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Some properties of fuzzy dual cones
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Some properties of fuzzy dual cones O.V. Baskov 1 Universitetskiy prosp., 35, Peterhof, Saint Petersburg, 198504, Russian Federation Received 3 October 2015; received in revised form 19 January 2017; accepted 5 February 2017
Abstract The notion of dual cone is generalized to fuzzy case. The definition of fuzzy dual cones is given. It is shown that they obey properties similar to those of crisp dual cones. Relationship between α-cuts of fuzzy sets and their fuzzy dual cones is also studied. © 2017 Elsevier B.V. All rights reserved. Keywords: Fuzzy cone; Convex fuzzy cone; Dual cone
1. Introduction Fuzzy sets were first introduced by L.A. Zadeh in [11]. They generalize the notion of set belonging. Since then many notions and problems have received generalizations to fuzzy case. Amongst them, convex sets and cones have been defined and thoroughly studied [5]. Dual cone is an important concept in operation research, which helps to solve decision making and optimization problems. Hence a generalization of the concept of dual cone to the fuzzy case appears to be of interest. This paper presents a definition of dual cone of a fuzzy set that naturally generalizes the notion of dual cone of an ordinary set. The generalization being natural implies two aspects. First, if one takes a fuzzy set that is actually a crisp set, i.e. all its elements have membership degrees zero or one, and constructs its fuzzy dual cone, then this fuzzy cone must be in fact a crisp dual cone of that crisp set. And second, fuzzy dual cones must obey properties similar to the properties of crisp dual cones [4]. Thus, we will show that fuzzy dual cones are always convex and closed, that if we take two fuzzy sets such that one contains the other, the fuzzy dual cone of the latter contains the fuzzy dual cone of the former, that the fuzzy dual cone of a fuzzy set is also the fuzzy dual cone of the closure or the conical hull of that fuzzy set. We will also prove that the fuzzy dual cone of the fuzzy dual cone of a fuzzy set is the closure of the conical hull of the fuzzy set. But since fuzzy dual cones have more complex structure than crisp cones, some properties that are unique to fuzzy dual cones arise. In particular, we will study the α-cuts of fuzzy dual cones and show that under certain conditions α-cut of the fuzzy dual cone of a fuzzy set is, as a crisp set, the dual cone of the closure of the strict (1 − α)-cut of the fuzzy set. E-mail address: [email protected]. 1 Supported by Russian Foundation for Basic Research, project No. 17-07-00371.
http://dx.doi.org/10.1016/j.fss.2017.02.002 0165-0114/© 2017 Elsevier B.V. All rights reserved.
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Our study is also motivated by recent developments in the multicriteria decision making. Given a set of possible options X, a decision maker wants to choose a subset C (X) of “best” options from his/her point of view. Each option is evaluated using m criteria fi : X → R, i = 1, . . . , m, and it is assumed that the greater values of the criteria are better. These criteria can be rewritten as a vector criterion f = (f1 , . . . , fm ) defined on X with values in Rm . It is generally impossible to find the “best” option that maximizes all the criteria simultaneously, so various approaches have been developed that allow to determine a reasonably “good” choice [6]. For instance, the criteria can be aggregated into a single criterion that is to be maximized [10]. Another approach is to let the decision maker perform the choice, and aid him/her by reducing the set of options. It utilizes the notion of a preference relation that models the preferences of the decision maker. We write x1 x2 if the decision maker deciding between the two alternatives x1 , x2 chooses x1 and not x2 . If certain axioms hold [7], it can be proved that C (X) is contained in the Pareto set Pf (X) with respect to the vector criterion f : C (X) ⊆ Pf (X). This provides an upper estimate for the set of chosen options. For further reduction of this estimate additional information is required. For example, if the decision maker agrees to gain some values for a group of more significant criteria at the cost of some losses for other group of criteria, such compromise can be described using an information quantum u ∈ Rm [9]. Given a set of information quanta u1 , . . . , up ∈ Rm , one can obtain a narrower estimate for the set C (X) according to [9]. The preference relation is conical, and its associated cone contains the basis vectors e1 , . . . , em and all the quanta u1 , . . . , up . If we denote the generators of the dual cone of the conical hull of these vectors by g1 ,. . . , gq , then we may define a new vector criterion g as follows: for every x ∈ X let g (x) = g1 · f (x) , . . . , gq · f (x) , where the components are scalar products of the generators and the old vector criterion. Then the set of chosen options C (X) must be contained in the Pareto set Pg (X) with respect to the new vector criterion g: C (X) ⊆ Pg (X) ⊆ Pf (X). Thus the problem of narrowing the Pareto set is reduced to the problem of finding the generators of a dual cone. The article [8] considers the case when the information provided by the decision maker is fuzzy. Then the cone associated with the preference relation also becomes fuzzy, so in order to solve the Pareto set reduction problem we need to introduce fuzzy dual cones. The rest of the paper is organized as follows. The second section gives the necessary definitions. In the third section fuzzy conical hulls are discussed. The definition of fuzzy dual cones and their properties appear in the fourth section. 2. Preliminaries Let X be a set of objects. Definition 1. A fuzzy set in X is characterized by a membership function λ (x) which associates with each object x in X a real number λ (x) in the interval [0; 1] where the value λ (x) of λ at x represents the degree of membership of x in the fuzzy set [11]. We will use the same symbol λ for the fuzzy set itself. Notation x|α will be used for describing an element x with its associated membership degree α in a fuzzy set implied from the context. As a set of objects the m-dimensional vector space Rm will be considered. The definition of a convex fuzzy set was given in [11]. Definition 2. A fuzzy set λ is convex if λ (γ x + (1 − γ ) y) min {λ (x) ; λ (y)}
(1)
for all x, y ∈ Rm and γ ∈ (0; 1). Example 1. Consider the fuzzy set λ in R2 given by ⎧ 2 ⎪ 0 < ϕ < π2 , ⎨ π ϕ, λ (x) = π2 (π − ϕ) , π2 < ϕ < π, ⎪ ⎩ 0, otherwise,
(2)
where ϕ is the polar angle of the vector x. We can easily see that this fuzzy set is not convex since the inequality (1) does not hold for x = (1; 1), y = (−1; 1), and γ = 12 . But if we redefine λ (x) = 1 for vectors x on the ray ϕ = π2 , the fuzzy set will become convex.
