decision-theoretic three-way approximations of fuzzy sets

Information Sciences xxx (2014) xxx–xxx

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Information Sciences

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Decision-theoretic three-way approximations of fuzzy sets

http://dx.doi.org/10.1016/j.ins.2014.04.0220020-0255/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +1 306 585 5226.E-mail addresses: [email protected] (X. Deng), [email protected] (Y. Yao).

Please cite this article in press as: X. Deng, Y. Yao, Decision-theoretic three-way approximations of fuzzy sets, Inform. Sci. (2014)dx.doi.org/10.1016/j.ins.2014.04.022

Xiaofei Deng, Yiyu Yao ⇑Department of Computer Science, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

a r t i c l e i n f o

Article history:Received 8 January 2013Received in revised form 20 September2013Accepted 8 April 2014Available online xxxx

Keywords:Approximations of fuzzy setShadowed setThree-way decisionTripartition

a b s t r a c t

A three-way, three-valued, or three-region approximation of a fuzzy set is constructedfrom a pair of thresholds ða; bÞ on the fuzzy membership function. An element whosemembership grade equals to or is greater than a is put into the positive region, an elementwhose membership grade equals to or is less than b is put into the negative region, and anelement whose membership grade is between b and a is put into the boundary region. Afundamental issue is the determination and interpretation of the required pair ofthresholds. In the framework of shadowed sets (i.e., an example of three-way approximationsof fuzzy sets), Pedrycz provides an analytic solution to computing the thresholds by searchingfor a balance of uncertainty introduced by the three regions. To gain further insights intothree-way approximations of fuzzy sets, we introduce an alternative decision-theoreticformulation in which the required thresholds are computed by minimizing decision cost.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Fuzzy sets extend the classical sets by allowing graded membership values so that one can describe a concept that hasan unsharp, gradually changing boundary [41]. The use of the unit interval ½0;1� as the set of membership grades has bothadvantages and shortcomings. On the one hand, an infinite set of values provides a high degree of flexibility and a greatexpressive power mathematically. One can distinguish objects at minute details by using many levels of membershipgrade. On the other hand, the infinity number of values leads to a difficulty in interpreting and understanding a fuzzymembership function in practice. In many situations, our perception of fuzziness is of a qualitative nature. We may beonly able to distinguish objects by using a few levels of fuzziness. A classical study of Miller [19] shows that humancan only process about seven plus or minus two units of information. More recent studies suggest that the actual numberis smaller and is around four [8]. The use of a few grade levels has cognitive advantages. In practical applications, it mayalso happen that we do not need to distinguish objects that are very similar to each other. An approximation may be suf-ficient, as a very precise membership value may not offer much more significant information. In addition, estimating andrepresenting uncertainty or vagueness by a numeric value usually associate with a cost of observation or an error of esti-mation. Obtaining a more precise numeric value usually leads to a higher cost although a lower error. There is a trade-offbetween accuracy and cost. Accordingly, approximating a fuzzy set by using several levels of grade is of practical impor-tance. An approximation of a fuzzy set by using only a few grade values may simplify a problem and, more often than not,is practically sufficient.

The concept of shadowed sets, proposed by Pedrycz [23,26–28], is an example of three-way, three-valued, or three-regionapproximations of a fuzzy set. It has received much attention in recent years in theory and applications

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2 X. Deng, Y. Yao / Information Sciences xxx (2014) xxx–xxx

[2,4,7,12–15,18,20,22–24,30–32,40,43,44]. Intuitively, the construction of a shadowed set is based on the followingprinciples. If the membership grade of an element is close to 1, it would be considered to be the same as 1 and is elevatedto 1; if the membership grade is close to 0, it would be considered to be the same as 0 and is reduced to 0; if the membershipgrade is neither close to 1 nor close to 0, it would be put into a shadowed region. The elevation and reduction operations usethresholds that provide semantically meaningful and acceptable levels of degree of closeness of membership value to 1 and0, respectively.

A fundamental issue of three-way approximations of fuzzy sets is the interpretation and determination of thresholds. Insearching of the optimal thresholds, Pedrycz [23] suggests a method by minimizing an objective function that provides abalance of uncertainty characterized by a fuzzy set. More specifically, the objective function is expressed as the differencebetween the shadowed area and the sum of the increased and decreased areas by the elevation and reduction operations.The approach works well computationally. On the other hand, we can improve the method from several aspects. The min-imization procedure provides an analytic solution when the thresholds are related, i.e., they add up to 1. For a pair of unre-lated thresholds, one may not have an analytic solution. Pedrycz’s objective function is one of many possible ways tointerpret and compute thresholds in terms of uncertainty. It is more useful to investigate the physical meaning of variousobjective functions based on more operational notions such as error rate and cost. By applying results from decision-theo-retic rough sets (DTRS) [39] and three-way decisions [35–37], this paper solves these difficulties by introducing a model ofdecision-theoretic three-way approximations of fuzzy sets.

A basic idea of three-way decisions is to classify a set of objects into three regions, called the positive, negative andboundary regions, by using an evaluation function and a pair of thresholds [37]. If we interpret the membership functionof a fuzzy set as an evaluation, we can immediately obtain a new interpretation of a three-way approximation of a fuzzyset. For each object, one can make one of three decisions: elevate the membership grade to 1, reduce the membership gradeto 0, or change the membership grade to a third intermediate value. The elevation of a membership grade to 1 means thatone accepts an object to be an instance of the concept represented by a fuzzy set, as its membership grade is close to 1. Thereduction of a membership grade to 0 means that one rejects the object to be an instance of the concept, as its membershipgrade is close to 0. For an object whose membership grade is neither close to 1 nor close to 0, one would like to make a non-commitment decision by using a third value. Each decision is associated with some errors and costs. Such a three-way deci-sion interpretation offers naturally a model for constructing three-way approximations by using the notions of error andcost. One can obtain an optimal pair of thresholds by minimizing overall error or cost of three-way approximations.

The rest of this paper is organized as follows. Section 2 reviews basic concepts of fuzzy sets, three-way approximations offuzzy sets and shadowed sets from the viewpoint of three-way decisions. Section 3 proposes an error-based interpretation ofshadowed sets. It shows the limitations of the existing formulation of shadowed sets and, therefore, motivates the introduc-tion of a decision-theoretic model. Section 4 is a detailed derivation of the decision-theoretic model of three-way approxi-mations of fuzzy sets. Section 5 discusses two special models, namely, the error-based ð0:75;0:25Þ-model and a symmetricða;1� aÞ-model.

