Research and Applications in Economics Volume 3, 2016 doi: 10.14355/rae.2016.03.001
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Ranking Fuzzy Numbers with Goodness Criteria and Its Applications in Stock Performance Evaluation Li Zhang*1, Zhenyuan Wang2 Department of Economics, University of Nebraska at Omaha, 6001 Dodge ST Omaha, NE, USA
*1
Department of Mathematics, University of Nebraska at Omaha, 6001 Dodge ST Omaha, NE, USA
2
lwestman@unomaha.edu; 2zhenyuanwang@unomaha.edu
*1
Abstract This work summarizes some ranking methods for fuzzy numbers. A point view of reference system is proposed as a general way to establish ranking methods on sets of fuzzy numbers. Four goodness criteria of ranking methods are established, which can be applied further to decision-making on environment with uncertainty. The idea of ranking fuzzy numbers can also be used for ranking stock performance. As an application, a set of real NASDAQ data is adopted in an example to show the details of the ranking method by lexicography that is a reference system as well. Keywords Rankings; Fuzzy Numbers; Ranking Criteria; Reference Systems; Stock Evaluation
Introduction Fuzzy number [9, 20] is the most important mathematical quantities concerning fuzziness and vagueness which is hard to be ordered directly and intuitively. In literature, the concepts of ranking and ordering are frequently confused. As a matter of fact,ranking and ordering can be clearly distinguished under a mathematical concept, called relation. Ranking, as a relation on nonempty sets, can be established on a set of fuzzy numbers. Rankings on sets of fuzzy numbers play a critical role in data analysis with uncertainty. To be clear and intuitive, some reference systems,whose rankings have been already defined, are required. The most common reference system is the set of all real numbers with the natural ordering. In literature, there are various ranking techniques widely used in many areas such as decision-making and artificial intelligence, however, many of them do not lead to the unique commonly-recognized results [1-8, 10-14, 16-19, 21]. To eliminate the unreasonable ranking methods and reduce the candidates of effective ranking method, a number of criteria to judge the goodness of ranking techniques are necessary. In this work, the methods of reference systems are established and discussed for constructing ranking methods based on fuzzy number rankings established by lexicography as one of reference systems. Four proposed criteria are used to judge various examples of ranking methods. In fact, some ranking indexes of fuzzy numbers based on the centroid of area between the curve of their membership function and horizontal axis seem problematic under the judgement of those criteria. It is well-known that the stock’s daily performance is measured by a few numerical indexes, just as rankings of fuzzy numbers are determined by their parameters. Thus, we attempt to use a reasonable ranking method for fuzzy numbers to evaluate the daily performance of stock, such as real NASDAQ data. After the introduction, this paper is arranged as follows. Section 2 provides the fundamental knowledge of relations on nonempty sets and concepts of fuzzy numbers. In section 3, establishing rankings on given sets of fuzzy numbers is presented with some examples. Section 4 newly introduces the criteria for goodness of the rankings. In section 5, a fuzzy number ranking method is shown torank the daily stock performances by lexicography. Finally, conclusions are summarized in Section 6.
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Preliminaries Relations on Nonempty Sets Let be a nonempty set. relation, denoted by , on is a subset of that is related to if , denoted by simply.
, i.e.,
. For any
,we say
To discuss some common types of relation, we need several charactonyms that describe their properties.Let relation on . Reflexivity.Relation
is reflexive iff
for every
Symmetry.Relation
is symmetric iff
implies
Antisymmetry.Relation Transitivity.Relation
is antisymmetric iff is transitive iff
Total comparability.Relation
. for any and
and
be a
.
imply
imply
forany
for any
is totally comparable iff either
or
. .
holds for any
.