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Definition 3. A fuzzy set λ is a fuzzy cone if λ (γ x) = λ (x) for all γ > 0 and x ∈ Rm [1]. Clearly, the fuzzy set (2) from the above example is a fuzzy cone. Note that we can define the value λ (0) arbitrarily, and λ will still remain a fuzzy cone. We will denote an α-cut of a fuzzy set λ by λα = {x ∈ Rm : λ (x) α}. A strict α-cut will be denoted by λ+ α = {x ∈ Rm : λ (x) > α}. Definition 4. A closure of a fuzzy set λ is the fuzzy set λ given by λ (x) = sup α,
(3)
α : x∈λα
where λα is the closure of the crisp set λα [3]. As usual, a fuzzy set λ is called closed if λ = λ. Note that from (3) it follows that λα =
λβ , so any α-cut of a
β<α
closed fuzzy set is a closed crisp set. 3. Fuzzy conical hulls Recall that cone {a1 , . . . , am } denotes a conical hull of vectors a1 , . . . , am ∈ Rm , that is, the minimal convex cone containing these vectors. A fuzzy conical hull is defined similarly [5]. Definition 5. A fuzzy conical hull of vectors a1 |α1 , . . . , ap |αp is the minimal convex fuzzy cone λ such that λ (ai ) αi for each i = 1, . . . , p.
We will denote λ = cone a1 |α1 , . . . , ap |αp . We will also assume λ (0) = 1 by definition. The fuzzy cone being minimal means that for any other convex fuzzy cone μ satisfying μ (ai ) αi for all i = 1, . . . , p and μ (0) = 1 it holds that λ (x) μ (x) for all x ∈ Rm .
Consider the fuzzy cone μ given by μ (0) = 1, μ (x) = 0 for all x ∈ / cone a1 , . . . , ap , and μ (x) =
max
min αi
(4)
p i : ϕi >0 ϕ1 ,...,ϕp : ϕi 0, x= ϕi ai i=1
otherwise. To verify cone is convex, take two nonzero vectors y, z for which μ (y) > 0, that this fuzzy
μ (z) > 0, that is, y, z ∈ cone a , and consider the vector x = γ y + − γ z for some γ ∈ 1). Let ϕ 1 , . . . , ϕp , . . . , a (1 ) (0; 1 p and ψ1 , . . . , ψp be sets of coefficients at which the maxima in (4) for μ (y) and μ (z) correspondingly are attained. p Then x = (γ ϕi + (1 − γ ) ψi ) ai . Since μ (x) is the maximum over all possible representations of x as a conical i=1
combination of the vectors a1 , . . . , ap , we have μ (x)
min
i : γ ϕi +(1−γ )ψi >0
αi . Now observe that γ ϕi +(1 − γ ) ψi > 0 iff
αi = min {μ (y) ; μ (z)}. Combining, we get μ (x) min {μ (y) ; μ (z)}
as desired. The cases when one or both of the vectors y, z are zero or do not belong to cone a1 , . . . , ap are trivial. The formula (4) gives the representation of the fuzzy conical hull λ = cone a1 |α1 , . . . , ap |αp . Indeed, take any p
nonzero vector x ∈ cone a1 , . . . , ap . Let ϕ1 , . . . , ϕp be the coefficients of the conical combination x = ϕi ai either ϕi > 0 or ψi > 0, hence
min
i : γ ϕi +(1−γ )ψi >0
i=1
at which the maximum in (4) for μ (x) is attained. Since λ is convex, λ (x) min λ (ai ) min αi = μ (x). And i : ϕi >0
i : ϕi >0
as λ by definition is the minimal convex fuzzy cone satisfying the necessary constraints, λ (x) μ (x). Therefore, λ (x) = μ (x). Other cases are trivial. Thus, any nonzero vector x with the membership degree λ (x) > 0 in the fuzzy conical hull λ = cone a1 |α1 , . . . , p
ap |αp can be represented as x = ϕi ai , ϕi 0. And vice versa, any vector x = ϕi ai with ϕi 0 has the i : αi λ(x)
membership degree λ (x) min αi . i : ϕi >0
i=1
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Next we define finitely-generated fuzzy cones. The definition is the same as for crisp cones [2]. Definition 6. A fuzzy cone that can be represented as a fuzzy conical hull of a finite number of vectors is called finitely-generated. The structure of finitely-generated fuzzy cones becomes clear if we consider the following result [2].
Assertion 1. Let λ = cone a1 |α1 , . . . , ap |αp , where α1 · · · αp . Then λ (x) =
max
αk for all nonzero
x ∈ Rm . Here the maximum over the empty set is assumed to be zero, so that λ (x) = 0 when x ∈ / cone a1 , . . . , ap . k : x∈cone{a1 ,...,ak }
Thus, a finitely-generated fuzzy cone can be viewed as an overlapping of cones with different associated degrees of membership. It also follows that finitely-generated fuzzy cones are always closed, as their α-cuts are closed for any α. In general case we can consider a fuzzy conical hull of an arbitrary fuzzy set. The following definition can be given [5]. Definition 7. A fuzzy conical hull (or fuzzy cone) of a fuzzy set λ is the minimal convex fuzzy cone containing λ. We will write μ = cone λ. In a similar way as above it can be shown that the formula μ (x) =
sup k
min λ (ai )
(5)
i
ϕi ai =x, ϕi >0
i=1
holds. 4. Fuzzy dual cones Recall that dual cone of some set M is defined as a set of vectors x such that xy 0 for all y ∈ M, where xy denotes the scalar product of these vectors. We can view it from the other side: the dual cone does not contain vectors x for which ∃y ∈ M : xy < 0. This formulation is used to generalize the notion of dual cone. Definition 8. Let λ be a fuzzy set. A fuzzy dual cone of λ is the fuzzy set μ given by μ (y) = for nonzero y
∈ Rm ,
and μ (0) = 1.
inf
x∈Rm : xy<0
(1 − λ (x))
Assertion 2. The definition is correct in that a fuzzy dual cone is indeed a fuzzy cone. Moreover, it is always convex and closed. Proof. In the notation of the definition, for all γ > 0 and nonzero y ∈ Rm we have μ (γ y) = λ (x)) =
inf
x∈Rm : xy>0
(1 − λ (x)) = μ (y), so the fuzzy set μ is a fuzzy cone.
