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ORIGINAL ARTICLE
On Admissible Total Orders for Interval-valued g Intuitionistic Fuzzy Membership Degrees CrossMark
LA. Da Silva - B. Bedregal • R.H.N. Santiago
Received: 24 July 2015/ Revised: 6 January 2016/ Accepted: 12 January 2016/
Abstract The linearity contained in the natural order of unit interval [0,1] plays an important role in many concepts and applications of fuzzy theory. Besides, it is very important in concepts like ordered weighted aggregation operators (OWA) and fuzzy decision making. However, this linearity is not inherited by most of fuzzy logics which extend the standard one. To recover the linearity for such frameworks it is required that its related partial order to be extended give rise to the notion of admissible order. In this paper, we study some admissible orders for the framework of interval-valued intuitionistic fuzzy logic.
Keywords Interval-valued intuitionistic fuzzy sets • Score function • Accuracy function • Admissible total orders
© 2016 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.
1. Introduction
Since Zadeh's seminal paper [28] several extensions of fuzzy set theory have been proposed [5]. Among them we stress the interval-valued fuzzy set theory [10, 29]
I. A. Da Silva
Programa de Pós-Gradua9áo em Engenharia Elétrica e de Computa9áo, Universidade Federal do Rio Grande do Norte, Campus Universitario s/n, 59078-970 Natal, Brazil B. Bedregal (/ ') ■ R.H.N. Santiago
Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Campus Univcrsitário s/n, 59078-970 Natal, Brazil email: bedregal@dimap.ufrn.br
Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research Society of China.
© 2016 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.
This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/jfiae.2016.06.003
and the intuitíonistic fuzzy set theory1 [1, 3]. The first captures the intrinsic difficulty to precisely determine the membership degree of an object with respect to some linguistic term; in this case, instead of an exact value, an expert provides an interval which expresses the intrinsic imprecision in his/her degree of assignment [4]. The second adds an extra degree (non-membership degree) to the usual fuzzy sets in order to model the hesitation and uncertainty about the assigned degree of membership. In fuzzy set theory, the non-membership degree is by default the complement of the membership degree, ha(x), i-e-, 1 - Ha(x). Both extensions were combined by Atanassov et al. [2], in this case the underlying imprecision during the process of assignment of membership and non-membership degrees is represented by a pair of intervals. This extension is known as interval-valued intuitionistic fuzzy sets (IV-IFSs). There are several applications of IVIFSs as well as some extensions of usual fuzzy concepts to this setting [12,17,22].
One of the main advantages of the usual degrees in [0,1] is that they are linearly ordered, i.e., every pair of such degrees are comparable. This property is fundamental in applications like decision making, where the maximum between any two results is required. Therefore it is interesting that linearity can be extended to the set of membership degrees of IVIFS — abbreviated here by IVIFMD. This is done, here, in such a way that the diagonal set of IVIFMD is order-isomorphic to [0,1].
Xu and Yager [26] proposed a total order for intuitionistic fuzzy membership degrees based on the notion of score and accuracy functions [11, 15]. The resulting order is admissible, i.e., it extends the natural partial order of intuitionistic fuzzy degrees [8]. The score and accuracy functions for IVIFMD can be extended in several ways, e.g. see [16,20,21,27]; here we propose anew accuracy function for IVIFMD. The notion of admissible total order [8,9] can be naturally generalized for any poset, in this paper we consider admissible total orders for IVIFMD with their usual partial order. Here we prove that, when we consider several extensions of score and accuracy functions for IVIFMD, the natural extensions of such linear order are not compatible with the natural one, i.e., it does not induce another linear order. Nevertheless, based on the proposal of Xu and Yager [26] we provide a new linear order which is very similar (but not equal) to the admissible total order introduced in [24] which extends the Xu and Yager order for IVIFMD. In addition, we also introduce a family of total orders for IVIFMD based on arbitrary total orders for interval-valued fuzzy degrees and analyze their admissibility.
