Shilkret integral based on binary-element sets and its application in the area of synthetic evaluation

The aim of this paper is to present a new framework for studying fuzzy measure and Shilkret integral which is alternative framework from usual fuzzy measure and Shilkret integral. First, we define a fuzzy measure based on binary-element sets, then, we propose Shilkret integral with respect to fuzzy measure based on binary-element sets, and we illustrate our framework by a practical example on synthetic evaluation.


INTRODUCTION
he classical measure and integral theory is based on the additivity of set functions. Fuzzy measures [15] (capacities [3] or non-additive measures [4]) are obtained by replacing the additive requirement of classical measures with weaker requirement of monotonicity (with respect to set inclusion). Based on the fuzzy measure, several non-additive integrals (the Choquet integral [3], the Shilkret integral [13], the Sugeno integral [14], etc.) have been introduced in the last 60 years. In recent years there has been increasing interest in the development of new integrals useful in decision analysis or modeling of engineering problems [1, 2,5,6,9,10,11,12].
In this paper, we first propose a new framework for studying fuzzy measure and Shilkret integral through introducing the notion of binary-element sets. Secondly, we illustrate our framework by a practical example on multiple criteria decision analysis in the area of synthetic evaluation of objects in terms of multiple quality factors.
The structure of the paper is as follows. In the next section we recall basic definitions of usual fuzzy measures and Shilkret integral. In Sections 3 and 4, we present our alternative framework ``fuzzy measures based on binaryelement sets'' and ``Shilkret integral based on binary-element sets''. In section 5, we illustrate our framework by a practical example on global evaluation of TV sets.
Throughout the paper, we denote R as the set of all real numbers, R+ the set of all non-negative real numbers, the universal set = {1, … , } denotes a finite set of n elements (states of nature, criteria, individuals, etc), and

Usual fuzzy measures and Shilkret integral
In this section we recall basic definitions of usual fuzzy measures and Shilkret integral. Usual fuzzy measures on some finite universe are special monotone set functions defined in the following way (see, e.g. [8] The Shilkret integral is intimately related to the concept of fuzzy measures. Its definition in the case of a finite universal set is as follows (see, e.g. [13]). with respect to , which is denoted by ( ℎ) ∫ defined as Where ⋁ is the maximum and is a permutation on such that � (1)� ≥ ⋯ ≥ ( ( )).

Fuzzy measures based on binary-element sets Bi-element sets
In the context of multi-criteria decision making problem, we shall consider a set = {1, … , } of criteria and a set of alternatives described Shilkret integral based on binary-element sets and its application in the area of synthetic evaluation according to these criteria. We further assume that to each alternative is described by a vector = ( 1 , ⋯ , , ⋯ , ), ∈ [0,1] (or R+) and expresses to which degree the alternative satisfies criterion ∈ . The problem is now to evaluate the alternatives with respect to criteria. Since the criteria have not always the same importance, and defining a fuzzy measure on a set of criteria can be seen as a way of modeling the interaction phenomena existing among these criteria. Hence, for every alternative, any element ∈ has either positive effect (i.e., is positively important criterion of weighted evaluation not only alone but also is interactive with other) or has negative effect (i.e., is negatively important criterion). Thus, for every alternative, we can represent the element as + whenever is positively important, and as + whenever is negatively important, and we call this element a binary-element (or simply bi-element). For this situation, we can describe the set of all possible combinations of binary elements (i.e. the possible outcomes of the alternative) of criteria given by which is corresponds to power set ( ) in the notation of classical set theory. The set of all bi-element sets ( ) can be identified with {0,1} 2 , hence | ( )| = 2 . Also, any bi-element set ∈ ( ) can be written as a binary alternative denoted by We introduce the cardinality, complement, inclusion relation ⊆, union, and intersection of bi-element sets of ( )as follows. Definition 3: The cardinality of the bi-element set ∈ ( ) is number of important elements + in , denoted by + . That is,

Definition 4:
For any bi-element set ∈ ( ), we define the complement of by Definition 5: Let ∈ ( ), be the set of all bi-element sets and , ∈ ( ). Then, ⊆ ∀ ∈ + ∈ + ∈ . … (2) Definition 6: Let ∈ ( ), be the set of all bi-element sets and , ∈ ( ). The union, ∪ of and is defined by, Shilkret integral based on binary-element sets and its application in the area of synthetic evaluation Definition 7: Let ∈ ( ), be the set of all bi-element sets and , ∈ ( ). The union, ∩ of and is defined by,

Fuzzy measures based on bi-element sets
Based on the notion of bi-element sets in the above subsection, the following definition gives an equivalent definition of usual fuzzy measures.
A fuzzy measure based on bi-element sets is said to be additive if it satisfies: The conjugate (or dual) of a fuzzy measure based on bi-element sets is a fuzzy measure ̅ defined by ̅ ( ) = 1 − ( ), ∈ ( ).