inf
(1 −
x∈Rm : x(γ y)>0
To verify convexity, take two vectors y , y ∈ Rm and some γ ∈ (0; 1). If both of these vectors are nonzero, observe that for any x ∈ Rm : x γ y + (1 − γ ) y < 0 either xy < 0 or xy < 0, hence
1 − λ (x) min inf inf (1 − λ (z)) ; (1 − λ (z)) z∈Rm : zy <0 z∈Rm : zy <0
= min μ y ; μ y , and since x is arbitrary, μ γ y + (1 − γ ) y = If at least one of the vectors cone.
x∈Rm :
y,
y
inf
x(γ y
+(1−γ )y )<0
(1 − λ (x)) min μ y ; μ y .
is zero, the necessary property immediately follows from the fact that μ is a fuzzy
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Finally, suppose that the fuzzy cone μ is not closed. Then there exists a vector y ∈ Rm such that μ (y) > μ (y). Obviously, y = 0. Since μ (y) = sup α > μ (y), we can find β > μ (y) for which y ∈ μβ . Then there exists a α : y∈μα
sequence of vectors yk ∈ μβ converging to y. Take any vector x ∈ Rm : xy < 0. Since yk → y, starting with some number k we have xyk < 0, therefore μ (yk ) 1 − λ (x). From y ∈ μβ we obtain that 1 − λ (x) μ (yk ) β. As x was chosen arbitrarily, inf (1 − λ (x)) β, that is, μ (y) β, which contradicts the choice of β. Thus, the m x∈R : xy<0
supposition was wrong, and the fuzzy cone μ is closed.
2
The following assertions give some basic properties of fuzzy dual cones. Assertion 3. Let λ1 and λ2 be fuzzy sets such that λ1 ⊆ λ2 , that is, λ1 (x) λ2 (x) for all x ∈ Rm . Let μ1 be the fuzzy dual cone of λ1 , and μ2 be the fuzzy dual cone of λ2 . Then μ1 ⊇ μ2 . Proof. Since λ1 (x) λ2 (x) for all x ∈ Rm , we have μ1 (y) =
inf
x∈Rm : xy<0
(1 − λ1 (x))
inf
x∈Rm : xy<0
for all y ∈ Rm , which means that μ1 ⊇ μ2 .
(1 − λ2 (x)) = μ2 (y)
2
Assertion 4. Let λ be a fuzzy set, and μ be its fuzzy dual cone. Then μ is also the fuzzy dual cone of λ. Proof. Denote the fuzzy dual cone of λ by ν. We need to show that μ = ν. From λ ⊆ λ and the previous assertion it follows that μ ⊇ ν, so it is sufficient to show that μ ⊆ ν. Suppose the contrary: ∃x ∗ ∈ Rm : μ (x ∗ ) > ν (x ∗ ). As ∗ 1 − λ (y) , for any ε > 0 we can find a vector yε ∈ Rm such that x ∗ yε < 0 and 1 − λ (yε ) inf ∗ ν (x ) = m y∈R : x y<0
ν (x ∗ ) + ε. Next, as λ (yε ) =
sup α : yε ∈λα
α, for any δ > 0 we can find αδ such that yε ∈ λαδ and λ (yε ) − δ αδ λ (yε ).
As x ∗ yε < 0, there exists a neighborhood U (yε ) of yε such that x ∗ y < 0 for all y ∈ U (yε ). And since yε ∈ λαδ , there exists y ∗ ∈ U (yε ) such that y ∗ ∈ λαδ . Then λ (y ∗ ) αδ λ (yε ) − δ 1 − ν (x ∗ ) − ε − δ. As y ∗ ∈ U (yε ), we have x ∗ y ∗ < 0, so μ (x ∗ ) 1 − λ (y ∗ ) ν (x ∗ ) + ε + δ. By supposition μ (x ∗ ) > ν (x ∗ ), so we can choose δ = ε = 13 (μ (x ∗ ) − ν (x ∗ )) > 0. Then we obtain μ (x ∗ ) 23 μ (x ∗ ) + 13 ν (x ∗ ), or μ (x ∗ ) ν (x ∗ ), which contradicts the supposition. Thus, μ ⊆ ν, and in conjunction with the opposite inclusion we get μ = ν. 2 Assertion 5. Let λ be a fuzzy set, and μ be its fuzzy conical hull. Then the fuzzy dual cones of λ and μ coincide. Proof. Denote the fuzzy dual cone of λ by ν. Since λ ⊆ μ, ν contains the fuzzy dual cone of μ, so it suffices to show that ν (x) inf (1 − μ (y)). If for all y ∈ Rm : xy < 0 it holds that μ (y) = 0, then the infimum m y∈R : xy<0
equals to 1, and ν (x) 1 is trivial. Consider some vector y ∈ Rm : xy < 0, μ (y) > 0. Then by (5) we have k min λ (ai ), so for any ε > 0 we can find the combination ϕi ai = y, ϕi > 0 for which μ (y) = sup k
i
i=1
ϕi ai =y, ϕi >0
i=1
μ (y) − ε min λ (ai ) μ (y). Now note that there exists at least one vector ai ∗ satisfying xai ∗ < 0, since othi
erwise we would have xy = ε
k
ϕi xai 0. Then μ (y) − ε min λ (ai ) λ (ai ∗ ), and 1 − μ (y) 1 − λ (ai ∗ ) − i
i=1
inf m
z∈R : xz<0
(1 − λ (z)) − ε = ν (x) − ε. Now let ε → 0 to obtain 1 − μ (y) ν (x). Since it is true for all
y ∈ Rm : xy < 0, we can write
inf
y∈Rm : xy<0
(1 − μ (y)) ν (x) as desired.
2
Next comes one of the main properties of dual cones. Recall that for crisp sets the dual cone of the dual cone of a set is the least convex closed cone containing the set. The same property holds for fuzzy dual cones.