2. Approximations of fuzzy sets with three-way decisions

This section reviews basic concepts of fuzzy sets, three-way approximations of fuzzy sets and shadowed sets within theframework of three-way decisions.

2.1. Three-way decisions

By observing a common practice of decision making across many disciplines, Yao [37] proposes a general framework ofthree-way decisions. Suppose U is a universal set of objects and C is a set of conditions called criteria. An object in U satisfies,does not satisfy the criteria or only satisfies the criteria to a certain degree. Three-way decisions deal with the classificationof objects based on the set of criteria. According to an evaluation of the satisfiability of objects, one can make three-way deci-sions as follows:

1. Accept an object as satisfying the set of criteria if its degree of satisfiability is at or above a certain level.2. Reject the object by treating it as not satisfying the criteria if its degree of satisfiability is at or below another level.3. Neither accept nor reject the object but opt for a noncommitment or deferment decision that needs further investigation.

With three-way decisions, objects are classified into three regions, called the positive, negative and boundary regions,respectively. Three-way decisions may be considered as a generalization of two-way decisions. The introduction of a thirdoption of noncommitment leads to more flexibility. Intuitively, the set of criteria and the notion of satisfiability may be inter-preted by using more practical and operational concepts of, for example, costs, risks, profits, rates of error and so on. Thevalues of the evaluation are the degrees of satisfiability or desirability of objects with respect to the set of conditions. Forthis purpose, the set of values must be equipped with an order relation. In this paper, we adopt a special model of three-way decisions that uses a totally ordered set in which any two elements are comparable [37].

Please cite this article in press as: X. Deng, Y. Yao, Decision-theoretic three-way approximations of fuzzy sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.04.022



X. Deng, Y. Yao / Information Sciences xxx (2014) xxx–xxx 3

Suppose ðL;�Þ is a totally ordered set, where � is a total order, that is, � is reflexive (8a 2 L; a � a holds), transitive(8a; b; c 2 L; a � b ^ b � c ) a � c), and comparable (8a; b 2 L, either a � b or b � a holds). For a pair of thresholds withb � a (i.e., b � a ^ :ða � bÞ), we construct the set of designated values for acceptance Lþ ¼ ft 2 Ljt � ag and the set of des-ignated values for rejection L� ¼ ff 2 Ljf � bg. By applying the pair of thresholds ða; bÞ to an evaluation v : U ! L, one canobtain the ða; bÞ-positive, negative and boundary regions as follows:

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POSða;bÞðvÞ ¼ fx 2 UjvðxÞ 2 Lþg ¼ fx 2 UjvðxÞ � ag;NEGða;bÞðvÞ ¼ fx 2 UjvðxÞ 2 L�g ¼ fx 2 UjvðxÞ � bg;BNDða;bÞðvÞ ¼ ðPOSða;bÞðvÞ [NEGða;bÞðvÞÞc ¼ fx 2 Ujb � vðxÞ � ag: ð1Þ

The three regions are pair-wise disjoint and their union is the entire universe U. They do not necessarily form a partition of U,as some of the regions may be empty. In this paper, for convenience, we call the family of three regions a tripartition of theuniverse, although some of the regions may be empty.

2.2. Three-way approximations of a fuzzy set

A fuzzy set represents a concept with an unsharp, gradually changing boundary [41]. Formally, a fuzzy set over a universalset U is defined by a membership function:

lA : U ! ½0;1�; ð2Þ

where ½0;1� is the unit interval. The numeric value lAðxÞ 2 ½0;1� is called the membership value or grade of x 2 U in the fuzzyset A. Intuitively, an object with a full membership grade of 1 is viewed as a typical instance of the concept, an object with amembership grade of 0 is viewed as a non-instance of the concept, and a higher membership grade implies that an objectbelongs more to the concept.

Qualitatively, one may use a pair of core and support to approximate a fuzzy set [11,17]:

COREðlAÞ ¼ fx 2 UjlAðxÞ ¼ 1g ¼ fx 2 UjlAðxÞP 1g;SUPPORTðlAÞ ¼ fx 2 UjlAðxÞ – 0g ¼ fx 2 UjlAðxÞ > 0g: ð3Þ

To be consistent with three-way decisions, we can approximate a fuzzy set by three regions, namely, the positive, negativeand boundary regions:

POSðlAÞ ¼ fx 2 UjlAðxÞP 1g ¼ COREðlAÞ;NEGðlAÞ ¼ fx 2 UjlAðxÞ 6 0g ¼ ðSUPPORTðlAÞÞ

c;

BNDðlAÞ ¼ fx 2 Uj0 < lAðxÞ < 1g ¼ SUPPORTðlAÞ � COREðlAÞ: ð4Þ

The qualitative approximation of a fuzzy set uses only the two extreme points of the unit interval ½0;1�, namely, the fullmembership grade 1 and the membership grade 0.

The qualitative approximation of a fuzzy set may be very restrictive. An object is not put into the positive region eventhough its membership grade is almost the same as 1, and an object is not put into the negative region even though its mem-bership grade is almost the same as 0. All objects with non-full or non-zero membership grades are put into the boundaryregion. To resolve this restriction, in his seminal paper Zadeh [41] suggests the following three-valued approximations of afuzzy set:

‘‘. . . one can introduce two levels a and b (0 < a < 1, 0 < b < 1;a > b) and agree to say that (1) ‘x belongs to A’ if fAðxÞP a;(2) ‘x does not belong to A’ if fAðxÞ 6 b; and (3) ‘x has an indeterminate status relative to A’ if b < fAðxÞ < a. This leads to athree-valued logic (Kleene, 1952) with three truth values: T (fAðxÞP a), F (fAðxÞ 6 b), and U (b < fAðxÞ < a).’’

In this quote, fAðxÞ : U ! ½0;1� denotes a fuzzy set and Kleene’s book is given by Ref. [16]. Quantitative approximations of afuzzy set by a pair of thresholds ða; bÞ can be viewed as a specific model of three-way decisions. The three regions are definedrespectively by:

POSða;bÞðlAÞ ¼ fx 2 UjlAðxÞP ag;NEGða;bÞðlAÞ ¼ fx 2 UjlAðxÞ 6 bg;BNDða;bÞðlAÞ ¼ fx 2 Ujb < lAðxÞ < ag: ð5Þ

Zadeh’s three cases correspond to the three regions of three-way decisions. For example, for case (1), we accept an object tobe an element of A with an understanding that its membership grade may not be 1, but is only close to 1.