Shilkret integral based on binary-element sets and its application in the area of synthetic evaluation
From the definition of the Shilkret integral with respect to the fuzzy measure based on bi-element sets we immediately see the following property.
Proposition 1 Let ( ) be the set of all bi-element sets and μ: B(X) →[0; 1] be a fuzzy measure based on bi-element sets. Then Shilkret integral of with respect to μ is given by

Illustrative Example (Practical Example Description of the example
Let us try to illustrate our approach by the example of evaluation of three TV sets adopted from [15]. We consider (for the sake of simplicity) only two quality factors : ``Picture'' and ``Sound''. These are denoted by P and S, respectively, and the corresponding weights are 1 = 0.6 and 2 = 0.4.
As an illustration, we give in Table 1, the scores for each factor and each TV set: &Tech.Journal, Vol. 33,Part (B), No.3,2015

Shilkret integral based on binary-element sets and its application in the area of synthetic evaluation
In the sequel, we will illustrate the two different approaches by applying them to above scores for global evaluation of three TV sets.
From the above results. It can be seen that the first TV set is the best. Such a result is hardly acceptable since it does not agree with our intuition: A TV set without any sound is not practical at all, even though it has an excellent picture. It is significant to realize that the cause of this counterintuitive result is not an improper choice of the weights. For example, if we chose 1 = 0.2 and 2 = 0.8, we would have obtained 1 = 1 × 1 + 2 × 0 = 0.2 2 = 1 × 0 + 2 × 1 = 0.8 3 = 1 × 0.5 + 2 × 0.5 = 0.5. Now, according to these results, the second TV set is identified as the best one, which is also counterintuitive: A TV set with good sound but no picture is not a real TV set, but just a radio.
We may conclude that, according to our intuition, the third TV set be identified as the best one: among the three TV sets, only the third one is really practical, even though neither picture nor sound are perfect. Thus, when using the method of weighted mean, no choice of the weights would lead to this expected result under the given scores.

Our approach
The method of weighted mean is based on an implicit assumption that the factors 1 , 2 ⋯ , are ``independent" of one another. That is, their effects are viewed as additive. This, however, is not justifiable in some real problems. In this example, the importance of the combination of picture and sound is much higher then the sum of importance associated with picture and sound alone. If we adopt our approach  Journal, Vol. 33,Part (B), No.3,2015 Shilkret integral based on binary-element sets and its application in the area of synthetic evaluation (a fuzzy measure based on binary-element sets) to characterize the importance of the two factors and, relevantly, use Shilkret integral based on binaryelement sets as a synthetic evaluator of the quality of the three TV sets, a satisfactory result may be obtained. We consider = { , }, the alternative consists of observing the factors where a TV has qualified. The possible outcomes of the alternative are the set of all possible combinations of binary elements of criteria ( , ). That is, For instance, the meaning of the bi-element set { + , − } is as follows: the set of factors in which picture is important while sound is unimportant.
We interpret a value ( ) of a fuzzy measure, as the ``degrees of importance'' of bi-element set ∈ ( ). Where, 1 , 2 and 3 characterize the scores given for the three TV sets: 1 = (1,0), 2 = (0,1), and 3 = (0.5. ,0.5). Hence, we get a reasonable conclusion ``the third TV set is the best'' which agrees with our intuition. Consequently, the result is satisfactory.

Conclusions
In this paper, we have proposed alternative framework of fuzzy measure and Shilkret integral with respect to fuzzy measure. First, we have presented a fuzzy measure through attaching each element of the domain on the subsets of which the set function is defined. Secondly, we have proposed a new model for study Shilkret integral based on the notion of binary-element sets.
Finally, practical example is given for clarifying the usefulness of the proposed framework.