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3. Fuzzy Set Operation (1)
We present a new class of fuzzy aggregation operators that we call fuzzy triangular aggregation operators. To do so, we focus on the situation where the available information cannot be assessed with exact numbers and it is necessary to use another approach to assess uncertain or imprecise information such as fuzzy numbers.
We also use the concept of triangular norms t-norms and t-conorms as pseudo-arithmetic operations. Main properties of these operators are discussed as well as their comparison with other existing ones.
The fuzzy triangular aggregation operators not only cover a wide range of useful existing fuzzy aggregation operators but also provide new interesting cases. Finally, an illustrative example is also developed regarding the selection of strategies.
The available information for the human knowledge is said to be precise crisp information or not fuzzy information. The aggregation operators resp. Indeed, in MCGDM problems, one generally considers a finite set of alternatives , from which we must select the best with respect to a finite set of criteria. For each criterion of alternative , an expert is consulted to assign a value or score. According to Marichal [ 1 ], the MCGDM procedure comprises three main steps: the modeling step, in which we look for appropriate models to represent available information scores and weights ; the aggregation step, in which we try to find an overall score for each alternative on the basis of the partial scores and the weights; the exploitation step, in which we transform the global information about the alternatives either into a complete ranking of the elements in , or into a global choice of the best alternatives in.
In the real world, we are sometimes confronted with situations where the available information cannot be assessed with exact numbers and it is necessary to use another approach to represent such information with high degree of uncertainty or imprecision. Several methods exist in this case, some of which include fuzzy numbers, interval numbers, and linguistic numbers. In this paper, as, for instance, in [S. Chen and S. Chen [ 3 ]], [Herrera et al. The aim of this paper is to develop a new class of fuzzy aggregation operators dealing with NFNs, which we call fuzzy triangular aggregation operators.
For these ones, investigations will be done such as the main properties of operators and an illustrative example. In order to do so, the paper is organized as follows. In Section 2 , we give some preliminary notions and present the main properties of fuzzy aggregations operators. In Section 3 , we review some fuzzy aggregation operators.
Section 4 develops our proposed fuzzy aggregation operators. Section 5 presents an illustrative example regarding the selection of investment strategies.
Finally, we summarize the main conclusions in Section 6. In this section, we briefly describe fuzzy numbers FNs and arithmetic operations related to it. The notion of FNs was originally introduced by Zadeh [ 7 ].
Since then, it has been studied and applied by a lot of authors, especially Dubois and Prade [ 8 ] and Kaufmann and Gupta [ 9 ]. Its main advantage is that it can represent, in a more complete way, information coming from human language. That is, it can consider the maximum and minimum values, and the possibility that the internal values may occur.
Definition 1 Zadeh , [ 7 , 10 ]. A fuzzy number is a fuzzy set the membership function of is denoted by of a universe of discourse the real line which is i convex, that is, and , ; ii normalized, that is,. For example, a trapezoidal FN TpFN can be characterized by a trapezoidal membership function defined by where , , , and are the real parameters of , with.
If , the FN is called an interval number see [ 11 ]. Also, if , the FN is reduced to a crisp value. Notice that we will denote the TpFN as. Assume that there are two TpFNs, and , with , for each. That is, , the set of fuzzy subsets of positive real numbers. The pseudo-arithmetic operations between and are defined as follows [Chen and Hwang [ 12 ], Kaufmann and Gupta [ 9 ]]:.
In order to rank FNs, a lot of methods exist in the literature. In fact, in [ 2 , 13 ], the authors recommend to rank and according to the following procedure: i ; ii ; iii. To aggregate FNs, a number of aggregation operators have been developed.
Before briefly giving some well-known fuzzy aggregation operators, let us present their main properties. We present some properties that are generally considered as relevant for aggregation in a fuzzy environment. In a general way, let us consider fuzzy aggregation operators defined as , with a natural number. In order words, is continuous if and only if it is a component-wise continuous operator.
Remark 2. Notice that P1 and P2 are the two fundamental properties that characterize general fuzzy aggregation operators. They must not be violated. In what follows, a vector is called a weighting vector if , , and. Moreover, without loss of generality, it is worth noting that our presentation deals with ,. Dong and Wong [ 14 ] are the first authors who investigated the WA weighted averaging operator when the available information cannot be assessed with exact numbers and it is necessary to use other techniques such as FNs.
Definition 3. The fuzzy weighted averaging operator, denoted by FWA, is the mapping , which has an associated weighting vector such that, where the operations and are defined in 2. Remark 4. Theorem 5. Let and let be a weighting vector. P1 Boundary conditions: P2 Monotonicity: let ,. P4 Commutativity: let be a permutation of. However, if , then P5 Idempotency: assume that for all. Then P6 Bounded: pose and. There are various versions of the fuzzy ordered weighted averaging FOWA operator.
But today the formulation on this operator is attributed to S. Chen [ 3 ]. As the FWA operator, the reason for using the FOWA operator is that sometimes the available information cannot be assessed with exact numbers and it is necessary to use other techniques such as FNs. Definition 6. The fuzzy ordered weighted averaging operator, denoted by FOWA, is the mapping , which has an associated weighting vector , such that where the operations and are defined in 2 ; is a permutation on such that.