This paper is organized as follows: Section 2 contains the notions of intuitionistic fuzzy sets, interval-valued fuzzy sets (IVFSs) and some related notions such as s core and accuracy indexes. Section 3 presents the notion of IVIFS as well as some notions and notations, particularly some extensions of score and accuracy indexes for IVIFMD are analyzed. Section 4 provides two total orders and a parameterized family of total orders for IVIFMD, and their properties are analyzed. Finally, some conclusions and future research are pointed out in Section 5.
1 This notion is not related with the fuzzy extension of the intuitionistic logic of Brouwer [18] proposed by Takcuti and Titani in [23]. In order to avoid this confusion, some authors have used the name "Atanassov intuitionistic fuzzy logic" but we opted to omit the name of Atanassov by considering that the readers are clear that here it is being used the approach of Atanassov.
2. Preliminaries
2.1. Intuitionistic Fuzzy Sets
Definition 2.1 [1] Let L* = {(x, y) e [0, l]2 | x + y < 1} and the partial order (x],y{) <i- (x2,yi) iff x i < x2 andyi > y2■ An intuitionistic fuzzy set (IFS), A, over a nonempty set X is a class given by
A = {(x,ha(x),va(x)) \xeX\,
where Qia(x),va(xJ) e L*.
The structure <L*,<z,.) is a complete lattice. Deschrijver and Kerre [13] proved that IFS, considering the complete lattice (L*, <l-), are Muzzy sets in the sense of Goguen [14]. Elements of V are called V-values. For every pair (x, y) e V the value x represents the degree of membership whereas the value y represents the degree of non-membership. The hesitation or indeterminacy inherent in V-values, (jc, >>), can be measured by the intuitionistic fuzzy index function k : L* —» [0,1]
jt(x,y) = 1 - x-y.
This function was introduced by Atanassov [1] and generalized by Bustince et al. [6].
The L*-values, (x, y), such that y = 1 - x are called diagonals. Clearly, the function <S(x) = (x, 1 - x) is a bijection between [0,1] and the set of diagonal elements of L", is denoted by £)L.. Notice that for any x e [0,1], n(6(x)) = 0, this reflects that the diagonal elements do not carry hesitation.
Chen and Tan [11] introduced the notion of score for V-values (x, y) as
s*(x,y) = x-y. (1)
In other words, the score measures the difference between membership and non-membership. Hong and Choi [15] introduced the notion of accuracy function for L"-values (x,y) as2
h'(x,y) = x + y. (2)
Bustince et al. [7] proposed another partial order for L*, namely,
(*i.yi) —L* {x2,y2) iff*, < x2 andyi < y2.
This order is consistent with the hesitation and accuracy measures of V-values, in the following sense:
Proposition 2.1 Let (xuyi),(x2,y2) e L*. If(xuyi) <l- (x2,y2), then 7t(x2,y2) < n(xuyi).
2 Notice that the range of s* is [-1,1], whereas the range of h* is [0,1].
Proof If Oi, y-i) <L. (X2,y>2), then (1) x\ < x2 and yi < y2, or (2) x\ < x2 and yi < y2. In both cases jq +y\ < x2 +y2. Therefore, 1 - (x2 + yd < 1 - (jci +yi) which implies thatjr(x2,y2) < n(x\,y\).
Corollary 2.1 Ifixuyi) <l- (x2,y2), then h*(x\,y\) < h*(x2,y2).
Proof If (JCi.yi) <l- (x2,y2), then by Proposition 2.1,1 - (x2 + y2) < 1 - (x\ + y\) and so x\ + yi < x2 + y2. Therefore, h*(x\,y\) < h*(x2,y2).
Xu and Yager [26] applied the score and accuracy on L* in order to obtain the linear order
As mentioned above, total orders are desirable in many situations, but a reasonable condition is that they extend the natural order, <£., on L*. These total orders are called of admissible [8]. The next proposition shows that Xu and Yager total order is admissible and it is also related with the partial order <l- proposed by Bustince and the usual order on [0,1].