From the definitions of above charactonyms, we can see that the total comparability implies reflexivity, i.e., the requirement of reflexivity is weaker than comparability. However, for some types of relations such as partial ordering, we only need to consider the weaker requirement. A relationsatisfyingthe reflexivity and the transitivity is called a quasi-ordering. A relation satisfyingthe reflexivity, the antisymmetry, and the transitivityis called a partial ordering. A relation satisfyingthe reflexivity, the symmetry, and the transitivity is referred toequivalence; while a total orderingis a relation satisfyingthe antisymmetry, the transitivity, and the total comparability. As for a ranking, it is atransitive and totally comparable relation. It is evident that any ranking must be reflexive. These relations are illustrated in Table 1. Any total ordering is a ranking. The difference between ranking and ordering, including partial ordering and total ordering, is just the antisymmetry that is only possessed by orderings. To define a ranking on an uncountable set, a common way is to use a reference system that is an existing ranked set (or totally ordered set). Generally, let be a nonempty sets and let (T, be a ranked set, where relation be a ranking on . To define a ranking on , we just need to establish a mapping from to T, denoted by . In fact, we may define a relation on by if and only if for any . It is easy to see that relation on is transitive, and totally comparable and, therefore, is a ranking on . Here, ranked set (T, is the reference system used in the procedure of defining a ranking on . However, if we want to define a total ordering on nonempty set , generally, a totally ordered set as the reference system and a one-to-one mapping from to the totally ordered set are needed. Let (T, be a totally ordered set, where relation be a total ordering on T and let be a one-to-one mapping from to T. Similarly, defining if and only if for any , we may obtain relation on . Relation is reflexive, antisymmetric, transitive, and totally comparable and, therefore, is a total ordering on . The antisymmetryof relation is guaranteed by the property “one-to-one” of mapping. Such a method of using reference systems to define a ranking for fuzzy numbers is adopted in Sections 3, 4, and 5 of this paper. Usually, more ordered reference systems are needed for establish the above-mentioned mappings. Rankingsof fuzzy numbers are applied to decision making in fuzzy environment widely. TABLE 1. ELATIONS AND THEIR PROPERTIES
Quasi ordering Ranking Partial ordering Total ordering Equivalence
Reflexive
Symmetric
Antisymmetric
Transitive
Totally comparable
Fuzzy Numbers Let be a nonempty set, called the universal set. A fuzzy subset of is identified by its membership function [ ] It is a generalization of the concept of classical sets. By taking R , the set of all real numbers,
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the concept of real numbers can be generalized to fuzzy numbers as follows. Definition 1.A fuzzy subset ofR, denoted by ̃ , is called a fuzzy number when its membership function satisfies the following conditions: (FN1)
Set
, the support set of ̃ (denoted by supp ), is bounded.
(FN2)
Set
, the α-cut of ̃ (denoted by
), is a closed interval, denoted as [
],for every
R
[
]
].
The strong α-cut of fuzzy number ̃ is scrip set , denoted by for ]. So, the support set of ̃ is just . For any fuzzy number ̃ , the left and right branches of its membership function are denoted by and respectively in the following form {
(1)
where is nondecreasing on while is nonincreasing on , where is called a core point of ̃ . The core point of a fuzzy number may not be unique. Generally, if there is no confusion, the membership function of a fuzzy number is still denoted by , without any subscript. Only when we need to specify whose the membership function is, a subscript is used to indicate the fuzzy number. In the above expression, point is called a kernel point (or, core point) of the fuzzy number. For a given fuzzy number, its kernel points may not be unique. The set of all kernel points of fuzzy number ̃ is called its kernel (or, core), denoted by , which is a closed interval, and is just its α-cut with α = 1, i.e., . The