Abstract
In this paper, we propose a partial ordering ⪯ on the set of ordered weighted averaging (OWA) operators. Based on this relation ⪯, we introduce the negation, conjunction and disjunction operations, and establish a bounded De Morgan lattice equipped with an involutive negation for OWA operators. Finally, we develop a similarity measure between OWA operators based on the ordering ⪯.
Keywords
Introduction
Ordered weighted averaging (OWA) operators were introduced by Yager [14]. Afterwards, various types of OWA operators have been proposed and a lot of successful applications have been carried out [1, 15]. Now, OWA operator has become a theoretic basis and an important information fusion tool for decision-making designing. As we all know, for any OWA operator Agg, it has
This paper is organized as follows. In Section 2, we briefly review the coordinatewise order, lexicographical order and OWA operators. In Section 3, we show that both the coordinatewise order and lexicographical order are not suitable for the ordering of weighted averaging (WA) and OWA operators. In Section 4, we introduce a new order for OWA operators and give the necessary and sufficient condition of Agg1 (
Preliminaries
Orders
Let L be a set, ≤ is a partial order on L if it has the following properties [3, 5]: for any α, β, γ ∈ L, Reflexivity: α ≤ α Antisymmetry: α ≤ β and β ≤ α ⇒ α = β Transitivity: α ≤ β and β ≤ γ ⇒ α ≤ γ Comparability: α ≤ β or β ≤ α
Then (L, ≤) is called a partially ordered set. Moreover, a partially ordered set L is a totally ordered set (other names are clain and linearly ordered set) if it also has the following property: for any α, β ∈ L
This condition means any two elements of L are comparable.
Let L1, ⋯ , L
n
be partially (or totally) ordered sets. The coordinatewise order on the Cartesian product L1 × ⋯ × L
n
is defined by
The lexicographic order on the Cartesian product L1 × ⋯ × L
n
is defined by
Let (w1, w2, ⋯ , w
n
) ∈
For a fixed weighting vector
The following are some special cases of OWA aggregations and their associated weighting vectors [14, 15]: Maximum operator: Minimum operator: Average operator: Window-OWA operators Orlike S-OWA operator
Andlike S-OWA operator
Coordinatewise order and lexicographical order are two orders on
Coordinatewise order on WA and OWA operators
Both WA and OWA operators are determined by its weighting vector. We can define the order of two Given two WA (or OWA) operators by the coordinatewise order of their associated weighting vectors. Because for any two weighting vectors
However, the coordinatewise order of
(⇒): We prove it by using reduction to absurdity.
This’s a contradiction of
From above Theorem 1, for any two weighting vectors
Similar to the orders of t-norm and fuzzy implications in [2, 9], we can define the order of WA operators by the following:
Unfortunately, Definition 1 of order on WA operators is equivalent to the coordinatewise order of their associated weighting vectors.
(⇒): Consedering the following special cases
By WA1 ≤ WA2, we get
WA1 (1, 0, 0, . . . , 0) ≤ WA2 (1, 0, 0, . . . , 0)
WA1 (0, 1, 0, . . . , 0) ≤ WA2 (0, 1, 0, . . . , 0)
⋯
WA1 (0, 0, 0, . . . , 1) ≤ WA2 (0, 0, 0, . . . , 1).
From these inequalities, we have
⋯
Then by Theorem 1, we have
From above two Theorems 1 and 2, for any two weighting vectors
First, the lexicographical order on the set of weighting vectors is a little different from that in Eq. (2)
(⇒) Because
From above result,
Then we show that the lexicographical order used on WA operators has the following problem.
Since
As we all known, for any two weighting vectors
But from above Theorem 3, we also have
Thus the lexicographical order of weighting vectors is not suitable for the ordering of their associated WA operators.
Similarly, the lexicographical order of weighting vectors is not suitable for the ordering of their associated OWA operators, as seen in the following example.
We propose a new order on the set of OWA operators in this section.
First, we can define the order of OWA operators by the following:
In above section 3.2, we know that the similar method is not suitable for defining order on WA operators (see Definition 1 and Theorem 3). But we demonstrate here that it is feasible for defining order on OWA operators.