Proposition 2.2 Let (jti.yi), (x2,y2) e L*. Then
1) if (*i>yi) —l* (x2,y2), then Oi,}>i) <xy (x2,y2);
2) if s*(xi,yi) = s*(x2,y2) and (xuyi) <l- (x2,y2), then (x^yi) <xy (.x2,y2)\ and
3) if x < y, then 6(x) <xy S(y). Proof Let , yi), (x2, y2) e L*. Then
1) if (xi,yi) <£. (x2,y2), then x\ < x2 andy2 < >>i, or xi < x2 and < y\. In both cases, we have that x\ + y2 < x2 + yi. Therefore, i*(*i,yi) < s"(x2,y2) and so (xuyi) <xy (x2,y2)\
2) if (x\,yi) <L. (x2,y2), then x\ < x2 andyi < y2, or jci < x2 andyi < y2. In both cases, we have that xi+yi <x2+ y2. Therefore, h'(xuyi) < h*(x2,y2) and so, because s*(ii,yi) = i'fe,^), we have that (^1,^1) <xy (x2,y2)\ and
3) if x < y, then 1 - y < 1 - x and so 5(x) <Xr S(y).
Notice that, in the items l)and2), case (x\,y\) = (X2, y2), then trivially (x\, yi) <Xy (xi,yi)
Remark 2.1 Nevertheless, the "reverse" proposition does not hold, i.e., there are cases such that (xi,yi) <xy (x2,y2), e.g. (0.3,0.5) <xy (0.2,0.1), but neither (xi,yi) <l' (x2,>>2)nor (i'*(xi,>>i) = s*(x2,y2) and (*i,yi) <l- (x2,y2)).
In the following, we propose a new admissible total order for L* which also extends the partial order <l- (with a small constraint).
Definition 2.2 Let (x\,y\), (x2,y2) e V.
(xuyi)<xy(x2,y2) iff
s'(xuyt)<S*(x2,y2), or s"(x1,yi)=s*(x2,y2) and h'(M,yi)<h'(x2,y2)
(x\,yi) (X2,y2) iffx 1 < x2 or (xi = x2 andy2 < yi).
Theorem 2.1 <£. is a total order. Proof Straightforward.
Proposition2.3 (*i,yi)<L'(Jt2.y2)iff(*i.yi) —L* (.x2,y2)or((xi.yi) <L- (x2,y2)and xi * x2).
Proof (=>) If (xi^i) <£. (x2,y2), then (i) xi < x2 or (ii) xi = x2 and y2 < yi. Case (i), then y2 < yi, and therefore (xi,;yi) <l- (x2,y2), or yi < y2, and hence (xi,y\) <f (x2,y2) (with x\ + x2). Case (ii), then it is straightforward to see that (xuyi) <L- (.x2,y2).
(<=) If (xi.yi) <l- (x2,y2), then xi < x2 and y2 < yi. Case xi < x2, then by Equation (4), (xi,y\) <l- (x2,y2). Case x\ = x2, then because y2 < yu we have that (.xuyi) <L- (x2,y2).
If(*i.yi) (x2,)>2)andxi + X2, then x\ < x2 and so, by Equation (4), (*i,yi)<L' ix2,y2).
Remark 2.2 The last item of Proposition 2.2 trivially holds whenever we consider <1,- instead of <xy- Moreover, in contrast with <xy, ^l- also extends the partial order
This remark and Proposition 2.3 relate <£• with <L, and <l- in a more closed way than <xy', c.f. Proposition 2.2 and Remark 2.1.
22. Interval-valued Fuzzy Sets
Let L = {[a,b] | 0 < a < b < 1} be the set of closed subintervals of [0,1]. An interval-valued fuzzy set (IVFS), A, over a set X is a class
A = {(x,nA(x)) | xeX],
s.t. [Ia : X -» L. Associated with L there are projections, n\,n2 : L —> [0,1], s.t.
7Ti([a, b\) = a and jt2([a, b\) = b.
For notational simplicity, given an arbitrary X e L, we will denote tti(X) and n2{X) by X and X, respectively.