A three-way approximation of a fuzzy set can be formally defined as a three-valued fuzzy set TlAðxÞ : U ! fn; b; pg,

TlA ðxÞ ¼p; x 2 POSða;bÞðlAÞ;n; x 2 NEGða;bÞðlAÞ;b; x 2 BNDða;bÞðlAÞ;

8><>:

ð6Þ

cite this article in press as: X. Deng, Y. Yao, Decision-theoretic three-way approximations of fuzzy sets, Inform. Sci. (2014), http://.org/10.1016/j.ins.2014.04.022




where p; n and b denote three membership grades corresponding to the positive, negative and boundary regions, respec-tively. Following discussion from three-valued logic [10], we consider two orderings of the three membership grades. Adegree-of-truth ordering is given by n �t b �t p, that is, n is the least membership grade, p is the greatest membership grade,and b is an intermediate membership grade. A degree-of-information is given by b �i n and b �i p. That is, membershipgrades n and p are more informative than b. In the light of the two orderings, the choice of a set of particular symbols isnot important, as long as it preserves the two orderings. We typically use 0 to represent n; 1

2 to represent b, and 1 to representp.

Banerjee and Pal [1] and Chakrabarty et al. [5] consider similar approximations of a fuzzy set by a pair of ordinary sets inthe framework of rough sets.

2.3. Shadowed sets

An unresolved issue with Zadeh’s proposal is that there is a lack of a theory for interpreting and determining the requiredpair of thresholds. Pedrycz [23–27,29] proposes a framework of shadowed sets that gives one solution to this problem.

A shadowed set S on U is defined as a mapping from U to the set f0; ½0;1�;1g, that is, S : U ! f0; ½0;1�;1g. Elements of Uwith membership grade 1 constitute the core of S and elements with membership grade ½0;1� form the shadow of S. It is pos-sible that either the core or the shadow of a shadowed set is empty. It may be commented that this definition of a shadowedset is not necessarily related to a fuzzy set.

For the application of a shadowed set as an approximation of a fuzzy set, Pedrycz [23] suggests a constructive method.Given a pair of thresholds ða; bÞ with 0 6 b < a 6 1, one can construct a shadowed set from a fuzzy set lA as follows [23]:

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SlA ðxÞ ¼1; lAðxÞP a;0; lAðxÞ 6 b;

½0;1�; b < lAðxÞ < a:

8><>:

ð7Þ

To be consistent with Zadeh’s formulation and the theory of three-way decisions, we use labels, notations and conventionthat are slightly different from ones used by Pedrycz. In terms of three-way decisions, a shadowed set can be convenientlyinterpreted as three regions: the positive region defined by membership grade 1, the negative region defined by membershipgrade 0, and the boundary region defined by membership grade ½0;1�. The boundary region is in fact the shadowed region.

An important contribution of shadowed sets is a new interpretation of approximations of a fuzzy set. The membershipfunction of a shadowed set can be viewed as a modification of a fuzzy membership function as depicted in Fig. 1. For anobject x, we elevate the membership grade from lAðxÞ to SlA ðxÞ ¼ 1, if lAðxÞP a; we reduce the membership grade fromlAðxÞ to SlA ðxÞ ¼ 0, if lAðxÞ 6 b; we change the membership grade from lAðxÞ to SlA ðxÞ ¼ ½0;1�, if b < lAðxÞ < a. Anothercontribution of shadowed sets is a systematic way to compute the pair of thresholds ða; bÞ. Pedrycz [23] introduces a methodby minimizing an objective function that characterizes the uncertainty of a shadowed set. Consider a shadowed set approx-imation of a fuzzy set as depicted in Fig. 1. By comparing membership functions of a fuzzy set and a shadowed set, we canidentify three areas as shown in Fig. 1: the elevated area of membership values, the reduced area of membership values, andthe shadow or shadowed area. The three areas represent changes from the fuzzy set membership function into a shadowedset membership function.

Based on the three areas, Pedrycz [23] suggests that an optimal pair of thresholds should satisfy the following condition:

Elevated Areaða;bÞðlAÞ þ Reduced Areaða;bÞðlAÞ ¼ Shadowða;bÞðlAÞ: ð8Þ

That is, the area of shadow is the sum of the elevated area and the reduced area. In practical situations, it may be difficult tofind a pair of thresholds that satisfies this condition, particularly, when U is a finite universe. Instead, Pedrycz [23] proposesto minimize the following absolute difference:

Fig. 1. A shadowed set approximation of a fuzzy set.





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V ða;bÞðlAÞ ¼ jElevated Areaða;bÞðlAÞ þ Reduced Areaða;bÞðlAÞ � Shadowða;bÞðlAÞj

¼X

lAðxÞPa

ð1� lAðxÞÞ þX

lAðxÞ6b

lAðxÞ � cardðfx 2 Ujb < lAðxÞ < agÞ

��

��; ð9Þ

where cardð�Þ denotes the cardinality of a set. For simplicity, we assume that universe U is finite. Hence, summation is used inEq. (9). An optimal pair of thresholds ða; bÞ can be obtained by taking the arguments that minimize the objective functionV ða;bÞðlAÞ:

arg minða;bÞ

V ða;bÞðlAÞ: ð10Þ

Minimization involving two parameters may have computational difficulties. Consequently, Pedrycz [23,25–27] furtherassumes that the pair of thresholds is related by aþ b ¼ 1. In this special case, the objective function (9) can be simplifiedinto:

V ða;1�aÞðlAÞ ¼X

lAðxÞPa


lAðxÞ61�a

lAðxÞ � cardðfx 2 Uj1� a < lAðxÞ < agÞ

��

��: ð11Þ

For many types of fuzzy membership functions, one can obtain analytic solutions of a. For more details, see references [23–27].

2.4. Two senses of the notion of shadowed sets

Pedrycz’s formulation of the notion of shadowed sets consists of two main steps. The first step is to define a shad-owed set as a mapping from U to f0; ½0;1�;1g, independent of a fuzzy set, which is equivalent to a tripartition of U. Thesecond step is to construct a shadowed set as an approximation of a fuzzy set. This leads to different views on themeaning of the notion of shadowed sets. Following Zadeh’s discussions on two senses of the notion of fuzzy logic[42], we consider two senses of the notion of shadowed sets. In a wide sense, we interpret the notion of shadowed setsbased only on the form and interpretation of a shadowed set. That is, a shadowed set is a three-valued fuzzy set, whichcan be used to approximate a fuzzy set. We focus more on the general idea rather than detailed formulation and par-ticular choice of the set of three levels of grade. In other words, an interpretation of the notion of shadowed sets in awide sense is based on the first step of Pedrycz’s formulation. This view has been used by Cattaneo and Ciucci [2–4] forstudying algebraic structure of shadowed sets and by Deng and Yao [9] in the framework of decision-theoretic shadowedsets (DTSS). In a narrow sense, we interpret the notion of shadowed sets according to Pedrycz’s exact formulation,namely, the choice of the set of membership grades f0; ½0;1�;1g and the objective function as defined by Eq. (9). Thatis, an interpretation of the notion of shadowed sets in a narrow sense is based on the both steps of Pedrycz’sformulation.

