The Alternative Representation of the Bipolar Sugeno Integral

In the context of decision support systems, bi-capacities were introduced as an extension of classical capacities. Many bipolar fuzzy integrals related to the bi-capacities have been presented in recent years. One of these integrals is the Sugeno integral concerning aggregation on bipolar scales. The paper aims to build an equivalent representation of the bipolar Sugeno integral. Therefore, we first employ in this paper the framework based on a ternary-criterion set for proposing an alternative formula of the bipolar Sugeno integral to be suitable for bipolar scales. Then, we discuss some basic properties and give an illustrative example of this representation. This representation is consistent as an extension of the representation concerning the classical capacities and aggregation on the Sugeno integral unipolar scales.

In the context of decision support systems, bi-capacities were introduced as an extension of classical capacities. Many bipolar fuzzy integrals related to the bicapacities have been presented in recent years. One of these integrals is the Sugeno integral concerning aggregation on bipolar scales. The paper aims to build an equivalent representation of the bipolar Sugeno integral. Therefore, we first employ in this paper the framework based on a ternary-criterion set for proposing an alternative formula of the bipolar Sugeno integral to be suitable for bipolar scales. Then, we discuss some basic properties and give an illustrative example of this representation. This representation is consistent as an extension of the representation concerning the classical capacities and aggregation on the Sugeno integral unipolar scales.
Many generalizations of the fuzzy integrals concerning aggregation on bipolar scales were presented in recent years [16][17][18][19]. Furthermore, bipolarity and its potential applications have been studied in many recent works of literature [20][21][22][23]. In this respect, alternative formulas for the case of the Choquet integral on bipolar scales have been proposed in [24][25][26], so these formulas allow for a generalization of several results around the bipolar Choquet integral.
The paper aims to build an equivalent representation of bipolar Sugeno integral different from the Sugeno integral framework introduced in [18]. Therefore, we first employ the framework based on a ternary-criterion set for proposing an alternative formula of the bipolar Sugeno integral in this paper. Then, we study the main properties of this representation. This representation is consistent as an extension of the representation concerning the classical capacities and aggregation on the Sugeno integral unipolar scales.
The following section recalls some basic concepts that we need in this contribution. In Section 3, we introduce the bicapacities defined in the new approach. Then, section 4 proposes an alternative representation of bipolar Sugeno integral with illustrated example. In Section 5, we study the main properties of this representation. Finally, in section 6, some conclusions are described.

Capacities and Sugeno Integral
In this paper, we denote by for a measurable space, where is a σalgebra of subsets of the universal set . A capacity [11] is an extension of a classical measure and is defined as follows: Definition 1: [10] Let be a measurable space. A capacity is a function that satisfies the requirements: i) ii) if , for all Let us denote by to a class of nonnegative real-valued input on measurable space . For any the Sugeno integral of related to is defined as follows.

Definition 2: [10] To any real-valued input
The Sugeno integral of with respect to capacity is given by

Bi-capacities
Although capacities can capture a wide range of decision-making applications, they are incompetent in some circumstances, especially when the capacities are defined on bipolar scales. Therefore, in several workable cases, it is normal to employ a scale that goes from bad ``negative'' to good ``positive'' values, which includes the middle neutral amount. This scale is called a bipolar scale because of encodes the bipolarity of the impact, and exemplary examples are R (unbounded cardinal), [−1, 1] (bounded cardinal), or {excellent, good, medium, bad, very bad} (ordinal). For simplicity, we use the [−1, 1] scale in this paper, with a neutral value of zero.
Considering that the independence between the negative and positive partitions does not hold, for this reason, we have to treat the triple alternatives ∪ , and give each of them a value in [−1, 1]. This value is denoted as , that is, a function with two arguments, the first argument being the set of criteria that are completely satisfied, the second being the set of criteria that are not completely satisfactory, and the remaining criteria being on the neutral level." Grabisch and Labreuche [16] gave the following definition and called it bi-capacity.