Consider the following partial order on L,
X<L 7 iff Z < y and X <Y.
The poset (L, <l> is a complete lattice and hence a Goguen L-fuzzy set. The score and accuracy functions for interval values are defined as
s(X) = v(X) - 1 and h(X) = 1 - w(X),
where v(X) = X + X and w(X) = X-X.
Remark 2.3 Let p : L -* L* be the well-known isomorphism between (L*, <£.} and (L, <l>, defined by p(X) = (X, 1 - X). Then i = s* o p and h = h* o p.
Equation (5) establishes a total order for intervals (a kind of lexicographic order) whereas Equation (6) states Xu and Yager version for <l. Both orders are total admissible orders in the sense that they extend the natural order <L; see [8].
X<L7iffX< 7or(X = 7andX< 7), (5)
X <XY 7 iff s(X) < s(Y) or (j(X) = s(7) and h(X) < h(Y)). (6)
Analogously to Proposition 2.3, we have the following characterization of <l:
X <L Y iff X <L Y or (7 C X and X * 7).
3. Interval-valued Intuitionistic Fuzzy Sets
Definition 3.1 [2] An interval-valued intuitionistic fuzzy set, A, over a nonempty set X is a class
A = {(x,na(x),vA(.x))\xeX\,
where ¡1a, va : X —> L with the condition ¡ia(x) + va(x) — [1.1].
Deschrijver and Kerre [13] provided an alternative approach for IFS's in terms of L-fuzzy sets [14]. We can also see IVIFS in the same terms by considering the complete lattice (L*, <l<>; where
L* = {(X, y)eLxL|X+7< 1}
(Xu 70 <L. (X2,72) iff Xj <L X2 and 7! >L 72. (7)
Elements of L* are called L*-values. There are some special L*-values; namely,
• Ql- = ([0,0], [1,1]) and 1L. = ([1,1], [0,0]).
• Semi-diagonal and diagonal elements:
A L*-value (X, 7) is a semi-diagonal element whenever X and 7 are degenerate intervals, i.e., X = X and 7 = 7. A semi-diagonal element (X, 7) is a diagonal element whenever X + 7 = [1,1]. The set of semi-diagonal elements is denoted by Ds whereas the set of diagonal elements by X>. Clearly, DC Ds and there is a bijection between [0,1] and D, given by
<P(x) = ([x,x], [1 - jc, 1 - *]); (8)
as well as between L* and Ds, given by
tff(x>y) = (\.x,x\,\y,.y]);
Fuzzy Inf. Eng. (2016) 8: 169-182_175
and between L and £>s, given by
t = i/iop. (10)
In other words, L* contains a copy of [0,1], L* and L. 3.1. Score and Accuracy Indexes for L*-values
Lee [16] and Xu [25]3 extended to L* the score measure proposed by Chen and Tan. In this paper we adopt the corresponding definition provided by Xu; namely, a function S : L* -> [-1,1] defined by
5« 7) = *^. (ID
For each (X, Y) e L*, S (X, Y) is called the score of (X, 7). The next remark justifies the suitability of this definition.
Remark 3.1 When S is applied to semi-diagonal elements, it works, up to isomorphism, in the same way as s*, i.e., S(ifr(x,yj) = s*(x,y) for any (x,y) e L". When S is applied to diagonal elements, it works as s when applied to degenerate intervals, i.e., for any x e [0,1], S(</>(x)) = 2x - 1 = i([x, x]). Moreover, S can be obtained from s and s*, as shown in Equation (12),
= (12)
Notice also that the divisor 2 is irrelevant to compare L*-values, however it is essential in order to guarantee the same range, [-1,1], between S and s". The score function induces a preorder on L"; namely,
X <s Y iff S (X) < S (Y)
for any X, Y e L*.