The interpretation in the wide sense allows us to generalize shadowed sets by considering different formulations. Theinterpretation in the narrow sense may unnecessarily limit further development of shadowed sets. In this paper, we willuse ‘‘three-way approximations of fuzzy sets’’ to denote the interpretation in the wide sense and use ‘‘shadowed sets’’ todenote the interpretation in the narrow sense. Thus, shadowed sets are an example of three-way approximations of fuzzysets.

3. An error-based interpretation of shadowed sets

This section provides a detailed analysis of Pedrycz’s objective function in terms of errors of approximations and intro-duces a new objective function by the total error of approximations. The results motivate the introduction of decision-the-oretic three-way approximations in the next section.

3.1. An analysis of Pedrycz’s objective function

In order to gain further insights into objective function (9), we can re-express it in terms of the errors introduced by ashadowed set approximation. Given an object x with membership grade lAðxÞ, the elevation operation changes the member-ship grade from lAðxÞ to 1, the reduction operation changes the membership grade from lAðxÞ to 0, and the errors inducedby elevation and reduction are given respectively by:

EeðlAðxÞÞ ¼ 1� lAðxÞ;ErðlAðxÞÞ ¼ lAðxÞ � 0 ¼ lAðxÞ: ð12Þ

By considering all objects in U, the elevated area gives the total error induced by elevation and the reduced area gives thetotal error induced by reduction, that is,





Pleasedx.doi

EeðlAÞ ¼ Elevated Areaða;bÞðlAÞ ¼X

lAðxÞPa

ð1� lAðxÞÞ;

ErðlAÞ ¼ Reduced Areaða;bÞðlAÞ ¼X

lAðxÞ6b

lAðxÞ: ð13Þ

Unfortunately, the meaningfulness of the shadow is not as clear. This is perhaps due to the fact that the unit interval ½0;1� isused, rather than a fixed value from ½0;1�. By computing the differences between lAðxÞ and the maximum value 1 and theminimum value 0 and summarizing them up, we have:

ð1� lAðxÞÞ þ ðlAðxÞ � 0Þ ¼ 1: ð14Þ

Thus, we can re-express the shadow in terms of errors as follows:

EsðlAÞ ¼ Shadowða;bÞðlAÞ ¼X

b<lAðxÞ<a


b<lAðxÞ<a

lAðxÞ ¼ Es"1 ðlAÞ þ Es#0 ðlAÞ: ð15Þ

The two terms can be interpreted, respectively, as the total error of elevation and reduction of membership grade of objectsin the shadowed region.

With the error-based interpretation of the three areas, we re-express the objective function in terms of errors by:

V ða;bÞðlAÞ ¼ jEeðlAÞ þ ErðlAÞ � EsðlAÞj

¼X

lAðxÞPa


lAðxÞ6b

lAðxÞ �X

b<lAðxÞ<a


b<lAðxÞ<a

lAðxÞ

0@

1A

��

��: ð16Þ

Thus, the objective function is a kind of trade-off of errors produced by three regions. However, the rationale for such a trade-off is not entirely clear. It is therefore necessary to further investigate the physical meaning of the objective function.

3.2. Three-way approximations defined by minimizing the total error

Recall that the choice of a particular symbol is not important, as long as we preserve both degree-of-truth ordering anddegree-of-information ordering. For example, Cattaneo and Ciucci [2,3] use 1

2 as the membership grade instead of the unitinterval ½0;1� as suggested by Pedrycz. For the choice of the unit interval ½0;1�, according to Pedrycz [26], elements in theshadow ‘‘for which we have assigned the unite interval are completely uncertain – we are not at position to allocate anynumeric membership grade. Therefore, we allow the usage of the unit interval which reflects uncertainty meaning thatany numeric value could be permitted here.’’ A question arises naturally: if we do want to use a single number in the interval½0;1� as the membership grade of elements in the shadow, which number should we use? Given that ½0;1� represents a stateof complete uncertainty, the only number in ½0;1� for such a state is 0:5.

The value 0:5 is the middle point of the unit interval ½0;1�, representing a membership grade of the highest uncertainty[23]. It is a semantically meaningful choice to represent the membership grades of objects in the boundary region. By replac-ing the unit interval ½0;1� of a shadowed set with 0:5, one can define a three-way approximation of a fuzzy set:0 6 b < 0:5 < a 6 1,

TlAðxÞ ¼1; lAðxÞP a;0; lAðxÞ 6 b;

0:5; b < lAðxÞ < a:

8><>:

ð17Þ

The three-way approximation TlA has the same form and structure as a shadowed set defined by Eq. (7).

Fig. 2. A three-way approximation of a fuzzy set.





As illustrated by Fig. 2, for the new definition the correspondences between areas of elevation and reduction and errors ofelevation and reduction remain to be the same. However, we need to revise the definition of the shadow and errors of theshadowed region:

Table 1Loss fun

Actio

ae

ar

as#as"

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Shadowða;bÞðlAÞ ¼ Es0:5 ðlAÞ ¼ Es#ðlAÞ þ Es"ðlAÞ ¼X

0:56lAðxÞ<a

ðlAðxÞ � 0:5Þ þX

b<lAðxÞ<0:5

ð0:5� lAðxÞÞ: ð18Þ

The value Es#ðlAÞ is the total error of reducing the membership grade from lAðxÞ to 0:5 for objects with 0:5 6 lAðxÞ < a. Thevalue Es"ðlAÞ is the total error of elevating the membership grade from lAðxÞ to 0:5 for objects with b < lAðxÞ < 0:5.

