 (the environmental impact analysis). DM evaluated the four alternatives by IVIFNs according to the 3 criteria and obtained the following matrix:
([0.4, 0.5],[0.3, 0.4]) ([0.6, 0.7],[0.2, 0.3]) M = ([0.3, 0.6],[0.3, 0.4]) ([0.7, 0.8],[0.1, 0.2])
([0.4, 0.6],[0.2, 0.4]) ([0.6, 0.7],[0.2, 0.3]) ([0.5, 0.6],[0.3, 0.4]) ([0.6, 0.7],[0.1, 0.3])
([0.1, 0.3],[0.5, 0.6]) ([0.4, 0.7],[0.1, 0.2]) ([0.5, 0.6],[0.1, 0.3]) ([0.3, 0.4],[0.1, 0.2])
(11)
Assume that the weights of C1, C2 and C3 are 0.35, 0.25 and 0.40, respectively. Step 2. Utilize Eq. (4) to calculate the weighted arithmetic average value (WAAV) of alternative Ai (i = 1, 2, 3, 4), considering that the values of weights are approximate, denoted by Fw (Ai):
Fw ( A1 ) = ([0.2944, 0.4590],[0.3325, 0.4704]); Fw ( A2 ) = ([0.5296, 0.7000],[0.1516, 0.2551]); Fw ( A3 ) = ([0.4375, 0.6000],[0.1933, 0.3565]); Fw ( A4 ) = ([0.5476, 0.6565],[0.1000, 0.2213]). Meanwhile, we use Eq. (6) to get the weighted geometric average value (WGAV) of alternative Ai (i = 1, 2, 3, 4), in contrast with WAAVs, denoted by Gw(Ai):
Gw ( A1 ) = ([0.2297, 0.4266],[0.4017, 0.4898]); Gw ( A2 ) = ([0.5102, 0.7384],[0.1614, 0.2616]); Gw ( A3 ) = ([0.4181, 0.6000],[0.2260, 0.3618]); Gw ( A4 ) = ([0.4799, 0.5864],[0.1000, 0.2236]). Step 3. Use Eq. (8) to calculate the accuracy R(P) of Aij depending on the WAAVs:
RAa1 ( P) = −1.4982 , RAa2 ( P) = 1.3474 , RAa3 ( P) = 0.2308 , RAa4 ( P) = 1.8742 . Based on the WGAVs, by Eq. (8) - 53 www.ivypub.org/msr
and Eq. (9), obtain the next values:
RAg1 ( P) = −1.7456 , RAg2 ( P) = 1.0894 , RAg3 ( P) = 0.0995 , RAg1 ( P) = 0.6225 ;
RAg1 ( N ) = 1.2316 , RAg2 ( N ) = 5.7585 , RAg3 ( N ) = 3.0353 , RAg4 ( N ) = 5.3341 . Step 4. It is obviously to see that alternatives A2 and A4 is better than A3; A3 are better than A1, according to the values of R(P). However, the difference between A2 and A4 is not significant, so we should consider the negative aspect with the function R(N) by Eq. (9) as: RA2 ( N ) = 5.7324 , RA4 ( N ) = 5.5972 . The comparisons based on the a
a
WGAVs have no confusion. Step 5. Determine the ranking order of all alternatives according to the WAAVs in decreasing order as the follows: A2 A4 A3 A1 , which emphasizes the negative aspect and corresponds with Ye’s (2009) results;
A4 A2 A3 A1 , which emphasizes the positive aspect and has some difference with Ye’s (2009) results. Depending on the EGAVs, the ranking derived from the positive accuracy function is in accordance with the one obtained by the negative side. Both of the two rankings are A2 A4 A3 A1 , which are the same with the conclusion adapted from (V.Lakshmana Gomathi Nayagam, S. Muralikrishnan, Geetha Sivaraman, 2011). By taking the EGAVs into consideration, we argue for another problem that for different weighted values choosing appropriate operator.