This relation is not a partial order, since it is not anti-symmetric. In fact, S ([0.2,0.3], [0.4,0.5]) = 5 ([0.1,0.2], [0.3,0.4]) = -0.2. However, it is possible to build a partial order from "<s" by defining an equivalence relation on L*-values in the following way:
or equivalently,
Y iff X <s Y and Y<iX
X =s Yiff S(X) = S(Y).
As usual we will denote by X <s Y when X<jY but X is Y.
In the literature, there are several extensions of accuracy functions for L*. For example,
3 Since this reference is written in Chinese, we build our work based on the definition in [24].
1) Accuracy function Hi : L* ->[0,1] defined by Xu [25]
= (13)
2) Accuracy function H2 : L* -> [-1,1] defined by Ye [27]
tf2(X,y) = v(X)-l + ^.
3) Accuracy function : L* —> [-1,1] defined by Nayagam, Muralikrishnan and Sivaram [20]
where "■" is the arithmetical product on intervals [19] restricted to L; namely: XY= \XY,XY].
4) Accuracy function H4 : L* -» [0,1] defined by Nayagam et al. [21]: Let 5 e [0,1]. Then
„,YV, v(X) + (2-v(X)-v(Y))-S /) = ---.
5) The novel accuracy function H5 : L* —» [0,1]
H5(X, Y) = \ - (w(X) + w(r)). (14)
Notice that the accuracy of diagonal elements of V, measured by h*, is 1, i.e., the maximal one. This is reasonable, since there is no hesitation (imprecision in the membership degree). So, it is also reasonable to extend this paradigm for the diagonal elements in L*. However, it only remains valid for Hi and — i.e., for each x e [0,1] and i = 1 or i = 5, Hi{ip(x)) = 1 = h([x, x]). On the other hand, the functions H2, H3 and H4 do not follow such paradigm, in fact Hi([x, x], [1 - x, 1 - *]) = x, for i = 2,4 and H3([x, x], [1 - x, 1 - x]) = -x2 + 3x - 1. Another negative characteristic of H2 and H3 is that their range is not the same of h*.
Analogously, with respect to score index S, when applied to semi-diagonals elements, Hi is equivalent to h*, since Hi (tf/(x, j)) = h'(x, y). But, Hk ° (fr ± h* for k = 2,-•■ ,5.
Thus, Hi is the most reasonable accuracy function for L*. Nevertheless, H5 could also be a good choice because it has not only most of Hi properties but also has an extra property; namely it can be built from h* and h in the same way as S can be obtained from s* and s — see Equation (12), i.e.,
Fuzzy Inf. Eng. (2016) 8: 169-182_177
4. Ranking L*-values
Analogously to what happens with the score index, the accuracy functions admit different L*-values with the same accuracy, e.g. for k = 1, k = 3 or k = 5, Hk(X, Y) = Hk(Y,X), whenever v(X) = v(Y); and for k = 2 or k = 4, Hk(Xu Y{) = Hk(X2, Y2), whenever v(Xi) = v(X2) and v(7i) = v(Y2). Therefore, none of these indexes could be individually used for ranking L*-values as suggested by Nayagam et al. [21]. But in order to rank any possible set of L*-values it is necessary to provide a linear order on L* as it is done by Xu and Yager [26] for L"-values — there, the order is based on the score and accuracy indexes. Following the same idea, but considering different accuracy functions, we define five binary relations on L*-values. For any Jfce {1,--- ,5} andX,Y e L*, let
— {Sil^Y, (15)
Denote by X =t Y whenever X <kY and Y <kX and denote by X <* Y whenever X<k YbutXi*Y.
Nevertheless, in the same way as noted by Wang et al. [24] for k = 1, these relations are not partial orders. The next example provides counter-examples for the anti-symmetry property of <k for each k= 1, ■ ■ ■ , 5.
