Generation of partial orders for intervals by means of the slope function
Introduction
Interval-valued fuzzy set [23], which is equivalent to intuitionistic fuzzy set [3], has been widely used in image processing, decision making, classification and so on. It is obvious that the interval ordering plays an important role in the arithmetic operation of intervals on the real line , especially on closed interval . Several order relations for intervals, the development of whose application has been witnessed by many publications such as [1], [2], [4], [7], [8], [9], [11], [12], [13], [15], [16], [17], [18], [19], [20], [21], have been introduced and discussed in [5], [6], [10], [14], [22].
On one hand, in 1990, Ishibuchi and Tanaka [14] introduced five partial orders and for intervals, in order to convert the maximization problem with the interval objective function into a multi-objective problem using the order relations. In 1996, Chanas and Kuchta [6] generalized known concepts of the solution of the linear programming problem with interval coefficients in the objective function based on preference relations between intervals. And a whole family of preference relations, introduced by them, comprises some partial orders as a special case. On the other hand, in 2006, Hu and Wang [10] defined a total order , which can compare any two intervals, and various properties were established for solving problems with uncertainties. At the same time, Xu and Yager [22] gave a total order which is different from for some geometric aggregation operators based on intuitionistic fuzzy sets. A method to build the admissible orders in terms of two aggregation functions was proposed by Bustince et al. [5] in 2013. The admissible order is a total order and refines the partial order . And it is proved that some of the most used examples of total orders that appear in the literature are specific cases of their construction.
Let us recall these well-known concepts about order relations for intervals. We only consider the order relations of intervals on real number set . And the set of all intervals on is denoted by . Let and be two intervals. The mid-point and half-width are defined respectively as follows. And these different methods of ranking intervals are shown as follows.
- (1)
iff (see [14]);
- (2)
iff (see [14]);
- (3)
iff (see [14]);
- (4)
iff (see [14]);
- (5)
iff (see [14]);
- (6)
iff (see [10]);
- (7)
iff (see [22]);
- (8)
iff (see [5]);
- (9)
iff (see [5]);
- (10)
iff (see [5]);
- (11)
iff (see [5]);
In our opinion the above orders work in a similar way although they are defined from various angles, so we need a tool to unify them by putting them in the same frame. Considering those classical orders and comparing the left endpoints of any two intervals, we consider the above classical orders in three aspects and then a new binary relation is constructed in the form of by the slope function . What is more, the binary relation can be proved to be a generalization of several classical interval order relations.
Section snippets
A new binary relation and its algebraic properties
In this section, we try to introduce a new binary relation. And it is used to encompass the above orders as some specific cases in the next section. Since the application of these interval orderings is discussed in [1], [2], [4], [7], [8], [9], [11], [12], [13], [15], [16], [17], [18], [19], [20], [21], our approach covers those methods.
In order to find certain qualities in common among those orders on , we consider the first three partial orders. For any given , if , then we can
A generalization of some classical order relations
As mentioned above, this paper studies interval orders in order to unify the classical orders. So we first discuss some basic conditions which are widely used in the definitions of those classical interval order relations.
Lemma 3.1 Some basic relations are all the special cases of . , where ; , where ; , where ; , where ; ,
The conditions of being a partial order
Theorem 3.3 means that nine classical orders are the special cases of . Now let us show under what conditions the binary relation is reflexive, antisymmetric and transitive, namely a partial order.
Lemma 4.1 The binary relation is reflexive, if and only if . Proof Let be any interval on . It follows from Definition 2.1 that Hence the binary relation is reflexive, if and only if . □
Lemma 4.2 The
The conditions of being an admissible order
Bustince et al. introduce the concept of an admissible order as a total order that extends the usual partial order between intervals, and they propose a method to build these admissible orders in terms of two aggregation functions.
Definition 5.1 (See Bustince et al., 2013, [5].) The order is called an admissible order, if is a total order on ; implies , for all , where is the set of all intervals on close interval .
In fact, the above definition can be generalized to
Conclusions
In this paper we have defined an order relation for intervals which proves to be a generalization of several classical interval order relations by means of the slope function. And the conditions for obtaining a partial order, a total order or even an admissible order have also been studied. While all analyses and discussions are made on , those conclusions can apply to without doubt.
Acknowledgements
The authors thank the anonymous reviewers and the Editor-in-Chief for their valuable suggestions in improving this paper. This research was supported by the National Natural Science Foundation of China (Grant No. 61179038).
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