Suppose
| ∩ Then, for any two disjoint pairs of sets the binary relation ⊑ is defined by: The Supremum (Sup) and Infimum (Inf) on the structure ⊑ are denoted by , respectively. The Sup is given by Inf is given by Hence, this ordered set becomes a lattice with the top being and the bottom being , (for more details, see [16]). The structure has another order relation introduced by Bilbao et al. [27]. This order relation is defined as follows: Inf is given by ∩ ∩ (6) and the Sup does not exist. Thus, this order relation on the structure is an Inf-semilattice with the bottom being (see, [16], [27] for more details). Definition 3: [16] A set function is called a bi-capacity on if satisfies the following requirements: i.

Bipolar Sugeno Integral Based on Bi-capacities With Order ⊑
The bipolar maximum of , denoted by ⋁ ⋁ ⋀ Note that the operator is the symmetric maximum introduced in [28]. Thus, the bipolar Sugeno integral based on bi-capacities with order ⊑ is defined as follows.
In this section, we begin by recalling the basic concepts of the ternary-criterion set and the equivalent definition of bicapacities (for more details, see [24][25][26]).
We consider every criterion that has either a positive impact, a negative, or no impact. So that we symbolize the criterion as whenever is positively significant, as whenever is negatively significant and as whenever is neutral, and we call this criterion a ternary-criterion. For all the ternary-criterion set is the set that contains only out of Hence, in this approach, we denote by { | { } for the set of all possible combinations of ternary criterion set of elements.
We have can be identified with , hence | | Also, simply remarked that for any ternary criterion set is an alternative to a ternary vector with .

Definition 5: For any two sets
The order relation ⊑ between ternary-criterion sets , Then ⊑ , The next definition is an alternative definition of bi-capacities defined on the approach of ternary-criterion sets.

Definition 6:
A mapping is called bi-capacity based on the ter-criterion sets if it satisfies the following requirements: 1) Bi-capacities are functions defined on the structure . Hence, we can introduce another order relation on the structure , we denote by , which is an alternative to the order relation in a bi-cooperative game [27].

Definition 7: For any two sets
The order relation ⊑ between ternary-criterion sets , Then, and (12)

The Alternative Representation of Bipolar Sugeno Integral
Let be a non empty finite set, the binary operators ⋀, ∨ on [0,1] is defined as follows: For any s, t [0,1] , s ∧ t := min{s, t} and s ∨ t := max{s, t}. According to [28,18], the symmetric minimum and the symmetric maximum are operations have been introduced as follows: For Here, we propose an alternative representation of bipolar Sugeno integral related to bi-capacity defined on the ternarycriterion set.
For an input vector ( ) We denote a ternary-criterion set with Therefore, we can define the alternative representation of bipolar Sugeno integral of real input related to bi-capacity as follows.

Definition 8: Let
be the set of all ternary-criterion sets and be a bi-capacity defined on a ternary-criterion set. Then, the alternative representation of bipolar Sugeno integral of related to is defined by where is a permutation on so that | | | |, and { ( ) ( ) } is ternary-criterion set For the sake of clarity, we give an illustrative example of the alternative representation of the bipolar Sugeno integral.

Example 1: Let us consider
and we define the bi-capacity values as shown in Table 1.

The function on defined by
That is, Then, the ternary criterion set corresponding to is We obtain,

Basic Properties of The Alternative Representation of Bipolar Sugeno Integral
Here, we introduce the basic properties satisfied by the alternative representation of bipolar Sugeno integral concerning the bi-capacity. So that, if then ∫ ∫ is proved inside the scope that the order of each element of and does not change.
Therefore, by repeating the above process 2 times at the point of the replacement of the order, if then, the following can be proved even in the range with the order replacement.

∫ ∫
Thus, by applying this procedure successively for each , the monotonic property can be proved.

Conclusion
In this contribution, we have proposed an alternative representation of bipolar Sugeno integrals related to bi-capacities. Then, we studied the basic properties of this representation. According to the definitions of bi-capacities and their integrals through the concept of ternary-criterion sets, we can see these definitions are significant because associating polarity with each criterion is easier than associating polarity with a set of criteria. Therefore, we look forward to future work on these definitions to allow an easy way to introduce new integrals related to the bi-capacities.

Author contribution
The contribution in the current work was equally made by all authors.

Funding
No grant is provided for the present work and given by any funding organization.

Data availability statement
Upon any demand made by the corresponding author, the data which supports the conclusions of the current work can be made available.

Conflicts of interest
There is no conflict of interest in the current work.