Example 4.1 Let Xi = ([0.1,0.3], [0.2,0.6]), X2 = ([0.2,0.2], [0.3,0.5]), X3 = ([0.1,0.4], [0.2,0.3]) and X, = ([0.2,0.3], [0.1,0.4]) be L*-values. Then, S(X,) = S(X2) = -0.2, S(X3) = S(X,) = 0, Hi(Xi) = Hi(X2) = 0.6, H2(Xi) = H2(X2) = -0.2, i/3(X3) = H,(X4) = 0.07, H4(X 1) = HA(X2) = 0.2 - 0.4<5 and H5(X3) = H5(X4) = 0.6. Therefore, Xi =k X2 for k e {1,2,4}. Analogously, X3 =k X4 for k e {3,5}. So, <k for any k e {1, ■ • ■ , 5} is not an anti-symmetric relation. However, Wang et al. [24] provided the following total orders4 for L*:
X<Y iff
fX<sY, or
X=SY and ffi(X)<ff!(Y), or
X=SY and H,(X)=H,(Y) and T(X)<T(Y), or
X=SY and Hi(X)=Hi(Y) and T(X)=T(Y) and G(X)<G(Y)
for any X, Y e L*. where T(X, Y) = w(X) - w(Y) and G(X, Y) = w(X) + w(Y).
Example 4.2 Consider the L*-values X;, with i = 1, • • • ,4 in Example 4.1. Then S(Xi) = S(X2) = -0.2 and S(X3) = S(Xt) = 0. Therefore, in order to rank those L*-values w.r.t. the relation "<", we need to calculate the accuracy index: //1 (X1 ) = Hi(X2) = 0.6 and i?i(X3) = Hi(X4) = 0.5; but this is not conclusive. So, we need to compute the index T: T(Xt) = T(X2) = TÇX4) = -0.2 and T(X3) = 0.2. So, it remains to decide whether Xi < X2 or the converse. To achieve that we calculate the index G: G(X 1) = 0.6, whereas G(X2) = 0.2. Therefore, the ranking of X; is
X2 < Xi < X4 < X3.
4 This was not claimed in [24], but the proof of Theorem can be adapted in order to prove that < is a total order.
Notice that index G is inversely proportional to H5. In the sequel we introduce a new linear order, like the total order "<', but which is based on the indexes: S, Hi, Hj and T. The underlying motivation for that is not to provide a "better" total order, but an order which, although similar, is more faithful than "<' to the idea of Xu and Yager total order for V, which is exclusively based on the notion of score and accuracy functions,
X< Yiff
X<s Y, or
X=sY and Hi(X)<Hi(Y), or
X=SY and H\(X)=Hi (Y) and H5(X)<H5(Y), or ( '
[X=SY and HiX)=HiY) and HJX)=HJY) and T(X)<T(Y).
Notice that this order can be rewritten as:
fX<! Y, or
X<Y iff \ X=iY and X<SY, or (18)
[x=iY and X=5Y and T(X) < T(Y)
for any X, Y 6 L*.
Theorem 4.1 < is a total order onL*.
Proof Clearly, < is reflexive and transitive, i.e., it is a preorder. Suppose that there exist X, Y e L* such that X # Y, but X ^ Y and Y < X. For notational simplicity consider that X = ([a, b], [c, d]) and Y = ([e, /], [g, h]). From Equations (17), (11), (13), (14) and definition of index T, we have the next equation system:
a + b —c —d = e + f — g — h, (19)
a + b + c + d = e + f + g + h, (20)
b-a + d-c = f-e + h-g, (21)
b-a + c-d = f-e + g-h. (22)
By Equation (19) we have b - c = e + f- g- h- a + d and, by Equation (21), b-c = f-e + h- g + a- d and so
e + d = a + h. (23)
By Equation (20) a + c = e + f + g + h- b- d and Equation (21) gives -(a + c) = f-e + h-g-b-d and so
b + d = f + h. (24)
By Equation (21) we have d-c = f-e-g + h + a-b and by Equation (22) we have -(d -c) = f-e + g-h + a-b and so
a + f = e + b.
If a < e, then by Equation (23), d < h and so by Equation (24) / < b and by Equation (25) e < a which is a contradiction. Therefore, < is antisymmetric5 and so < is a partial order.
