The Importance and Interaction Indices of Bi-Capacities Based on Ternary-Element Sets

Grabisch and Labreuche have recently proposed a generalization of capacities, called the bi-capacities. Recently, a new approach for studying bi-capacities through introducing a notion of ternary-element sets proposed by the author. In this paper, we propose many results such as bipolar Mobius transform, importance index, and interaction index of bi-capacities based on our approach.


Introduction:
The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [1] as a generalization of capacity [2] (fuzzy measure [3,4,5,6,7] or non-additive measure [8]) in the context of decision making, who consider the case where scores are expressed on a bipolar scale, i.e. having a central neutral level, usually 0. Grabisch and Labreuche [1] have laid down the basis for the main concepts around bi-capacities, among them the Mobius representation, importance index, and interaction index.Other remarkable works on bi-capacities include the one of Fujimoto and Murofushi [9], who defined the Mobius transform of bi-capacities under the name of bipolar Mobius transform in order to avoid the complicated expression of the Choquet integral in terms of the Mobius transform given in [10].In [11], the author has proposed a new approach for studying bi-capacities through introducing a notion of ternaryelement sets.This approach is alternative approach from that defined by Grabisch and Labreuche [1] and allows a simple way to prove new results on bi-capacities as it was done for capacities.Consequently, we propose in this paper many results such as bipolar Mobius transform, importance index, and interaction index based on our approach.The structure of the paper is as follows: in the next section we recall the definition of bi-capacities based on ternary-element sets.Section 3 presents the bipolar Mobius transform based on our approach.In sections 4, we propose Open Access importance and interaction indices of bicapacities based on ternary element sets.The paper ends with some conclusions.Throughout the paper, the universal set * + denotes a finite set of elements (states of nature, criteria, individuals, etc), and we will consider R or , with 0 as neutral level that will be considered as prototypical bipolar scales.

Bi-Capacities Baed on Ter-Element Sets
In this section, we begin by recalling the basic concepts of ternary-elementset (or simply ter-element set) and the equivalent definition of bi-capacities based on ternary element sets (for more details, see [11]).We consider every element that has either a positive effect (i.e., is positively important criterion of weighted evaluation not only alone but also is interactive with others ), or a negative effect (i.e., is negatively important criterion), or has no effect (i.e., is criterion at neutral level).Hence, we represent the element as whenever is positively important, as whenever is negatively important, and as whenever is neutral, and we call this element a ternary-element (or simply ter-element).The ternary-element set (or simply ter-element set) is the set which contains only out of for all * + Thus, in our model we consider the set of all possible combinations of ternary elements of criteria given by ( ) We have ( ) can be identified with * + , hence | ( )| Also, simply remarked that for any terelement set ( ) is equivalent to a ternary alternative ( ) with .We introduce the order relation between ter-element sets of ( ) as follows.
Definition 1 Let ( ) be the set of all ter-element sets and ( ) Then, ( ) The following definition is equivalent definition of bi-capacities based on notion of ter-element sets.
Definition 2 Let ( ) be the set of all ter-element sets.A set function ( ) , is called bi-capacity based on the ter-element sets if it satisfies the following requirements: Bi-capacities are functions defined on the structure of the underlying partially ordered set [12].There are several orders on the structure ( ) that have been introduced by Grabisch and Labreuche [1] and Bilbao et al. [13].
Here, we introduce an order on the structure ( ) different from the order described in definition 1.We consider the following definition of an order on ( ) which is equivalent to Bilbao order on bi-cooperative game [13].For convenience, we denote by the order relation defined on ( ) as in the classical order relation, and we will use the order on ( ) to establish our next results of this research.

Proof:
From definition 4, we have ( ) ∑ ( ) ( ) where the order is defined by Equation (2), and the cardinality of the ter-element sets is defined by Equation (3).

Importance and Interaction Indices Based on Ter-Element Sets
The Shapley interaction index related to a capacity, among a combination of criteria has been introduced by Grabisch [16] as a natural generalization of the Shapley importance value [17].Later, Grabisch and Labreuche [1] generalized the Shapley interaction index to a bicapacity on ( ) In this subsection, we propose equivalent definition of the Shapley interaction index for bi-capacities based on ter-element sets.The definition of Shapley interaction index with respect to a bi-capacity based on ter-element set on ( ) is as follows.
Definition 6 Let be a bi-capacity based on ter-element set on ( ) For ( ) the interaction index with respect to is defined by The following numerical example illustrates the definition of interaction index with respect to a bi-capacity based on ter-element set on ( ) The special case of the interaction index on singleton of ter-element set (i.e. when | | ) is the importance index related to a bi-capacity of terelement set ( ) Therefore, we define the importance index of when it is positively important or when it is negatively important as follows.
Definition 7 Let be a bi-capacity based on ter-element set on ( ) The important index of * + is defined by