support set of a fuzzy number is also an interval, but is not necessarily a closed interval. Any fuzzy number is a convex fuzzy subset ofR suchthat [ ] for any Rwith . This is equivalent to the condition that the α-cut is an interval for all ]. The sets of all fuzzy numbers is denoted by NF.From now on, we may omit the waive sign in the notation of fuzzy numbers for convenience. There are some special but common types of fuzzy numbers. For example, any rectangular fuzzy number, , can be denoted by vector [ ]with . Its kernel is closed interval [ ]. In fact, its α-cut is interval [ ] for all ]. The set of all rectangle fuzzy numbers is denoted by NI.Another example of common types is triangular fuzzy numbers. A triangular fuzzy number, ̃ , can bedenoted by vector [ ]. Its kernel is singleton { }and .The set of all triangular fuzzy numbers is denoted byNT. Generalizingboth rectangular fuzzy numbers and triangular fuzzy numbers, trapezoidal fuzzy numbers are commonly used as a practical type of fuzzy numbers. A trapezoidal fuzzy number, ̃ , can be identified by vector [ ].Its kernel is closed interval [ ]. The set of all trapezoidal fuzzy numbers is denoted byNP. Any set among NF,NI,NT, andNP is closed underoperators of addition and scalar multiplication[15] for fuzzy numbers. For example, the sum of two triangular fuzzy numbers is still a triangular fuzzy number. The set of nonnegative fuzzy numbers (i.e., with membership function satisfying for all ), denoted by N+, is closed under the fuzzy number operators addition and nonnegative scalar multiplication. Rankings on Sets of Fuzzy Numbers Ranking fuzzy numbers is a generalization of ordering real numbers. It should be consistent with the natural ordering of the real numbers which originates from the natural ordering of the nonnegative integers and serves as the most basic reference system for ranking fuzzy numbers. Let be a set of fuzzy numbers closed under the addition and the (nonnegative) scalar multiplication and let T be a nonempty set with ranking .Ranked set ( , ) serves as a reference system. Usually, we take R with its natural ordering as T. Any mapping can determine a ranking on , in other words, ranking a set of fuzzy numbers just requires us to finda mapping from set to the real line as mentioned in Section 3, and then define relation on by if and only if for any . Mapping is not necessarily one-to-one. Relation is called the ranking introduced by mapping and ̃ is called the rank of fuzzy number ̃ for any ̃ . Let us see some examples of constructing rankings on sets of fuzzy numbers. Example 1. For any fuzzy number ̃ with membership function , its kernel
=
, the α-cut of ̃ when
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], . Defining mapping
, is a closed interval, denoted by[ NF
̃
(2)
̃ whenever
We may get a ranking from (2),which names ̃
, i.e.,
,defined on NF.
Example 2. For any fuzzy number ̃ with membership function , its support set Supp with end points . Defining mapping
is an interval
NF ̃
(3)
we may get a ranking defined on NFby (3) as well. Example 3. Defining mapping N+ ̃
(4)
where ranking index , i.e., the rank of nonnegative fuzzy number ̃ , is the distance from the origin to the centroid of the area between the curve of its membership function and the horizontal axis, we may get a ranking [14] defined on setN+. Fig. 1 illustrates index d of triangular fuzzy number[1 2 6], where
√
.
Example 4. The expansion center (medium) of a fuzzy number can be used as a ranking index. For any given fuzzy number ̃ with membership function , its expansion center, similar to the medium of probability distribution in statistics, is a real number such that ∫ [17]. A ranking on NFcan be introduced by ∫ mapping NF ̃
(5)
Example 5. For any given fuzzy number ̃ with membership function , the w-center of ̃ is defined as [21], where is a core point of ̃ . A ranking on NF can be introduced by mapping ∫ ∫ NF ̃ is independent of the choice of
(6)
when the core points are not unique.
The point view of mapping used for establishing ranking can be generalized to the case that the core domain of the mapping may be high-dimensional Euclidian space. It is just the lexicography.
1
(3,
d 0 1 2 3 4 56789 xx
FIGURE 1.THE MEMBERSHIP FUNCTION OF ̃ IN EXAMPLE 3.
