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Formal Introduction to Fuzzy Implications

100%
Grabowski A.
Formalized Mathematics
|
2017
|
tom 25
|
nr 3
241-248
EN
In the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets. This work is a continuation of the development of fuzzy sets in Mizar; it could be used to give a variety of more general operations, and also it could be a good starting point towards the formalization of fuzzy logic (together with t-norms and t-conorms, formalized previously).
2
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Basic Formal Properties of Triangular Norms and Conorms

100%
Grabowski A.
Formalized Mathematics
|
2017
|
tom 25
|
nr 2
93-100
EN
In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2]. After defining corresponding Mizar mode using four attributes, we introduced the following t-norms: minimum t-norm minnorm (Def. 6), product t-norm prodnorm (Def. 8), Łukasiewicz t-norm Lukasiewicz_norm (Def. 10), drastic t-norm drastic_norm (Def. 11), nilpotent minimum nilmin_norm (Def. 12), Hamacher product Hamacher_norm (Def. 13), and corresponding t-conorms: maximum t-conorm maxnorm (Def. 7), probabilistic sum probsum_conorm (Def. 9), bounded sum BoundedSum_conorm (Def. 19), drastic t-conorm drastic_conorm (Def. 14), nilpotent maximum nilmax_conorm (Def. 18), Hamacher t-conorm Hamacher_conorm (Def. 17). Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm). This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].
3
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ReSySTER: A hybrid recommender system for Scrum team roles based on fuzzy and rough sets

88%
Colomo-Palacios R., González-Carrasco I., López-Cuadrado J., García-Crespo Á.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
|
nr 4
801-816
EN
Agile development is a crucial issue within software engineering because one of the goals of any project leader is to increase the speed and flexibility in the development of new commercial products. In this sense, project managers must find the best resource configuration for each of the work packages necessary for the management of software development processes in order to keep the team motivated and committed to the project and to improve productivity and quality. This paper presents ReSySTER, a hybrid recommender system based on fuzzy logic, rough set theory and semantic technologies, aimed at helping project leaders to manage software development projects. The proposed system provides a powerful tool for project managers supporting the development process in Scrum environments and helping to form the most suitable team for different work packages. The system has been evaluated in a real scenario of development with the Scrum framework obtaining promising results.
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