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Assertion 6. The fuzzy dual cone of the fuzzy dual cone of a fuzzy set is the closure of the fuzzy conical hull of that fuzzy set. Proof. Denote the fuzzy set by η. Let λ be the closure of its conical hull: λ = cone η. Denote the fuzzy dual cone of η by μ. By previous assertions μ is also the fuzzy dual cone of λ. Next, let ν be the fuzzy dual cone of μ. We need to show that ν = λ. Consider some nonzero vector x ∈ Rm . For any vector y ∈ Rm such that xy < 0 we have μ (y) = inf (1 − m z∈R : zy<0
λ (z)) 1 − λ (x), or λ (x) 1 − μ (y). Since it is true for any y ∈ Rm : xy < 0, we may write λ (x) inf (1 − μ (y)) = ν (x). Thus, λ ⊆ ν. m y∈R : xy<0
Suppose that λ = ν, that is, ∃x ∗ ∈ Rm : λ (x ∗ ) < ν (x ∗ ). Choose a number α such that λ (x ∗ ) < α < ν (x ∗ ). By definition we demanded that all the fuzzy cones in question contain the zero vector with the membership degree / λα . Since λ is convex and closed, so is λα . Then we can apply equal to one, so x ∗ = 0. Next, by supposition x ∗ ∈ Farkas’s lemma and find a nonzero vector n ∈ Rm such that nx ∗ < 0 while ny 0 for all y ∈ λα . As for all y ∈ Rm : λ (y) α we have ny 0, if we take z ∈ Rm : nz < 0, then necessarily λ (z) < α, or 1 − λ (z) > 1 − α. Thus, μ (n) = inf (1 − λ (z)) 1 − α. But from nx ∗ < 0 we obtain ν (x ∗ ) 1 − μ (n) α, which contradicts the m z∈R : zn<0
supposition. Thus, λ = ν.
2
Example 2. As an illustration, consider the fuzzy cone λ given by (2). If we use the definition to construct its fuzzy dual cone μ, we will get ⎧ 2 ⎪ 0 < ϕ π2 , ⎨ π ϕ, μ (y) = π2 (π − ϕ) , π2 < ϕ < π, ⎪ ⎩ 0, otherwise, where ϕ is the polar angle of y. Next, by definition it is easy to verify that the fuzzy dual cone of μ is μ itself, that is, μ is self-dual. And as we have already discussed, the least convex fuzzy cone containing λ is indeed μ. Finally, let us return to the fact that fuzzy cones can be viewed as overlappings of crisp cones. The following assertion investigates relations between layers in these overlappings in fuzzy cones that are dual to each other. Assertion 7. Let λ and μ be two fuzzy cones dual to each other. Then the crisp cones λ+ α and μ1−α are dual to each other for all α ∈ [0; 1). + Proof. Let C be the dual cone of λ+ α . Take some vector x ∈ C. Then all vectors y ∈ λα satisfy xy 0. So for any vecm + / λα , or λ (z) α, or 1 − λ (z) 1 − α. Then μ (x) = inf tor z ∈ R such that xz < 0 we have z ∈ (1 − λ (z)) m
1 − α, that is, x ∈ μ1−α . And vice versa, take x ∈ μ1−α . From μ (x) =
inf
z∈Rm :
zx<0
z∈R : zx<0
(1 − λ (z)) 1 − α we obtain
1 − λ (z) 1 − α for all z ∈ Rm : zx < 0. Then for all y ∈ λ+ α , as λ (y) > α, or 1 − λ (y) < 1 − α, we get xy 0. This + means that x ∈ C. Thus, μ1−α is the dual cone of λα , and hence the dual cone of λ+ α . To complete the proof, note that + + the dual cone of μ1−α is the least convex cone containing λα , which is exactly λα . 2 The case α = 0 deserves special attention if we recall that the sets λ+ 0 and λ1 have special names: the support and the kernel of the fuzzy set λ, respectively. Corollary 1. If two fuzzy cones are dual to each other, then the closure of the support of one fuzzy cone is the dual cone of the kernel of the other fuzzy cone. Furthermore, one may have noticed that in the first part of the proof the use has not been made of the fact that λ is a convex fuzzy cone. Therefore, we can formulate the following corollary.
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Corollary 2. Let μ be the fuzzy dual cone of some fuzzy set λ. Then the crisp cone μ1−α is the dual cone of λ+ α for all α ∈ [0; 1). Unfortunately, the same cannot be said of μ+ α and λ1−α . To illustrate this, consider the following example. Example 3. Let λ be the fuzzy cone in R2 given by ⎧2 0 < ϕ < π2 , ⎪ π ϕ, ⎪ ⎪ ⎨ 2 (π − ϕ) , π < ϕ < π, 2 λ (x) = π ⎪ 1, ϕ = π, ⎪ ⎪ ⎩ 0, otherwise, where ϕ is the polar angle of x. It is easy to see that λ is neither convex nor closed. If we construct the fuzzy dual cone of λ, we get 2 (π − ϕ) , π2 ϕ π, μ (y) = π 0, otherwise, where ϕ is the polar angle of y. Now note that the kernel of λ consists solely of the ray ϕ = π , and the closure of the support of μ equals to the second quadrant π2 ϕ π , while the dual cone of the kernel of λ is the left halfplane π 2
ϕ
3π 2 .
Thus, μ+ 0 is not the dual cone of λ1 .