Under the new interpretation of a shadowed set, we propose that the total error of a three-way approximation is a seman-tically meaningful objective function:

Eða;bÞðlAÞ ¼ EeðlAÞ þ ErðlAÞ þ Es0:5 ðlAÞ: ð19Þ

We can minimize the total errors of the three areas instead of searching for a trade-off between different areas. In contrast toobjective function (9), we can easily express the total error given by Eq. (19) as the summation of errors of all objects asfollows:

Eða;bÞðlAÞ ¼Xx2U

Eða;bÞðlAðxÞÞ; ð20Þ

where

Eða;bÞðlAðxÞÞ ¼

1� lAðxÞ; lAðxÞP a;lAðxÞ � 0:5; 0:5 6 lAðxÞ < a;0:5� lAðxÞ; b < lAðxÞ < 0:5;lAðxÞ � 0; lAðxÞ 6 b:

8>>><>>>:

ð21Þ

It follows that the total error will be minimized if the error of each individual object is minimized. Therefore, we can searchfor a pair of thresholds ða; bÞ such that Eða;bÞðlAðxÞÞ is minimized for every object.

To minimize the error for each object, we consider the following three actions and associated errors:

elevate to 1 : 1� lAðxÞ;reduce to 0 : lAðxÞ � 0;reduce=elevate to 0:5 : jlAðxÞ � 0:5j:

In other words, the associated errors are the absolute differences between lAðxÞ and three values 1;0 and 0:5, respectively. Aminimized difference is obtained if lAðxÞ is changed into a value that is closest to lAðxÞ. For objects with 0:75 6 lAðxÞ 6 1;1is the closest value to lAðxÞ. For objects with 0 6 lAðxÞ 6 0:25;0 is the closest value. For objects with 0:25 < lAðxÞ < 0:75,0:5 is the closest value. Therefore, we have a ¼ 0:75 and b ¼ 0:25. That is, the pair of thresholds ð0:75;0:25Þ minimizes thetotal error.

The three-way approximation obtained by minimizing the total error is in fact a three-valued fuzzy set that is closest tothe fuzzy set according to the differences between the membership functions of the two fuzzy sets. The three regions canalso be interpreted as three ordinary sets. From this view, three-way approximations are an example of approximating afuzzy set by one or more of its closest ordinary sets, an important problem studied by many authors [5,6,14,21,22,30].

By minimizing the total error, we obtain a specific model of three-way approximations defined by the pair of thresholdsða ¼ 0:75; b ¼ 0:25Þ. This suggests that one needs to consider other information in order to obtain more generalized models.In the next section, we show that a consideration of different costs or risks of actions can achieve such a goal.

4. A decision-theoretic formulation of three-way approximations

By considering various costs of the actions of elevation and reduction, this section introduces a framework of decision-theoretic three-way approximation. The formulation uses a similar technique as used in decision-theoretic rough sets[33,34,39].

ction and errors of actions.

n Fuzzy set membership grade Three-way membership grade Error Loss (cost)

lAðxÞ 1 1� lAðxÞ ke

lAðxÞ 0 lAðxÞ � 0 kr

lAðxÞP 0:5 0:5 lAðxÞ � 0:5 ks#lAðxÞ < 0:5 0:5 0:5� lAðxÞ ks"





4.1. Cost-sensitive three-way approximations of a fuzzy set

A three-way approximation of a fuzzy set uses three membership grades of 0;0:5 and 1. For an object with a fuzzy mem-bership grade lAðxÞ, we take one of three actions: (a) elevate the membership grade to 1, (b) reduce the membership gradeto 0, and (c) change the membership grade to 0:5. For case (c), we further divide it into two subcases, namely, reduce themembership grade to 0:5 when lAðxÞP 0:5, and elevate the membership grade to 0:5 when lAðxÞ < 0:5. Each action mayincur some error. Furthermore, we assume that the costs for different actions are not necessarily the same. Table 1 summa-rizes information about three-way approximations of a fuzzy set.

In a decision-theoretic framework, the set of actions fae; ar; as#; as"g describes four possible actions on changing the mem-bership grade. For simplicity, we also use fe; r; s#; s"g to denote the four actions. The elevation action ae elevates the mem-bership grade of x from lAðxÞ to 1, the reduction action ar reduces the membership grade from lAðxÞ to 0, the reductionaction as# reduces the membership grade from lAðxÞP 0:5 to 0:5, and the elevation action as" elevates the membershipgrade from lAðxÞ < 0:5 to 0:5. The fuzzy membership grade lAðxÞ represents the state of the object. The three-way member-ship grades represent the results of various actions. The errors of different actions are given in the fourth column. The lossesof different actions are given in the last column. The values of the loss function k depends on practical applications and canbe given by a user.

Each of the four losses ke; kr , ks# and ks" provides the unit cost. The actual cost of each action is weighted by the magnitudeof its error. Let RðajxÞ ¼ kaEaðlAðxÞÞ denote the loss for taking action a 2 fe; r; s#; s"g. For an object x, the losses of four actionscan be computed as:

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RðaejxÞ ¼ keEeðlAðxÞÞ ¼ ð1� lAðxÞÞke;

Rðar jxÞ ¼ krErðlAðxÞÞ ¼ ðlAðxÞ � 0Þkr ¼ lAðxÞkr ;

Rðas#jxÞ ¼ ks#Es#ðlAðxÞÞ ¼ ðlAðxÞ � 0:5Þks#; lAðxÞP 0:5;Rðas"jxÞ ¼ ks"Es"ðlAðxÞÞ ¼ ð0:5� lAðxÞÞks"; lAðxÞ < 0:5: ð22Þ

For each object x, only one of the four actions is taken.In constructing a three-way approximation, suppose an action aðxÞ is taken for object x. The total loss of the approxima-

tion can be computed by:

R ¼Xx2U

RðaðxÞjxÞ ¼Xx2U

kaðxÞEaðxÞðlAðxÞÞ: ð23Þ

An optimal approximation can be obtained by minimizing the total loss R. This can be easily achieved by taking an action sðxÞthat minimizes the loss RðaðxÞjxÞ for every object. That is, sðxÞ is a solution to the following minimization problem:

arg minaðxÞ2action

RðaðxÞjxÞ; ð24Þ

where action ¼ fae; ar; as#; as"g. In case of a tie, one may apply a tie-breaking rule.For an object x, according to the values lAðxÞ, we have two groups of decision rules for obtaining three-way approxima-

tions of a fuzzy set: for lAðxÞP 0:5,

ðE1Þ If RðaejxÞ 6 Rðar jxÞ ^ RðaejxÞ 6 Rðas#jxÞ;then take action ae; i:e:; TlA ðxÞ ¼ 1;

ðR1Þ If Rðar jxÞ 6 RðaejxÞ ^ RðarjxÞ 6 Rðas#jxÞ;then take action ar ; i:e:; TlA ðxÞ ¼ 0;

ðS1Þ If Rðas#jxÞ 6 RðaejxÞ ^ Rðas#jxÞ 6 Rðar jxÞ;then take action as#; i:e:; TlAðxÞ ¼ 0:5;

for lAðxÞ < 0:5,

ðE2Þ If RðaejxÞ 6 Rðar jxÞ ^ RðaejxÞ 6 Rðas"jxÞ;then take action ae; i:e:; TlA ðxÞ ¼ 1;

ðR2Þ If Rðar jxÞ 6 RðaejxÞ ^ RðarjxÞ 6 Rðas"jxÞ;then take action ar ; i:e:; TlA ðxÞ ¼ 0;

ðS2Þ If Rðas"jxÞ 6 RðaejxÞ ^ Rðas"jxÞ 6 Rðar jxÞ;then take action as"; i:e:; TlAðxÞ ¼ 0:5:

Tie-breaking rules can be used so that only one action is taken for each object. The resulting three-way approximation hasthe minimum loss.