Let X, Y e L*. If 5(X) * S(Y) or H,(X) * Hi(Y) or HS(X) * H5(Y) or T(X) + r(Y), then clearly X < Y or Y < X. Otherwise, X <Y and Y <X and therefore, by symmetry, X = Y. Therefore, < is a total order.
Example 4.3 Consider the L*-values X, for i = 1,• • • ,4 in Example 4.1. Then, S(Xi) = S(X2) = -0.2 and S(X3) = S(X4) = 0. So we know that Xi as well as X2 are less than X3 and X4. They do not determine the order between Xi and X2 as well as between X3 and X4. Therefore, we need to calculate the accuracy index H\: H^Xx) = i/i(X2) = 0.6 and ffi(X3) = HiQL,) = 0.5 and so this does not provide new information on the order between Xi and X2 as well as X3 and X4. So, we need to compute the index Hs: H5(X 1) = 0.4, H5(X2) = 0.8 and H5(X3) = H5(X4) = 0.6. Thus, it remains to determine the order between X3 and X4; to achieve that we calculate the index T: T(X3) = 0.2 and TQCt) = -0.2. Hence,
Xi < X2 ^ X4 ^ x3.
If 5, Hi, Hj and T are interchanged in Equation (17) they provide another linear order. Therefore, such linear orders could also be used to rank IVIFMD, however we will consider only the order "<", since it is more faithful with respect to Xu and Yager total order. In fact, we use as first criteria the score index, as second the more reasonable extension of accuracy index, as third the second more reasonable extension of accuracy index and as the last criteria the index T.
In what follows we provide a parametric family composed by total orders for L*.
Theorem 4.2 Let "<" be a total order on L. Then the binary relation <* on L*, defined for any (Xi, Y-f), (Yu Y2) e L*, by
(•XiJD^C^.iy iff Xi < X2 or (Xi = X2 and Y2 < 7,) (26)
is a total order.
Proof The proof that <* is reflexive, antisymmetric and transitive, although extensive, is trivial. Since "<" is a total order, then for any (X,, K,), (K,, Y2) e L*, we have that Xi < X2 and in such case (X\,Y\)<*(X2,Y2), or Xi > X2 and in such case (X2,Y2)<*(Xi,Y{), or Xi = X2 and in such case we have again three alternatives: 1) Y, < Y2 and therefore (X2,r2)<*(Xi,yi), 2) Y2 < Yi and therefore (X1,y1)<*(X2,72), or 3) Yi = Y2 and so (Xi,yi)=(X2,y2). Hence, <* is total.
We will consider the total orders on L* which are obtained in this way and are based on Xu and Yager total orders "<xy" or the lexicographical order "<l".
Example 4.4 Consider the L*-values X„ with i = 1, ■ • • , 4 in Example 4.1. In order to provide a ranking for these L*-values, w.r.t. <*XY order, we compute the score of the interval-valued membership degree, i.e., s([0.1,0.3]) = i([0.2,0.2]) = -0.6
5 This part of the proof was strongly inspired in [24, Theorem 3.1] which is related to total order <
and i([0.1,0.4]) = i([0.2,0.3]) = -0.5. Now, we need to calculate their accuracy: /i([0.1,0.3]) = 0.8, /¡([0.2,0.2]) = 1, /¡([0.1,0.4]) = 0.7 and /¡([0.2,0.3]) = 0.9. Therefore: [0.1,0.3] <xr [0.2,0.2] <XY [0.1,0.4] <XY [0.2,0.3] and so
Xl <XY X2 ^xy X3 —xy X4.
On the other hand, the ranking of these same IVIFMD considering the total order <£ is the following:
Xl <lX3<£.X2<'X4.
Remark 4.1 The first three total orders « <XY) on L* are correct extensions for Xu and Yager order on L* and the usual order on [0,1], i.e., for any R e {< <xr], x < y iff <t>(x) R <p(y) and x <XY y iff i/r(x) R y). On the other hand, the total order " is correct with respect to the usual order on [0,1] and the "<£•" order.