5. Conclusion The given definition of fuzzy dual cones can be viewed as a natural generalization of dual cones to the fuzzy case. This is confirmed by the fact that dual fuzzy cones obey the same basic properties as crisp dual cones. Some new properties also arise due to more complex structure of fuzzy cones. References [1] Elsaid E. Ammar, Some properties of convex fuzzy sets and convex fuzzy cones, Fuzzy Sets Syst. 106 (3) (1999) 381–386. [2] O.V. Baskov, Algorithm for recalculating the generatrix lines of a finitely generated fuzzy cone after adding a generatrix to its dual cone, Comput. Math. Math. Phys. 55 (2015) 206–211. [3] Loredana Biacino, An extension principle for closure operators, J. Math. Anal. Appl. 198 (1996) 1–24. [4] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, 2004. [5] R. Lowen, Convex fuzzy sets, Fuzzy Sets Syst. 3 (1980) 291–310. [6] Kaisa M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, 1999. [7] V.D. Noghin, Estimation of the set of nondominated solutions, Numer. Funct. Anal. Appl. 12 (5–6) (1992) 507–515. [8] V.D. Noghin, Upper estimate for a fuzzy set of nondominated solutions, Fuzzy Sets Syst. 67 (3) (1994) 303–315. [9] V.D. Noghin, O.V. Baskov, Pareto set reduction based on an arbitrary finite collection of numerical information on the preference relation, Dokl. Math. 83 (3) (2011) 418–420. [10] J. Ramik, M. Vlach, Aggregation functions and generalized convexity in fuzzy optimization and decision making, Ann. Oper. Res. 195 (1) (2012) 261–276. [11] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Some properties of fuzzy dual cones O.V. Baskov 1 Universitetskiy prosp., 35, Peterhof, Saint Petersburg, 198504, Russian Federation Received 3 October 2015; received in revised form 19 January 2017; accepted 5 February 2017
Abstract The notion of dual cone is generalized to fuzzy case. The definition of fuzzy dual cones is given. It is shown that they obey properties similar to those of crisp dual cones. Relationship between α-cuts of fuzzy sets and their fuzzy dual cones is also studied. © 2017 Elsevier B.V. All rights reserved. Keywords: Fuzzy cone; Convex fuzzy cone; Dual cone
1. Introduction Fuzzy sets were first introduced by L.A. Zadeh in [11]. They generalize the notion of set belonging. Since then many notions and problems have received generalizations to fuzzy case. Amongst them, convex sets and cones have been defined and thoroughly studied [5]. Dual cone is an important concept in operation research, which helps to solve decision making and optimization problems. Hence a generalization of the concept of dual cone to the fuzzy case appears to be of interest. This paper presents a definition of dual cone of a fuzzy set that naturally generalizes the notion of dual cone of an ordinary set. The generalization being natural implies two aspects. First, if one takes a fuzzy set that is actually a crisp set, i.e. all its elements have membership degrees zero or one, and constructs its fuzzy dual cone, then this fuzzy cone must be in fact a crisp dual cone of that crisp set. And second, fuzzy dual cones must obey properties similar to the properties of crisp dual cones [4]. Thus, we will show that fuzzy dual cones are always convex and closed, that if we take two fuzzy sets such that one contains the other, the fuzzy dual cone of the latter contains the fuzzy dual cone of the former, that the fuzzy dual cone of a fuzzy set is also the fuzzy dual cone of the closure or the conical hull of that fuzzy set. We will also prove that the fuzzy dual cone of the fuzzy dual cone of a fuzzy set is the closure of the conical hull of the fuzzy set. But since fuzzy dual cones have more complex structure than crisp cones, some properties that are unique to fuzzy dual cones arise. In particular, we will study the α-cuts of fuzzy dual cones and show that under certain conditions α-cut of the fuzzy dual cone of a fuzzy set is, as a crisp set, the dual cone of the closure of the strict (1 − α)-cut of the fuzzy set. E-mail address: [email protected]. 1 Supported by Russian Foundation for Basic Research, project No. 17-07-00371.
http://dx.doi.org/10.1016/j.fss.2017.02.002 0165-0114/© 2017 Elsevier B.V. All rights reserved.
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Our study is also motivated by recent developments in the multicriteria decision making. Given a set of possible options X, a decision maker wants to choose a subset C (X) of “best” options from his/her point of view. Each option is evaluated using m criteria fi : X → R, i = 1, . . . , m, and it is assumed that the greater values of the criteria are better. These criteria can be rewritten as a vector criterion f = (f1 , . . . , fm ) defined on X with values in Rm . It is generally impossible to find the “best” option that maximizes all the criteria simultaneously, so various approaches have been developed that allow to determine a reasonably “good” choice [6]. For instance, the criteria can be aggregated into a single criterion that is to be maximized [10]. Another approach is to let the decision maker perform the choice, and aid him/her by reducing the set of options. It utilizes the notion of a preference relation that models the preferences of the decision maker. We write x1 x2 if the decision maker deciding between the two alternatives x1 , x2 chooses x1 and not x2 . If certain axioms hold [7], it can be proved that C (X) is contained in the Pareto set Pf (X) with respect to the vector criterion f : C (X) ⊆ Pf (X). This provides an upper estimate for the set of chosen options. For further reduction of this estimate additional information is required. For example, if the decision maker agrees to gain some values for a group of more significant criteria at the cost of some losses for other group of criteria, such compromise can be described using an information quantum u ∈ Rm [9]. Given a set of information quanta u1 , . . . , up ∈ Rm , one can obtain a narrower estimate for the set C (X) according to [9]. The preference relation is conical, and its associated cone contains the basis vectors e1 , . . . , em and all the quanta u1 , . . . , up . If we denote the generators of the dual cone of the conical hull of these vectors by g1 ,. . . , gq , then we may define a new vector criterion g as follows: for every x ∈ X let g (x) = g1 · f (x) , . . . , gq · f (x) , where the components are scalar products of the generators and the old vector criterion. Then the set of chosen options C (X) must be contained in the Pareto set Pg (X) with respect to the new vector criterion g: C (X) ⊆ Pg (X) ⊆ Pf (X). Thus the problem of narrowing the Pareto set is reduced to the problem of finding the generators of a dual cone. The article [8] considers the case when the information provided by the decision maker is fuzzy. Then the cone associated with the preference relation also becomes fuzzy, so in order to solve the Pareto set reduction problem we need to introduce fuzzy dual cones. The rest of the paper is organized as follows. The second section gives the necessary definitions. In the third section fuzzy conical hulls are discussed. The definition of fuzzy dual cones and their properties appear in the fourth section. 2. Preliminaries Let X be a set of objects. Definition 1. A fuzzy set in X is characterized by a membership function λ (x) which associates with each object x in X a real number λ (x) in the interval [0; 1] where the value λ (x) of λ at x represents the degree of membership of x in the fuzzy set [11]. We will use the same symbol λ for the fuzzy set itself. Notation x|α will be used for describing an element x with its associated membership degree α in a fuzzy set implied from the context. As a set of objects the m-dimensional vector space Rm will be considered. The definition of a convex fuzzy set was given in [11]. Definition 2. A fuzzy set λ is convex if λ (γ x + (1 − γ ) y) min {λ (x) ; λ (y)}
(1)
for all x, y ∈ Rm and γ ∈ (0; 1). Example 1. Consider the fuzzy set λ in R2 given by ⎧ 2 ⎪ 0 < ϕ < π2 , ⎨ π ϕ, λ (x) = π2 (π − ϕ) , π2 < ϕ < π, ⎪ ⎩ 0, otherwise,
(2)
where ϕ is the polar angle of the vector x. We can easily see that this fuzzy set is not convex since the inequality (1) does not hold for x = (1; 1), y = (−1; 1), and γ = 12 . But if we redefine λ (x) = 1 for vectors x on the ray ϕ = π2 , the fuzzy set will become convex.