4.2. Derivation of three-way approximations

In order to obtain an analytic solution defining a three-way approximation, we consider loss functions satisfying certainproperties. In this study, we make the following assumptions,

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ðc1Þ ke > 0; kr > 0; ks" P 0; ks# P 0;

ðc2Þ ks# 6 kr;

ðc3Þ ks" 6 ke:

The rationale of these conditions can be explained as follows. Condition (c1) requires that all costs are nonnegative and isused for convenience. Since 0 and 1 represent two membership grades of complete certainty in the unit interval ½0;1�, weassume a non-zero cost if we change any membership grade in ð0;1Þ into 0 or 1. According to condition (c2), reducing amembership value lAðxÞP 0:5 to 0.5 represents a smaller adjustment than reducing it to 0. A smaller cost is associated withaction as#. Condition (c3) is interpreted in a similar way.

Under assumptions (c1)–(c3), we can simplify the decision rules. For lAðxÞP 0:5, the two conditions in (E1) can beexpressed as:

RðaejxÞ 6 RðarjxÞ () ð1� lAðxÞÞke 6 lAðxÞkr () ðke þ krÞlAðxÞP ke () lAðxÞPke

ke þ kr¼ c;

RðaejxÞ 6 Rðas#jxÞ() ð1� lAðxÞÞke 6 ðlAðxÞ � 0:5Þks# () ðke þ ks#ÞlAðxÞP ke þ 0:5ks#

() lAðxÞP2ke þ ks#

2ðke þ ks#Þ¼ a:

ð25Þ

The two conditions in (R1) are expressed by:

Rðar jxÞ 6 RðaejxÞ () lAðxÞ 6 c;

Rðar jxÞ 6 Rðas#jxÞ () lAðxÞkr 6 ðlAðxÞ � 0:5Þks# () ðkr � ks#ÞlAðxÞ 6 �0:5ks# () lAðxÞ 6�ks#

2ðkr � ks#Þ¼ c�: ð26Þ

For rule (R1), we use a stronger version of condition (c2), namely, kr > ks#. The reason for not considering the special casekr ¼ ks# is given as follows. The condition Rðar jxÞ 6 Rðas#jxÞ is equivalent to ks# 6 0, if kr ¼ ks#. By condition (c1), we must havekr ¼ ks# ¼ 0. This contradicts to the condition kr > 0. It follows that, when kr ¼ ks#, the cost of ar is always greater than as# andrule (R1) cannot be used. In this case, we do not need rule (R1). Instead, other rules must be used. Finally, the two conditionsin (S1) are expressed as:

Rðas#jxÞ 6 RðaejxÞ () lAðxÞ 6 a;

Rðas#jxÞ 6 Rðar jxÞ () lAðxÞP c�: ð27Þ

The analysis introduces three parameters, c;a and c�, for expressing the conditions on the membership grades.By assumptions (c1) and (c2), we have:

0 < c < 1; 0:5 < a 6 1; c� 6 0;c < a; c� < c:

Consequently, we can simplify conditions for the three rules as follows:

ðE1Þ lAðxÞP c ^ lAðxÞP a() lAðxÞP a;

ðR1Þ lAðxÞ 6 c ^ lAðxÞ 6 c� () lAðxÞ 6 c�;

ðS1Þ lAðxÞ 6 a ^ lAðxÞP c� () lAðxÞ 6 a:

Because c� 6 0 contradicts with the assumption lAðxÞP 0:5, it is impossible to apply rule (R1) for reducing membershipvalues. When lAðxÞ ¼ a, we break the tie by choosing rule (E1). The two remaining rules can be simply expressed as: forlAðxÞP 0:5,

ðE1Þ If lAðxÞP a; then TlA ðxÞ ¼ 1;

ðS1Þ If 0:5 6 lAðxÞ < a; then TlA ðxÞ ¼ 0:5:

A single threshold 0:5 < a 6 1:0 is used to decide either elevating a membership grade to 1 or reducing it to 0.5.Consider now the case of lAðxÞ < 0:5. For rule (E2), the two conditions can be expressed as:

RðaejxÞ 6 Rðar jxÞ () lAðxÞP c;

RðaejxÞ6Rðas"jxÞ()ð1�lAðxÞÞke6 ð0:5�lAðxÞÞks" ()ðke�ks"ÞlAðxÞP ke�0:5ks" ()lAðxÞPke�0:5ks"

ke�ks"¼ cþ: ð28Þ





For a similar reason discussed earlier with respect to rule (R1), for rule (E2), we use a stronger version of (c3), namely,ke > ks". The two conditions in (R2) are expressed as:

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Rðar jxÞ 6 RðaejxÞ () lAðxÞ 6 c;

Rðar jxÞ 6 Rðas"jxÞ () lAðxÞkr 6 ð0:5� lAðxÞÞks" () ðkr þ ks"ÞlAðxÞ 6 0:5ks" () lAðxÞ 6ks"

2ðkr þ ks"Þ¼ b: ð29Þ

Finally, two conditions in (S2) are expressed by:

Rðas"jxÞ 6 RðaejxÞ () lAðxÞ 6 cþ;

Rðas"jxÞ 6 RðarjxÞ () lAðxÞP b: ð30Þ

The analysis introduces three parameters for expressing the conditions on the membership grades.By assumptions (c1) and (c3), we have:

0 < c < 1; 0 6 b < 0:5; cþ P 1;b < c; c < cþ:

Consequently, we can simplify conditions for each rule as follows:

ðE2Þ lAðxÞP c ^ lAðxÞP cþ () lAðxÞP cþ;

ðR2Þ lAðxÞ 6 c ^ lAðxÞ 6 b() lAðxÞ 6 b;

ðS2Þ lAðxÞ 6 cþ ^ lAðxÞP b() lAðxÞP b:

The condition cþ P 1 contradicts with the assumption lAðxÞ < 0:5; it is impossible to apply rule (E2) for elevating. WhenlAðxÞ ¼ b, we can break the tie by choosing rule (R2). The two remaining rules can be simply expressed as: for lAðxÞ < 0:5,

ðR2Þ If lAðxÞ 6 b; then TlA ðxÞ ¼ 0;

ðS2Þ If b < lAðxÞ < 0:5; then TlA ðxÞ ¼ 0:5:

A single threshold 0 6 b < 0:5 is used to decide either elevating the membership to 0.5 or reducing it to 0.By combining the two sets of rules and applying tie-breaking rules, we immediately have a set of three rules:

ðEÞ If lAðxÞP a; then TlAðxÞ ¼ 1;ðRÞ If lAðxÞ 6 b; then TlA ðxÞ ¼ 0;ðSÞ If b < lAðxÞ < a; then TlA ðxÞ ¼ 0:5;

where

a ¼ 2ke þ ks#

2ðke þ ks#Þ;

b ¼ ks"

2ðkr þ ks"Þ; ð31Þ

and 0 6 b < 0:5 < a 6 1. The results are a three-way approximation with membership grades 0; 0:5 and 1. The required pairof thresholds are computed systematically from the costs of various actions.

4.3. Semantics issues

So far, our discussions on the decision-theoretic three-way approximations of fuzzy sets have been mainly focused on themathematical formulation by assuming that a loss function is given. In order to increase chances of successful applicationsand to prevent misuses of the model, we must investigate semantics issues. As examples, three semantics issues are consid-ered in this subsection.

4.3.1. Interpretations of loss functionsA fundamental notion of the decision-theoretic model is a loss function that defines the costs of different actions. The pair

of thresholds depends crucially on the choice of loss functions. Once a user provides a loss function, the pair of thresholds canbe computed. On the other hand, in many applications, a user may give a pair of thresholds directly. Our formulation showsthat the thresholds can in fact be interpreted in terms of a loss function. If such an interpretation is explained to a user, a usermay provide a better estimation of the thresholds. That is, the decision-theoretic model enables us to give an interpretationand explanation of a common practice. We must also point out that the decision-theoretic model is only one of possibleexplanations. Shadowed sets are another explanation with a different objective function.

In theory, we can discuss some desired properties of a loss function. For example, conditions (c1)–(c3) are needed toensure that the required thresholds of three-way approximations are well defined. They are the conditions underwhich three-way approximations are meaningful. Except for conditions such as (c1), in theory it is difficult to specify the





meaningfulness of some properties and decide what is a good choice of loss functions. One must consider a particular appli-cation in order to interpret the meaning of loss functions. Empirical investigations of loss functions are an important issuewhen applying the model of decision-theoretic three-way approximations.

4.3.2. Relationships to shadowed setsAs approximations of a fuzzy set lA, the three-way approximation TlA and the shadowed set approximation SlA

have the same form. That is, their corresponding three regions are defined through a pair of thresholds ða; bÞ. However,their interpretations of thresholds are different. For shadowed sets, the objective function (9) is given with respect tothe membership function, lA. Different classes of membership functions will produce different thresholds. The samemathematical fuzzy set (i.e., fuzzy membership function) would have the same three-valued approximation independentof different applications. In contrast, the objective function (23) of the decision-theoretic framework is given indepen-dent of any particular fuzzy membership function. Instead, the objective function is given with respect to loss functions.Different loss functions will produce different thresholds. Unlike the model of shadowed sets, in the decision-theoreticmodel, the same mathematical fuzzy set may have different three-way approximations in different applicationsby using different loss functions. That is, decision-theoretic three-way approximations are adaptive to differentapplications.

Shadowed set approximations and decision-theoretic three-way approximations may be viewed as two examples of amore general model of three-way approximations of fuzzy sets. Each of them considers only one of two aspects of approx-imations, namely, the structure of membership function and costs of various actions. Shadowed set model considers only thestructure of a membership function and decision-theoretic model considers only the costs of different actions. In general, onemay have a framework by considering both the membership function and the loss functions [38], which may enable us tocombine advantages of shadowed set model and decision-theoretic model.

4.3.3. Relationships to decision-theoretic rough setsIn developing decision-theoretic three-way approximations of fuzzy sets, we adopt the main ideas from decision-theo-

retic rough sets [39]. Since a rough membership function can be viewed as a fuzzy membership function, a reviewer ofthe paper suggests an alternative way to compute the costs of various actions, in a similar way as decision-theoretic roughsets:

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RðaejxÞ ¼ keð1� lAðxÞÞ;Rðar jxÞ ¼ krðlAðxÞ � 0Þ;RðasjxÞ ¼ ks1 ð1� lAðxÞÞ þ ks0 ðlAðxÞ � 0Þ; ð32Þ

where ks0 and ks1 denote the actions of changing membership grade lAðxÞ to the end points 0 and 1 of the unit interval ½0;1�,respectively. The results correspond to a special case of decision-theoretic rough sets. An advantage of this new formulationis the omission of 0:5 in our formulation. The new formulation is more related to the interpretation of membership grade½0;1� in shadowed sets.

Eq. (32) enables us to establish a connection between three-way approximations of fuzzy sets and decision-theoreticrough sets in form. There are major differences between the two models [38]. For decision-theoretic rough sets, we dealwith a two-state three-way decision problem. An object x is in one of the two possible states, namely, either in a set or notin the set. Three-way approximations are due to uncertainty regarding the actual state of x. A rough membership functiondenotes the probability that an object is in the set. On the other hand, three-way approximations of fuzzy sets are a many-state decision problem. The set of possible states of x is the unit interval ½0;1�. A fuzzy membership grade denotes theactual state of an object. Three-way approximations are required because we want to simplify a many-state problem. Thatis, we use three states to approximate many states. Those semantic differences must be considered when applying a par-ticular model.

4.4. Examples

We use examples to illustrate the main ideas of decision-theoretic three-way approximations. The examples are gener-ated by using MATLAB. We form the universe of objects by randomly selecting a finite set of objects according to the distri-bution of a membership function. The membership functions used are from the MATLAB fuzzy logic toolbox.