Corollary 4.1 (L*, <}, <L*, <) and (L*, <XY) are bounded lattices.
Proof Since each total ordered set is trivially a lattice, then we just need to prove that (L*, <}, (L*, <) and (L*, <"XY) are bounded. But, it is an easy calculation to prove that 0l> and 1l> are the unique L*-values with score equal -1 and 1, respectively. Therefore, 0l* and 1l* are the bottom and top of the lattices (L*, =<) and (L*, <). It is also trivial in the case of <L*, <XY), since [0,0] and [1,1] are the minimal and the maximal elements of L with respect to the total order of Xu and Yager.
A fundamental property for a total order on a set, when there is not a natural total order, but just a partial one, as is the case of V, L and L*, is that it must extend the natural partial order. This property was called "admissible order" in [8] for the case of L. In fact, the two total orders on L* (and by isomorphism also L) are admissible (see Proposition 2.2 and Remark 2.2).
Theorem 4.3 If X <L. Y, then X < Y, X < Y and X <• Y.
Proof Let (Xi, y,), (X2, ¥2) e L* such that (Xi, y,) <L. (X2, Y2). Then, by Equation (7), Xi <l X2 and y2 <l . Without loss of generality we can suppose that (Xi, ) # (X2, y2) and therefore v(Xi) - v(yi) < v(X2) - v(y2), which implies that 5(Xt, y,) < S(X2,Y2). So, (Xi.Fi) < (X2,y2) and (Xi,yi) < (X2,y2). On the other hand, by Proposition 2.3, Xi <L X2 and y2 <l Yi. Therefore, by Equation (26), we have that (Xi,yi) <l(X2,Y2).
Thus, < and < are admissible total orders with respect to the usual order on L", i.e., <l>. Nevertheless, the total order <*XY, defined in Equation (26) by considering the order <xy, is not admissible. In fact, consider the next values: X = ([0.2,0.5], [0.2,0.3]) and Y = ([0.1,0.4], [0.3,0.4]). Then i([0.1,0.4]) = -0.5 < -0.3 = ¿'([0.2,0.5]) and therefore [0.1,0.4] <XY [0.2,0.5]. Hence, X <*XY Y. But, because [0.1,0.4] <L [0.2,0.5] and [0.2,0.3] <L [0.3,0.4], we have that Y <L. X. So the admissibility of "<" is not a sufficient condition for "<*" to be admissible, instead, as we will prove in the following, it is a necessary condition.
Fuzzy Inf. Eng. (2016) 8: 169-182_181
Proposition 4.1 Let <bea total order on L. If <* is an admissible total order on L", then < is an admissible order on L.
Proof Let X, Y e L such that X <L Y. Then, by Equation (7), (X, [0,0]) <L-(Y.i [0,0]). Thus, because <* is admissible, then (X, [0,0]) <* (Y, [0,0]). So, by Equation (26), X < Y or (X = Y and [0,0] < [0,0]) and therefore X < Y.
Open Problem: How could we provide a characterization of the admissible total order < on L such that <* is an admissible total order on L*.
5. Conclusion
In this paper, we provided a variant for the total order proposed by Wang, Li and Wang in [24] and also introduce a method to obtain total orders for L* from any total order on L. In particular, in this family we consider those which are generated by Xu and Yager and the lexicographical order in interval-valued fuzzy set context. We prove that these new orders are correct in the sense that they extend, up to the isomorphisms, the usual orders on [0,1] and L*. Finally, we also proved that three of the four total orders considers are admissible.
In [9] it is provided a way to obtain admissible total orders on L which are based on two aggregation functions. This construction can be extended in order to obtain new admissible total orders defined on L*. As a future work, we intend to answer the open problem stated at the end of the previous section.
Acknowledgments
This work was partially supported by the CNPq (Conselho Nacional de Desenvolvi-mento Científico e Tecnológico) Brazilian funding agency, under the Proc. No. 307681/2012-2. The authors are grateful to the anonymous reviewers for their constructive comments and suggestions on the improvement of our paper.
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