6 CONCLUSION In this paper, we propose a novel kind of accuracy function whose special characteristic defines in two aspects and establishes a method based on the new function for multi-criteria fuzzy decision making problems. This development contributes to the situation when the famous accuracy function proposed by Xu (2007) is confused with comparing between some two IVIFNs such as the examples shown in section 3. Utilizing the method introduced in this paper, we deal with the same example with Ye’s (2009). However, the conclusion is a little different with Ye’s, for the reason that we consider the IVIFNs with two aspects: the positive side and negative. Under the tendency of that people used to combining the membership degree and non-membership degree together for comparisons, since every coin has two sides, sometimes we should better think them separately, that maybe closer to the real life. Therefore this development contributes to the multi-criteria decision making method in a degree.
ACKNOWLEDGMENT This research was supported in part by National Natural Science Foundation of China (71272148), in part by the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039).
REFERENCES [1]
Herrera F., E. Herrera-Viedma. “Linguistic decision analysis: Steps for solving decision problems under linguistic information.”
[2]
F. Ye. “An extended TOPSIS method with interval-valued intuitionistic fuzzy numbers for virtual enterprise partner selection.”
[3]
J. Wu, Chiclana. “A social network analysis trust-consensus based approach to group decision-making problems with interval-
[4]
J. Ye. “Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy
[5]
K. Atanassov. “Intuitionistic fuzzy sets.” Fuzzy sets and Systems. 20 (1986): 87-96
[6]
K. Atanassov, G. Gargov. “Interval-valued intuitionistic fuzzy sets.” Fuzzy sets and systems. 31(1989): 343-349.
[7]
L. Zadeh. “Fuzzy sets.” Information and Control. 8 (1965): 338-353
[8]
L. Zadeh. “The concept of a linguistic variable and its applications in approximate reasoning.” Information Science. 8 (1975): 199-
Fuzzy Sets and Systems. 115 (2000): 67-82 Expert Systems with Applications. 37 ( 2010): 7050-7055 valued fuzzy reciprocal preference relations.” Knowledge-based systems. 59 (2014 ): 97-107 environment.” Expert Systems with Applications. 36 (2009): 6899-6902
249 [9]
M. Medineckene, E.K. Zavadskas, F. Bjork and Z. Turskis,. “Multi-criteria decision-making system for sustainable building - 54 www.ivypub.org/msr
assessment.” Archives of civil and mechanical engineering, 15 (2015): 11-18 [10] N. Chen, Z. Xu and M. Xia. “Interval-valued hesitant preference relations and their applications to group decision making.” Knowledge-Based Systems. 37 (2013): 528-540 [11] S. De, R. Biswas and A. Roy. “An application of intuitionistic fuzzy sets in medical diagnosis.” Fuzzy sets and systems. 117 (2001): 209-213 [12] V. Lakshmana Gomathi Nayagam, S. Muralikrishnan and Geetha Sivaraman. “Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets.” Expert Systems with Applications. 38 (2011): 1464-1467 [13] Y. Jiang, Z. Xu and X. Yu. “Group decision making based on incomplete intuitionistic multiplicative preference relations Information Sciences.” 295 (2015): 33-52 [14] Z. Xu. “A method based on distanve measure for interval-valued intuitionistic fuzzy group decision making. Information Sciences.” 180 (2010): 181-190 [15] Z. Xu, R. Yager. “Some geometric aggregation operators based on intuitionistic.” International Journal of General System. 35 (2006): 417-433 [16] Z. Xu. “Intuitionistic fuzzy aggregation operators.” IEEE Transactions on Fuzzy Systems. 15 (2007): 1179-1187. [17] Z. Xu. “Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making”. Control and Decision. 22 (2007): 215-219 [18] Z. Wang, Kevin W. Li and W. Wang. “An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights.” Information Science. 179 (2009): 3026-3040 [19] Z. Xu, M. Xia. “Hesitant fuzzy entropy measures and their use in multi-attribute decision making.” International Journal of Intelligent Systems. 27 (2012): 799-822
AUTHORS X.You (July, 1988). Study in Tianjin
T.Chen Professor of Tianjin University and major in
University for a doctorate of Management
Management and Economics.
and
Economics
in
management
and
decision making field, which will be received in 2018. Email: xinshangyou@163.com
- 55 www.ivypub.org/msr