Remark 7. There are several other particular cases with respect to the analysis of the weighting vector. Theorem 8. Since is a permutation of , we have ,.
And then. The two well-known and most important fuzzy aggregation operators presented above have been generalized using the concepts of quasi-arithmetic means and generalized means. The fuzzy weighted quasi-arithmetic averaging Quasi-FWA is an aggregation operator that generalizes the FWA operator by using quasi-arithmetic means while the fuzzy ordered weighted quasi-arithmetic averaging Quasi-FOWA is an aggregation operator that generalizes the FOWA operator by using quasi-arithmetic means.
Definition 9 Wang and Luo [ 15 ]. The fuzzy weighted quasi-arithmetic averaging operator, denoted by Quasi-FWA, is the mapping , which has an associated weighting vector , such that where is a strictly continuous monotone function and the arithmetic operations and are defined in 2. Theorem The fuzzy ordered weighted quasi-arithmetic averaging operator, denoted by Quasi-FOWA, is the mapping , which has an associated weighting vector , such that where is a strictly continuous monotone function, the arithmetic operations and are defined in 2 , and is a permutation on such that.
Remark In this section, we present a new method to construct fuzzy aggregation operators based on triangular norms t-norms and t-conorms. This new class of aggregation operators is called fuzzy triangular aggregation operators. We deal with the situation where the values to be aggregated are expressed as NFNs.
The operational laws that we use are based on triangular norms t-norms and t-conorms. They are appropriate extensions of logical connectives AND and OR in the case when the valuation set is the unit interval rather than. So, t-norms resp.
Definition 14 Dubois and Prade [ 17 ]. Example Several important nonparametrized and parametrized families of t-norms and t-conorms exist but the three prototypical and most used examples are given as follows [ 17 — 21 ]: 1 ; ; Zadeh [ 10 ] 2 ; , deduced to probabilistic theory 3 ;.
Giles [ 22 ]. Motivated by the arithmetic operations investigated in [S. Chen and J. Chen [ 23 ], Kaufmann and Gupta [ 9 ], Xu [ 24 ], Zhao et al. Definition Let be a t-conorm and be a t-norm. Let be a t-conorm. A vector , with , is called a weighting vector associated with if and only if the equality is verified:.
It is important to stress that i If , then ; ii If , then ; iii If , then. Note that, in this paper, we do not enter in the problem of using FNs in the weighting vector. Nevertheless, if the weighting vector is presented with NFNs, then, instead of converting these fuzzy weights into representative exact numbers by using a method for doing so as recommended by some authors, our recommendation is, for example, to use the fuzzy weighting vector model defined as follows: let be a t-conorm.
A fuzzy vector with is called a fuzzy weighting vector associated with if and only if the equality is verified: Contrary to other approaches, our method is more informative since it uses all the information and therefore leads to complete results.
Fuzzy Triangular Aggregation Operators
In mathematics , a t-norm also T-norm or, unabbreviated, triangular norm is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic , specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces. The monoidal operation of any partially ordered Abelian monoid L is therefore by some authors called a triangular norm on L. T-norms are a generalization of the usual two-valued logical conjunction , studied by classical logic, for fuzzy logics. Indeed, the classical Boolean conjunction is both commutative and associative.
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We present a new class of fuzzy aggregation operators that we call fuzzy triangular aggregation operators. To do so, we focus on the situation where the available information cannot be assessed with exact numbers and it is necessary to use another approach to assess uncertain or imprecise information such as fuzzy numbers. We also use the concept of triangular norms t-norms and t-conorms as pseudo-arithmetic operations. Main properties of these operators are discussed as well as their comparison with other existing ones. The fuzzy triangular aggregation operators not only cover a wide range of useful existing fuzzy aggregation operators but also provide new interesting cases.
Vladik Kreinovich: Presentations at Professional Meetings
Objectives Introduces various operations of fuzzy sets Introduces the concepts of disjunctive sum, distance, difference, conorm and t conorm operators. Example 2 Given two fuzzy sets A and B a. Represent A and B fuzzy sets graphically b. Calculate the of union of the set A and set B c. Calculate the intersection of the set A and set B d.
Two critical tasks in multi-criteria group decision making MCGDM are to describe criterion values and to aggregate the described information to generate a ranking of alternatives. A flexible and superior tool for the first task is q-rung orthopair fuzzy number qROFN and an effective tool for the second task is aggregation operator. So far, nearly thirty different aggregation operators of qROFNs have been presented. Each operator has its distinctive characteristics and can work well for specific purpose. However, there is not yet an operator which can provide desirable generality and flexibility in aggregating criterion values, dealing with the heterogeneous interrelationships among criteria, and reducing the influence of extreme criterion values. To provide such an aggregation operator, Muirhead mean operator, power average operator, partitioned average operator, and Archimedean T-norm and T-conorm operations are concurrently introduced into q-rung orthopair fuzzy sets, and an Archimedean power partitioned Muirhead mean operator of qROFNs and its weighted form are presented and a MCGDM method based on the weighted operator is proposed in this paper. The generalised expressions of the two operators are firstly defined.
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