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Definition 3. A fuzzy set λ is a fuzzy cone if λ (γ x) = λ (x) for all γ > 0 and x ∈ Rm [1]. Clearly, the fuzzy set (2) from the above example is a fuzzy cone. Note that we can define the value λ (0) arbitrarily, and λ will still remain a fuzzy cone. We will denote an α-cut of a fuzzy set λ by λα = {x ∈ Rm : λ (x) α}. A strict α-cut will be denoted by λ+ α = {x ∈ Rm : λ (x) > α}. Definition 4. A closure of a fuzzy set λ is the fuzzy set λ given by λ (x) = sup α,
(3)
α : x∈λα
where λα is the closure of the crisp set λα [3]. As usual, a fuzzy set λ is called closed if λ = λ. Note that from (3) it follows that λα =
λβ , so any α-cut of a
β<α
closed fuzzy set is a closed crisp set. 3. Fuzzy conical hulls Recall that cone {a1 , . . . , am } denotes a conical hull of vectors a1 , . . . , am ∈ Rm , that is, the minimal convex cone containing these vectors. A fuzzy conical hull is defined similarly [5]. Definition 5. A fuzzy conical hull of vectors a1 |α1 , . . . , ap |αp is the minimal convex fuzzy cone λ such that λ (ai ) αi for each i = 1, . . . , p.
We will denote λ = cone a1 |α1 , . . . , ap |αp . We will also assume λ (0) = 1 by definition. The fuzzy cone being minimal means that for any other convex fuzzy cone μ satisfying μ (ai ) αi for all i = 1, . . . , p and μ (0) = 1 it holds that λ (x) μ (x) for all x ∈ Rm .
Consider the fuzzy cone μ given by μ (0) = 1, μ (x) = 0 for all x ∈ / cone a1 , . . . , ap , and μ (x) =
max
min αi
(4)
p i : ϕi >0 ϕ1 ,...,ϕp : ϕi 0, x= ϕi ai i=1
otherwise. To verify cone is convex, take two nonzero vectors y, z for which μ (y) > 0, that this fuzzy
μ (z) > 0, that is, y, z ∈ cone a , and consider the vector x = γ y + − γ z for some γ ∈ 1). Let ϕ 1 , . . . , ϕp , . . . , a (1 ) (0; 1 p and ψ1 , . . . , ψp be sets of coefficients at which the maxima in (4) for μ (y) and μ (z) correspondingly are attained. p Then x = (γ ϕi + (1 − γ ) ψi ) ai . Since μ (x) is the maximum over all possible representations of x as a conical i=1
combination of the vectors a1 , . . . , ap , we have μ (x)
min
i : γ ϕi +(1−γ )ψi >0
αi . Now observe that γ ϕi +(1 − γ ) ψi > 0 iff
αi = min {μ (y) ; μ (z)}. Combining, we get μ (x) min {μ (y) ; μ (z)}
as desired. The cases when one or both of the vectors y, z are zero or do not belong to cone a1 , . . . , ap are trivial. The formula (4) gives the representation of the fuzzy conical hull λ = cone a1 |α1 , . . . , ap |αp . Indeed, take any p
nonzero vector x ∈ cone a1 , . . . , ap . Let ϕ1 , . . . , ϕp be the coefficients of the conical combination x = ϕi ai either ϕi > 0 or ψi > 0, hence
min
i : γ ϕi +(1−γ )ψi >0
i=1
at which the maximum in (4) for μ (x) is attained. Since λ is convex, λ (x) min λ (ai ) min αi = μ (x). And i : ϕi >0
i : ϕi >0
as λ by definition is the minimal convex fuzzy cone satisfying the necessary constraints, λ (x) μ (x). Therefore, λ (x) = μ (x). Other cases are trivial. Thus, any nonzero vector x with the membership degree λ (x) > 0 in the fuzzy conical hull λ = cone a1 |α1 , . . . , p
ap |αp can be represented as x = ϕi ai , ϕi 0. And vice versa, any vector x = ϕi ai with ϕi 0 has the i : αi λ(x)
membership degree λ (x) min αi . i : ϕi >0
i=1
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Next we define finitely-generated fuzzy cones. The definition is the same as for crisp cones [2]. Definition 6. A fuzzy cone that can be represented as a fuzzy conical hull of a finite number of vectors is called finitely-generated. The structure of finitely-generated fuzzy cones becomes clear if we consider the following result [2].
Assertion 1. Let λ = cone a1 |α1 , . . . , ap |αp , where α1 · · · αp . Then λ (x) =
max
αk for all nonzero
x ∈ Rm . Here the maximum over the empty set is assumed to be zero, so that λ (x) = 0 when x ∈ / cone a1 , . . . , ap . k : x∈cone{a1 ,...,ak }
Thus, a finitely-generated fuzzy cone can be viewed as an overlapping of cones with different associated degrees of membership. It also follows that finitely-generated fuzzy cones are always closed, as their α-cuts are closed for any α. In general case we can consider a fuzzy conical hull of an arbitrary fuzzy set. The following definition can be given [5]. Definition 7. A fuzzy conical hull (or fuzzy cone) of a fuzzy set λ is the minimal convex fuzzy cone containing λ. We will write μ = cone λ. In a similar way as above it can be shown that the formula μ (x) =
sup k
min λ (ai )
(5)
i
ϕi ai =x, ϕi >0
i=1
holds. 4. Fuzzy dual cones Recall that dual cone of some set M is defined as a set of vectors x such that xy 0 for all y ∈ M, where xy denotes the scalar product of these vectors. We can view it from the other side: the dual cone does not contain vectors x for which ∃y ∈ M : xy < 0. This formulation is used to generalize the notion of dual cone. Definition 8. Let λ be a fuzzy set. A fuzzy dual cone of λ is the fuzzy set μ given by μ (y) = for nonzero y
∈ Rm ,
and μ (0) = 1.
inf
x∈Rm : xy<0
(1 − λ (x))
Assertion 2. The definition is correct in that a fuzzy dual cone is indeed a fuzzy cone. Moreover, it is always convex and closed. Proof. In the notation of the definition, for all γ > 0 and nonzero y ∈ Rm we have μ (γ y) = λ (x)) =
inf
x∈Rm : xy>0
(1 − λ (x)) = μ (y), so the fuzzy set μ is a fuzzy cone.