Suppose that a user provides the following values of a loss function:

ke ¼ 0:35;kr ¼ 0:28;ks" ¼ 0:17;ks# ¼ 0:20:

The loss function satisfies conditions (c1)–(c3). According to Eq. (31), we can compute an optimal pair of thresholds asfollows:





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a ¼ 2ke þ ks#

2ðke þ ks#Þ¼ 2 � 0:35þ 0:2

2 � ð0:35þ 0:2Þ ¼ 0:8182;

b ¼ ks"

2ðkr þ ks"Þ¼ 0:17

2 � ð0:28þ 0:17Þ ¼ 0:1889: ð33Þ

In contrast to the original shadowed sets, one advantage of decision-theoretic three-way approximations is that the optimalpair of thresholds ða; bÞ is independent of particular membership functions. Fig. 3 gives decision-theoretic three-wayapproximations of four fuzzy sets with different types of membership functions, i.e., the Gaussian membership function,the bell-shaped membership function, the triangular-shaped membership function and the trapezoidal-shaped membershipfunction. Using the three-way decision rules (E), (R) and (S), we obtain three-way approximations represented by squares inthe figure. The a-threshold is represented by a dashed horizontal line and the b threshold is represented by a solid horizontalline.

In order to illustrate how to make a three-way decision for a particular object, we use the fuzzy set in Fig. 3(a) as anexample. The membership function is a Gaussian membership function:

lAðxÞ ¼ e�ðx�cÞ2

2r2 ; ð34Þ

where the parameters r ¼ 2 and c ¼ 5. For an object x ¼ 2:5510, its membership grade is lAðxÞ ¼ 0:4725 < 0:5. The losses oftaking actions ae; as" and ar of x are:

RðaejxÞ ¼ keEeðlAðxÞÞ ¼ 0:35 � ð1� 0:4725Þ ¼ 0:1846;Rðas"jxÞ ¼ ks"Es"ðlAðxÞÞ ¼ 0:17 � ð0:5� 0:4725Þ ¼ 0:0047;RðarjxÞ ¼ krErðlAðxÞÞ ¼ 0:28 � ð0:4725� 0Þ ¼ 0:1323:

Fig. 3. Decision-theoretic three-way approximations of fuzzy sets.





The loss of elevation action as" has the minimum cost Rðas"jxÞ ¼ 0:0047. That is, rule (S) should be used, sincelAðxÞ ¼ 0:4725 > b ¼ 0:1889 and lAðxÞ ¼ 0:4725 < a ¼ 0:8182.

5. Two special models of three-way approximations

As examples to demonstrate the generality of the decision-theoretic model, we derive two special models by consideringloss functions satisfying additional properties.

5.1. Error-based ð0:75; 0:25Þ-model

Considering a loss function satisfying the following condition:

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ðc4Þ ke ¼ kr ¼ ks" ¼ ks# > 0:

That is, the unit cost of elevating to 1, the unit cost of reducing to 0, the unit cost of elevating to 0:5 and the unit cost ofreducing to 0:5 are the same. It can be verified that ðc4Þ ) ððc1Þ; ðc2Þ; ðc3ÞÞ. Thus, we can obtain a special model of three-way approximations. According to Eq. (31), the pair of thresholds ða; bÞ is given by:

a ¼ 2ke þ ks#

2ðke þ ks#Þ¼ 3

4¼ 0:75;

b ¼ ks"

2ðkr þ ks"Þ¼ 1

4¼ 0:25:

We therefore obtain the error-based model introduced in Section 3.2. That is, error-based model is characterized by actionswith the same loss.

5.2. Symmetric ða; 1� aÞ-model

By imposing the condition aþ b ¼ 1, we can specialize the general ða; bÞ decision-theoretic model into a special casecalled the symmetric ða;1� aÞ-model.

By applying the condition aþ b ¼ 1 to Eq. (31), we have:

aþ b ¼ 1() 2ke þ ks#

2ðke þ ks#Þþ ks"

2ðkr þ ks"Þ¼ 1() 2ð2ke þ ks#Þðkr þ ks"Þ þ 2ks"ðke þ ks#Þ ¼ 4ðke þ ks#Þðkr þ ks"Þ

() ks"ke ¼ ks#kr ()ks"

kr¼ ks#

ke:

ð35Þ

Thus, the condition aþ b ¼ 1 for the symmetric ða;1� aÞmodel can be equivalently expressed in terms of loss functions bycondition (c5):

ðc5Þ ks"

kr¼ ks#

ke:

The condition (c5) can be considered as a constraint on a loss function, which transfers a two-parameter-model into a one-parameter symmetric model.

6. Conclusion

The theory of three-way decisions provides a framework for explaining many decision problems. Shadowed sets are amodel of three-way decisions for approximating a fuzzy set. A shadowed set can be viewed as elevating membership gradesaround 1 to 1, reducing membership grades around 0 to 0, and changing membership grades around 0:5 to the unit interval½0;1�, respectively, based on a pair of thresholds. By adopting the general idea of shadowed sets, we propose an alternativemodel of decision-theoretic three-way approximations. By minimizing the overall cost of three-way approximations, we cansystematically compute the required thresholds. From the general model, we can obtain the error-based ð0:75;0:25Þ-modeland symmetric ða;1� aÞ-model.

The decision-theoretic model offers several advantages. First, it provides a precise and semantically meaningful interpre-tation of objective function, which is the overall cost of three-way approximations. Second, the minimization of the overallcost immediately leads to an analytic solution to the required pair of thresholds in terms of various decision costs. Third,shadowed sets give an example to support the theory of three-way decisions. The decision-theoretic three-way interpreta-tion suggests a new viewpoint for future research on enlarging the domain of shadowed sets, that is, moving from a narrowsense of the notion of shadowed sets to a wide sense. One may examine new objective functions for deriving other types ofthree-way approximations.





Acknowledgements

This work is partially supported by a Discovery Grant from NSERC Canada. The authors would like to thank the reviewersfor their constructive comments and suggestions. One reviewer raises the question regarding the meaning of shadowed sets,which leads to Section 2.4 on two senses of the notion of shadowed sets.

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Please cite this article in press as: X. Deng, Y. Yao, Decision-theoretic three-way approximations of fuzzy sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.04.022

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http://dx.doi.org/10.1016/S1571-0661(04)80706-0

http://dx.doi.org/10.1016/S1571-0661(04)80706-0






















http://dx.doi.org/10.1109/CISDA.2012.6291542

http://dx.doi.org/10.1109/CISDA.2012.6291542













http://digitalcommons.utep.edu/cs_techrep/491

http://digitalcommons.utep.edu/cs_techrep/491




















http://dx.doi.org/10.1142/S0218001412500103

http://dx.doi.org/10.1142/S0218001412500103
























http://dx.doi.org/10.5402/2012/929085







decision-theoretic three-way approximations of fuzzy sets

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