OWA1 (1, 0, 0, . . . , 0) ≤ OWA2 (1, 0, 0, . . . , 0)
OWA1 (1, 1, 0, . . . , 0) ≤ OWA2 (1, 1, 0, . . . , 0)
⋯
OWA1 (1, 1, 1, . . . , 1) ≤ OWA2 (1, 1, 1, . . . , 1).
From these inequalities, we get
⋯
Thus Eq. (15) holds.
(⇐): This can be proved by the recurrence method. If Eq. (15) does not hold for some 1 ≤ k ≤ n - 1, i.e.,
From above two Theorem, for any two weighting vectors
Thus OWA1 ≤ OWA2 is equivalent to
⋯
By
(
Let
If
Further, we demonstrate that ⪯ on
If
Moreover, if k ≥ 2, then Window-OWA operators are weaker than average operator, i.e.,
This is very interesting, for example, when
(1): OWA1 (
(2): if OWA1 (
(3): if OWA1 (
(4): ⪯ on
We have explored the orders of WA and OWA operators based on the Definitions 1 and 2 respectively. The results can be summarized as follows (also see Table 1).
Orders on WA and OWA operators
(i) Coordinatewise order is not suitable for the ordering of WA and OWA operators (see Theorems 1 and 2);
(ii) Lexicographical order is not suitable for the ordering of WA and OWA operators (see Theorem 3 and Example 1);
(iii) The proposed order is suitable for the ordering of OWA operators (see Theorem 4).
Since it is meaningful to study the algebraic structures on the set of OWA operators based on the ⪯ on
Negation on OWA operators
First, we define a negation operation on
(i) h (w1, ⋯ , wn-1, w
n
) is a weighting vector; h is an inverted order mapping, i.e.,
⋯
Then we have
⋯
Thus
⋯
Then
Then
Because
(i) ¬ (ii) ¬ (iii) ¬ (¬ (iv) ¬ is an inverted order mapping, i.e.,
(iv)
⋯
Then we have
⋯
Since
⋯
Thus
From above result, ¬ is a strong negation on the partially ordered set (
For any two weighting vectors
if and only if
Second, we define the conjunction and disjunction operations on
(2) First, we give the proof of
⋯
Thus
Second, we give the proof of
⋯
Thus ¬ (
Moreover, the negation ¬, conjunction ∧ and disjunction ∨ operations on
the following properties hold:
where
where
(
(
¬ (
¬ (
Since 0.7 < 0.8 and 0.7 + 0.1 < 0.8 + 0.1, then (0.7, 0.1, 0.2) ⪯ (0.8, 0.1, 0.1). Thus we have
By the concept of ¬ in Lemma 4, then
Moreover, we have
Clearly, (¬ OWA1 ∧ ¬ OWA2) = ¬ (OWA1 ∨ OWA2) and (¬ OWA1 ∨ ¬ OWA2) = ¬ (OWA1 ∧ OWA2)
Thus (
From above Theorems, we can see that (
Thus (
In this section we describe the degree of similarity of OWA operators based on the ordering ⪯.
0 ≤ S ( S ( S ( If
Based on above definition of similarity measure of
(S2) If
(S3) Since
(S4) If
In this paper, the ordering problem of OWA operators has been investigated, and our conclusion can be summarized as following: Both the coordinatewise order ≤
c
and lexicographic order ≤
l
are not suitable for WA and OWA operators. Order of two OWA operators in Definition 2 is equivalent to ⪯ of their associated weighting vectors, i.e.,
⪯ is a relationship between ≤
c
and ≤
l
, i.e.,
( ( ( A similarity measure of
Our conclusion is contributing to a deeper understanding on the ordering problem of OWA operators. Meanwhile, we realise that the ordering of OWA operators is only considered on [0, 1] in this paper. A further discussion about the ordering of OWA operators on other aspects, such as convex poset [8] or complete lattices [11], are also meaningful in the future research. Other problems, such as how to define t-norms for OWA operators, are also worthwhile for investigating based on our proposed order of OWA operators.
Footnotes
Acknowledgments
This project was supported by the National Science Foundation of China under Grant No. 62006168 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ21A010001.