inf
(1 −
x∈Rm : x(γ y)>0
To verify convexity, take two vectors y , y ∈ Rm and some γ ∈ (0; 1). If both of these vectors are nonzero, observe that for any x ∈ Rm : x γ y + (1 − γ ) y < 0 either xy < 0 or xy < 0, hence
1 − λ (x) min inf inf (1 − λ (z)) ; (1 − λ (z)) z∈Rm : zy <0 z∈Rm : zy <0
= min μ y ; μ y , and since x is arbitrary, μ γ y + (1 − γ ) y = If at least one of the vectors cone.
x∈Rm :
y,
y
inf
x(γ y
+(1−γ )y )<0
(1 − λ (x)) min μ y ; μ y .
is zero, the necessary property immediately follows from the fact that μ is a fuzzy
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Finally, suppose that the fuzzy cone μ is not closed. Then there exists a vector y ∈ Rm such that μ (y) > μ (y). Obviously, y = 0. Since μ (y) = sup α > μ (y), we can find β > μ (y) for which y ∈ μβ . Then there exists a α : y∈μα
sequence of vectors yk ∈ μβ converging to y. Take any vector x ∈ Rm : xy < 0. Since yk → y, starting with some number k we have xyk < 0, therefore μ (yk ) 1 − λ (x). From y ∈ μβ we obtain that 1 − λ (x) μ (yk ) β. As x was chosen arbitrarily, inf (1 − λ (x)) β, that is, μ (y) β, which contradicts the choice of β. Thus, the m x∈R : xy<0
supposition was wrong, and the fuzzy cone μ is closed.
2
The following assertions give some basic properties of fuzzy dual cones. Assertion 3. Let λ1 and λ2 be fuzzy sets such that λ1 ⊆ λ2 , that is, λ1 (x) λ2 (x) for all x ∈ Rm . Let μ1 be the fuzzy dual cone of λ1 , and μ2 be the fuzzy dual cone of λ2 . Then μ1 ⊇ μ2 . Proof. Since λ1 (x) λ2 (x) for all x ∈ Rm , we have μ1 (y) =
inf
x∈Rm : xy<0
(1 − λ1 (x))
inf
x∈Rm : xy<0
for all y ∈ Rm , which means that μ1 ⊇ μ2 .
(1 − λ2 (x)) = μ2 (y)
2
Assertion 4. Let λ be a fuzzy set, and μ be its fuzzy dual cone. Then μ is also the fuzzy dual cone of λ. Proof. Denote the fuzzy dual cone of λ by ν. We need to show that μ = ν. From λ ⊆ λ and the previous assertion it follows that μ ⊇ ν, so it is sufficient to show that μ ⊆ ν. Suppose the contrary: ∃x ∗ ∈ Rm : μ (x ∗ ) > ν (x ∗ ). As ∗ 1 − λ (y) , for any ε > 0 we can find a vector yε ∈ Rm such that x ∗ yε < 0 and 1 − λ (yε ) inf ∗ ν (x ) = m y∈R : x y<0
ν (x ∗ ) + ε. Next, as λ (yε ) =
sup α : yε ∈λα
α, for any δ > 0 we can find αδ such that yε ∈ λαδ and λ (yε ) − δ αδ λ (yε ).
As x ∗ yε < 0, there exists a neighborhood U (yε ) of yε such that x ∗ y < 0 for all y ∈ U (yε ). And since yε ∈ λαδ , there exists y ∗ ∈ U (yε ) such that y ∗ ∈ λαδ . Then λ (y ∗ ) αδ λ (yε ) − δ 1 − ν (x ∗ ) − ε − δ. As y ∗ ∈ U (yε ), we have x ∗ y ∗ < 0, so μ (x ∗ ) 1 − λ (y ∗ ) ν (x ∗ ) + ε + δ. By supposition μ (x ∗ ) > ν (x ∗ ), so we can choose δ = ε = 13 (μ (x ∗ ) − ν (x ∗ )) > 0. Then we obtain μ (x ∗ ) 23 μ (x ∗ ) + 13 ν (x ∗ ), or μ (x ∗ ) ν (x ∗ ), which contradicts the supposition. Thus, μ ⊆ ν, and in conjunction with the opposite inclusion we get μ = ν. 2 Assertion 5. Let λ be a fuzzy set, and μ be its fuzzy conical hull. Then the fuzzy dual cones of λ and μ coincide. Proof. Denote the fuzzy dual cone of λ by ν. Since λ ⊆ μ, ν contains the fuzzy dual cone of μ, so it suffices to show that ν (x) inf (1 − μ (y)). If for all y ∈ Rm : xy < 0 it holds that μ (y) = 0, then the infimum m y∈R : xy<0
equals to 1, and ν (x) 1 is trivial. Consider some vector y ∈ Rm : xy < 0, μ (y) > 0. Then by (5) we have k min λ (ai ), so for any ε > 0 we can find the combination ϕi ai = y, ϕi > 0 for which μ (y) = sup k
i
i=1
ϕi ai =y, ϕi >0
i=1
μ (y) − ε min λ (ai ) μ (y). Now note that there exists at least one vector ai ∗ satisfying xai ∗ < 0, since othi
erwise we would have xy = ε
k
ϕi xai 0. Then μ (y) − ε min λ (ai ) λ (ai ∗ ), and 1 − μ (y) 1 − λ (ai ∗ ) − i
i=1
inf m
z∈R : xz<0
(1 − λ (z)) − ε = ν (x) − ε. Now let ε → 0 to obtain 1 − μ (y) ν (x). Since it is true for all
y ∈ Rm : xy < 0, we can write
inf
y∈Rm : xy<0
(1 − μ (y)) ν (x) as desired.
2
Next comes one of the main properties of dual cones. Recall that for crisp sets the dual cone of the dual cone of a set is the least convex closed cone containing the set. The same property holds for fuzzy dual cones.