Example 6. Lexicography can be used for ranking any fuzzy numbers. For a give fuzzy number ̃ with kernel, [
4
], denote its midpoint
by
and support set 〈
〉, where the pair of parentheses 〈
〉 may be any
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one among ( ), ( ], [ ), and [ ]. Two reference systems are used here for defining a ranking: one is the set of real numbers with the natural ordering, i.e., the totally ordered set(R, another is the totally ordered set ( , where are regarded as the ranking parameters. Then a ranking defined on the set of all fuzzy numbers can be established as follows.Let ̃, ̃ be two fuzzy numbers with ranking parameters , , , and , , , respectively. We say that ̃ ̃ iff that , or that when , or that when and ; we say that ̃ ̃ iff that , and ; we say that ̃ ̃ iff ̃ ̃ and ̃ ̃ . Then relation is a ranking on the set of all fuzzy numbers. Criteria for Goodness of Rankings on Sets of Fuzzy Numbers Let be a nonempty set of fuzzy numbers and ranking be introduced by mapping . For any fuzzy number ̃ , ̃ is its rank.In this section, anyone among NF,NI, NT, and NP may be taken as . On these sets of fuzzy numbers, binary operators such as addition, subtraction, multiplication, and division are well defined [15].We also know that anyone among NF,NI, NT, and NP is closed under addition, subtraction, and scalar multiplication; while NF and NI are also closed under any operator among multiplication, division, maximum, and minimum respectively [15]. Before introducing some criteria for measuring the goodness of rankings on sets of fuzzy numbers, we define the strong convergence of a fuzzy number sequence in NF and a partial ordering on NI, the set of all rectangular fuzzy numbers, i.e., all closed intervals. Definition 2. Let ̃ be a sequence of fuzzy numbers, simply denoted by ̃ . Sequence ̃ is strongly convergent to fuzzy number ̃ iff { and { converge to and , respectively, with respect to ] uniformly.In other words, for any small real number , there exists positive integer such that and for all and ]. Fuzzy number ̃ is called the limit of fuzzy number sequence ̃ . ̃
Example 7. Let
be a sequence of triangular fuzzy numbers with ̃
[
],
,
(7)
and ̃ , which is regarded as a special fuzzy number. Then ̃ strongly converges to ̃ . However, the sequence of their membership function, }, does not uniformly, with respect to R, converge to the membership function of ̃ , the characteristic function of singleton consisting of one real number 1, i.e., { though
does converge to
Example 8. Let
̃
,
R,
(8)
everywhere in R. be a sequence of fuzzy numbers with membership functions {
[
] R
(9)
Then, the sequence of their membership function, }, uniformly converges to the membership function of fuzzy number ̃ . However, does not converge to strongly. From Examples 7 and 8, we can see that the concept of strong convergence of a sequence of fuzzy numbers is different from the concept of uniform convergence of its membership function sequence. The following are some criteria for goodness of rankings on sets of fuzzy numbers. Continuity:We say that ranking on is continuous if for any fuzzy number sequence ̃ ̃ strongly converges to fuzzy number ̃ the corresponding ranking sequence ̃ converges to
that ̃ .
Geometric intuition:We say that ranking on is geometrically intuitive if, for any two fuzzy numbers ̃ and ̃ in , for all ] implies ̃ ̃ . Intuitively, geometric intuition means that, a fuzzy number precedes another if the nonzero part of the membership function of this fuzzy number is totally on the left (including the same location) of another’s. Satisfying the requirement of geometric intuition is consistent with the natural ordering of the real numbers.
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̃ and ̃
Hereditability for addition: ̃
̃ imply ̃
̃ + ̃ for any ̃ ̃ , ̃ ̃
̃
̃ implies ̃
Hereditability for nonnegative scalar multiplication: ̃ number c.
̃ for any ̃ ̃
. and any nonnegative real
Rankings shown in Examples 1, 2, and 5 are continuous, geometrically intuitive, and hereditable for both addition and nonnegative scalar multiplication.The ranking shown in Example 6 is geometrically intuitive and hereditable for both addition and nonnegative scalar multiplication. It is not difficult to know that the ranking method in Example 6 is also continuous when the definition of continuity is generalized to high-dimensional Euclidian space. Therefore, the ranking shown in example 6 is feasible and can be applied to stock performance evaluation shown in section 5. Rankings shown in Example 3 is hereditable for both addition and nonnegative scalar multiplication. However, the ranking shown in Example 3 is neither continuous nor geometrically intuitive. Some counterexamples are presented in the next three examples. Example 9. Triangular fuzzy number sequence ̃ to fuzzy number ̃ The centroid of ̃ is point (1, numbers in this sequence,
shown in (10) is adopted now. It strongly converges for all . According to (4), ranking indices of fuzzy
(the distance from the origin to the centroid of the area between the curve of its
membership function and the horizontal axis), are always √
= √
1.054. However, since the centroid of
√
= √
1.118. So,
fuzzy number ̃
is point (1,
, its ranking index is
does not converge to d.