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Assertion 6. The fuzzy dual cone of the fuzzy dual cone of a fuzzy set is the closure of the fuzzy conical hull of that fuzzy set. Proof. Denote the fuzzy set by η. Let λ be the closure of its conical hull: λ = cone η. Denote the fuzzy dual cone of η by μ. By previous assertions μ is also the fuzzy dual cone of λ. Next, let ν be the fuzzy dual cone of μ. We need to show that ν = λ. Consider some nonzero vector x ∈ Rm . For any vector y ∈ Rm such that xy < 0 we have μ (y) = inf (1 − m z∈R : zy<0
λ (z)) 1 − λ (x), or λ (x) 1 − μ (y). Since it is true for any y ∈ Rm : xy < 0, we may write λ (x) inf (1 − μ (y)) = ν (x). Thus, λ ⊆ ν. m y∈R : xy<0
Suppose that λ = ν, that is, ∃x ∗ ∈ Rm : λ (x ∗ ) < ν (x ∗ ). Choose a number α such that λ (x ∗ ) < α < ν (x ∗ ). By definition we demanded that all the fuzzy cones in question contain the zero vector with the membership degree / λα . Since λ is convex and closed, so is λα . Then we can apply equal to one, so x ∗ = 0. Next, by supposition x ∗ ∈ Farkas’s lemma and find a nonzero vector n ∈ Rm such that nx ∗ < 0 while ny 0 for all y ∈ λα . As for all y ∈ Rm : λ (y) α we have ny 0, if we take z ∈ Rm : nz < 0, then necessarily λ (z) < α, or 1 − λ (z) > 1 − α. Thus, μ (n) = inf (1 − λ (z)) 1 − α. But from nx ∗ < 0 we obtain ν (x ∗ ) 1 − μ (n) α, which contradicts the m z∈R : zn<0
supposition. Thus, λ = ν.
2
Example 2. As an illustration, consider the fuzzy cone λ given by (2). If we use the definition to construct its fuzzy dual cone μ, we will get ⎧ 2 ⎪ 0 < ϕ π2 , ⎨ π ϕ, μ (y) = π2 (π − ϕ) , π2 < ϕ < π, ⎪ ⎩ 0, otherwise, where ϕ is the polar angle of y. Next, by definition it is easy to verify that the fuzzy dual cone of μ is μ itself, that is, μ is self-dual. And as we have already discussed, the least convex fuzzy cone containing λ is indeed μ. Finally, let us return to the fact that fuzzy cones can be viewed as overlappings of crisp cones. The following assertion investigates relations between layers in these overlappings in fuzzy cones that are dual to each other. Assertion 7. Let λ and μ be two fuzzy cones dual to each other. Then the crisp cones λ+ α and μ1−α are dual to each other for all α ∈ [0; 1). + Proof. Let C be the dual cone of λ+ α . Take some vector x ∈ C. Then all vectors y ∈ λα satisfy xy 0. So for any vecm + / λα , or λ (z) α, or 1 − λ (z) 1 − α. Then μ (x) = inf tor z ∈ R such that xz < 0 we have z ∈ (1 − λ (z)) m
1 − α, that is, x ∈ μ1−α . And vice versa, take x ∈ μ1−α . From μ (x) =
inf
z∈Rm :
zx<0
z∈R : zx<0
(1 − λ (z)) 1 − α we obtain
1 − λ (z) 1 − α for all z ∈ Rm : zx < 0. Then for all y ∈ λ+ α , as λ (y) > α, or 1 − λ (y) < 1 − α, we get xy 0. This + means that x ∈ C. Thus, μ1−α is the dual cone of λα , and hence the dual cone of λ+ α . To complete the proof, note that + + the dual cone of μ1−α is the least convex cone containing λα , which is exactly λα . 2 The case α = 0 deserves special attention if we recall that the sets λ+ 0 and λ1 have special names: the support and the kernel of the fuzzy set λ, respectively. Corollary 1. If two fuzzy cones are dual to each other, then the closure of the support of one fuzzy cone is the dual cone of the kernel of the other fuzzy cone. Furthermore, one may have noticed that in the first part of the proof the use has not been made of the fact that λ is a convex fuzzy cone. Therefore, we can formulate the following corollary.
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Corollary 2. Let μ be the fuzzy dual cone of some fuzzy set λ. Then the crisp cone μ1−α is the dual cone of λ+ α for all α ∈ [0; 1). Unfortunately, the same cannot be said of μ+ α and λ1−α . To illustrate this, consider the following example. Example 3. Let λ be the fuzzy cone in R2 given by ⎧2 0 < ϕ < π2 , ⎪ π ϕ, ⎪ ⎪ ⎨ 2 (π − ϕ) , π < ϕ < π, 2 λ (x) = π ⎪ 1, ϕ = π, ⎪ ⎪ ⎩ 0, otherwise, where ϕ is the polar angle of x. It is easy to see that λ is neither convex nor closed. If we construct the fuzzy dual cone of λ, we get 2 (π − ϕ) , π2 ϕ π, μ (y) = π 0, otherwise, where ϕ is the polar angle of y. Now note that the kernel of λ consists solely of the ray ϕ = π , and the closure of the support of μ equals to the second quadrant π2 ϕ π , while the dual cone of the kernel of λ is the left halfplane π 2
ϕ
3π 2 .
Thus, μ+ 0 is not the dual cone of λ1 .
5. Conclusion The given definition of fuzzy dual cones can be viewed as a natural generalization of dual cones to the fuzzy case. This is confirmed by the fact that dual fuzzy cones obey the same basic properties as crisp dual cones. Some new properties also arise due to more complex structure of fuzzy cones. References [1] Elsaid E. Ammar, Some properties of convex fuzzy sets and convex fuzzy cones, Fuzzy Sets Syst. 106 (3) (1999) 381–386. [2] O.V. Baskov, Algorithm for recalculating the generatrix lines of a finitely generated fuzzy cone after adding a generatrix to its dual cone, Comput. Math. Math. Phys. 55 (2015) 206–211. [3] Loredana Biacino, An extension principle for closure operators, J. Math. Anal. Appl. 198 (1996) 1–24. [4] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, 2004. [5] R. Lowen, Convex fuzzy sets, Fuzzy Sets Syst. 3 (1980) 291–310. [6] Kaisa M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, 1999. [7] V.D. Noghin, Estimation of the set of nondominated solutions, Numer. Funct. Anal. Appl. 12 (5–6) (1992) 507–515. [8] V.D. Noghin, Upper estimate for a fuzzy set of nondominated solutions, Fuzzy Sets Syst. 67 (3) (1994) 303–315. [9] V.D. Noghin, O.V. Baskov, Pareto set reduction based on an arbitrary finite collection of numerical information on the preference relation, Dokl. Math. 83 (3) (2011) 418–420. [10] J. Ramik, M. Vlach, Aggregation functions and generalized convexity in fuzzy optimization and decision making, Ann. Oper. Res. 195 (1) (2012) 261–276. [11] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.