This means that the ranking of fuzzy numbers shown in Example 3 is not continuous.
1
1
0
0
2
4
6
8
10
12
14
0
16
18
20
2
4
6
8
10
12
14
16
18
20
x
x
0 FIGURE 2.THE MEMBERSHIP FUNCTIONS OF ̃ and ̃ IN EXAMPLE 10.
Example 10.A rectangular fuzzy number ̃ centroid of ̃ is point the ranking index of ̃is
[
]
while the centroid of ̃ is point √
√
triangular fuzzy number ̃
[
] are adopted. The
. According to the definition shown in Example 3, , while the ranking index of ̃ is
√
. However, from Fig. 2 we may see that the nonzero part of the membership function of ̃ is totally on the left of ̃ ’s. So, the ranking shownin Example 3 is not geometrically intuitive. √
Evaluation of Stock Performanceby Fuzzy Numbers’ Ranking Method Similar to the comparison of the magnitude of fuzzy numbers by a few parameters, stock performance can be evaluated by a few indexes. Thus, a corresponding relationship can be established from stock performance indexes of interest such as close, high, and low to parameters of fuzzy number, the middle point of core, right endpoint of the support set, and left endpoint of the support set, respectively. Therefore, methods for ranking fuzzy numbers shown in section 3, especially in example 6, can be applied to evaluate stock performance such as NASDAQ composite prices (in a week), retrieved from http://finance.yahoo.com/q/hp?s=%5Eixic+historical+prices, as shown in Table 2. To establish the above-mentioned corresponding relationship, the data should be pre-processed, i.e., normalized, to be indexes for close, high, low, denoted by They can be calculated as follows:
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(10) After rounding the results (take four significance digits), these indexes converted by (10) are listed in Table 3. TABLE 2.NASDAQ HIGH-LOW-CLOSE INDEXES FROM APRIL 17 TO APRIL 24, 2015
Date ( ) Apr 24, 2015 (5) Apr 23, 2015 (4) Apr 22, 2015 (3) Apr 21, 2015 (2) Apr 20, 2015 (1) Apr 17, 2015 (0)
High ( ) 5,100.37 5,073.09 5,040.65 5,028.22 5,000.20 4,974.09
Low ( ) 5,081.21 5,019.29 4,992.62 5,009.51 4,952.68 4,912.33
Close ( 5,092.08 5,056.06 5,035.17 5,014.10 4,994.60 4,931.81
TABLE 3.NORMALIZED DATA FROM TABLE 2.
Date (t) Apr 24, 2015 (5) Apr 23, 2015 (4) Apr 22, 2015 (3) Apr 21, 2015 (2) Apr 20, 2015 (1)
0.0071 0.0042 0.0042 0.0039 0.0127
0.0088 0.0075 0.0053 0.0067 0.0139
0.0050 -0.0031 -0.0043 0.0030 0.0042
Those three indexes can be made correspondence to three critical parameters of fuzzy numbers shown in Example 6as , respectively. Thus, the lexicography shown in Example 6 can be used here to rank stock performance. According to lexicography, should be compared first. From Table 3, we have .
(11)
Since , their corresponding should be compared. In fact, . Consequently, a reasonable ranking for NASDAQ performance from April 20 to April 24 in 2015 can be expressed as .
(12)
Conclusions This work presents some ranking methods on sets of fuzzy numbers. Regarding rankings as relations on the set of fuzzy numbers, a general view point and methods concerning reference systems are adopted. Series of ranking criteria are newly introduced to ensure the goodness of ranking method on sets of fuzzy numbers. Besides, this work attempts to apply a similar rankinginto stock performance evaluation by lexicography, which is a feasible method and may extend further applications in economic and financial areas. Further, the proposed criteria may generally be applied in solving various ranking problems in financial and economic areas. REFERENCES
[1]
Abbasbandy,Saeid.,and BabakAsady.“Ranking of fuzzy numbers by sign distance.” Information Sciences 176(16), 24052416, 2006.
[2]
[2Bortolan,Giovanni.,and Degani, Rosanna.“Areview of some methods for ranking fuzzy numbers.” Fuzzy Sets and Systems 15(1), 1-19, 1985.
[3]
Cheng,Ching H.“A new approach for ranking fuzzy numbers by distance method.” Fuzzy Sets and Systems 95(3), 307-317, 1998.
[4]
Chu,Ta C.,and Chung T. Tsao.“Ranking fuzzy numbers with an area between the centroid point and original point.” Computers and Mathematics with Applications 43(1-2), 111-117, 2002.
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www.seipub.org/rae
[5]
Research and Applications in Economics Volume 3, 2016
Dubois,Didier.,and Henri Prade. “Ranking fuzzy numbers in the setting of possibility theory.” Information Sciences 30(3), 183-224, 1983.
[6]
Farhadinia,Bahram.“Ranking fuzzy numbers on lexicographical ordering.”International Journal of Applied Mathematics and Computer Sciences 5(4), 248-251, 2009.
[7]
Furukawa, N. “A parametric total order on fuzzy numbers and a fuzzy shortest route problem.” Optimization 30(4), 367377, 1994.
[8]
HassasiNadar.,and Rahim Saneifard. “On the central value of fuzzy numbers.” Australian Journal of Basic and Applied Sciences 7(9), 1146-1152, 2011.
[9]
Klir,George. J.,and Bo Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, 1995.
[10] Liou,Tian.S.,and Mao.J. Wang. “Ranking fuzzy numbers with integral value.” Fuzzy Sets and Systems 50(3), 247-255, 1992. [11] Nasseri,Seyed. H.,and M. Sohrabi. “Hadi’s method and its advantage in ranking fuzzy numbers.” Australian Journal of Basic Applied Sciences 4(10), 4630-4637, 2010. [12] Ramík,Jaroslav.,and Josef Řim{nek.“Inequality relation between fuzzy numbers and its use in fuzzy optimization.” Fuzzy Sets and Systems 16(2), 123-138, 1985. [13] Wang, Yu. J., and Hsuan. SLee.“The revised method of ranking fuzzy numbers with an area between the centroid point and original point.” Computers and Mathematics with Applications 55(9), 2033-2042, 2008. [14] Wang, Ying. M., Jian. B Yang., Dong. L. Xu., and Kwai. S. Chin.“On the centroid of fuzzy numbers.” Fuzzy Sets and Systems 157(7), 919-926, 2006. [15] Wang, Zhen. Y., YangRong., and K. S. Leung. Nonlinear Integrals and Their Applications in Data Mining, World Scientific, 2010. [16] Wang,Zhen. Y.,and Li Zhang-Westman.“The cardinality of the set of all fuzzy numbers.” Proc. IFSA, 1045-1049, 2013. [17] Wang,Zhen. Y.,and Li Zhang-Westman. “Ranking fuzzy numbers by their expansion center.” Proc. IEEE-WCCI, 2533-2536, 2014. [18] Wu, Hsien. C.“Decomposition and construction of fuzzy sets and their applications to the arithmetic operations on fuzzy quantities.” Fuzzy Sets and Systems 233,1-25, 2013. [19] Yao,Jing.S. and KweimeiWu.“Ranking fuzzy numbers based on decomposition principle and signed distance.” Fuzzy Sets and Systems 116(2), 275-288, 2000. [20] Zadeh, Lotfi.A., Fuzzy sets, Information and Control 8, 338-353, 1965. [21] Li Zhang-Westman.,and Zhen. Y. Wang.“Ranking fuzzy numbers by their left and right wingspans.” Proc. IFSA, 1039-1044, 2013.
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