Resolution and simplification of Dombi-fuzzy relational equations and latticized optimization programming on Dombi FREs

In this paper, we introduce a type of latticized optimization problem whose objective function is the maximum component function and the feasible region is defined as a system of fuzzy relational equalities (FRE) defined by the Dombi t-norm. Dombi family of t-norms includes a parametric family of continuous strict t-norms, whose members are increasing functions of the parameter. This family of t-norms covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. Since the feasible solutions set of FREs is non-convex and the finding of all minimal solutions is an NP-hard problem, designing an efficient solution procedure for solving such problems is not a trivial job. Some necessary and sufficient conditions are derived to determine the feasibility of the problem. The feasible solution set is characterized in terms of a finite number of closed convex cells. An algorithm is presented for solving this nonlinear problem. It is proved that the algorithm can find the exact optimal solution and an example is presented to illustrate the proposed algorithm.Comment: arXiv admin note: text overlap with arXiv:2206.09716, arXiv:2207.0637

An exact algorithm for linear optimization problem subject to max-product fuzzy relational inequalities with fuzzy constraints

Fuzzy relational inequalities with fuzzy constraints (FRI-FC) are the generalized form of fuzzy relational inequalities (FRI) in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy constraints enable us to attain optimal points (called super-optima) that are better solutions than those resulted from the resolution of the similar problems with ordinary inequality constraints. This paper considers the linear objective function optimization with respect to max-product FRI-FC problems. It is proved that there is a set of optimization problems equivalent to the primal problem. Based on the algebraic structure of the primal problem and its equivalent forms, some simplification operations are presented to convert the main problem into a more simplified one. Finally, by some appropriate mathematical manipulations, the main problem is transformed into an optimization model whose constraints are linear. The proposed linearization method not only provides a super-optimum (that is better solution than ordinary feasible optimal solutions) but also finds the best super-optimum for the main problem. The current approach is compared with our previous work and some well-known heuristic algorithms by applying them to random test problems in different sizes.Comment: 29 pages, 8 figures, 7 table

Some mathematical aspects of fuzzy systems

In this work, three topics which are important for the further development of fuzzy systems are chosen to be investigated. First, the mathematical aspects of fuzzy relational equations (FREs) are explored. Solving FREs is one of the most important problems in fuzzy systems. In order to identify the algebraic information of the fuzzy space, two new tools, called fuzzy multiplicative inversion and additive inversion, are proposed. Based on these tools, the relationship among fuzzy vectors in fuzzy space is studied. Analytical expressions of maximum and mean solutions for FREs, and an optimal algorithm for calculating minimum solutions are developed. Second, the possibility of applying functional analysis theory to Takagi-Sugeno (T-S) fuzzy systems design is investigated. Fuzzy transforms, which are based on the generalised Fourier transform in functional analysis, are proposed. It is demonstrated that, mathematically, a T-S fuzzy model is equivalent to a fuzzy transform. Hence the parameters of a T-S fuzzy system can be identified by solving equations constructed using the inner product between membership functions and a given target function. The functional point of view leads to an insight into the behaviour of a fuzzy system. It provides a theoretical basis for exploring improvements to the efficiency of T-S fuzzy modelling. Third, the mathematical aspects of model-based fuzzy control (MBFC) are investigated. MBFC theory is not suitable for general nonlinear systems, due to an implicit linearity assumption. This assumption limits fuzzy controller design to a special case of linear time-varying systems control. To apply MBFC in general nonlinear control, a new stability criterion for general nonlinear fuzzy system is proposed. The mathematical aspects investigated in this research, provide a systematic guidance on issues such as efficient fuzzy systems modelling, balanced "soft" and "hard" computing in fuzzy system design, and applicability of fuzzy control to general nonlinear systems. They serve as a theoretical basis for further development of fuzzy systems.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Perfectionism Search Algorithm (PSA): An Efficient Meta-Heuristic Optimization Approach

This paper proposes a novel population-based meta-heuristic optimization algorithm, called Perfectionism Search Algorithm (PSA), which is based on the psychological aspects of perfectionism. The PSA algorithm takes inspiration from one of the most popular model of perfectionism, which was proposed by Hewitt and Flett. During each iteration of the PSA algorithm, new solutions are generated by mimicking different types and aspects of perfectionistic behavior. In order to have a complete perspective on the performance of PSA, the proposed algorithm is tested with various nonlinear optimization problems, through selection of 35 benchmark functions from the literature. The generated solutions for these problems, were also compared with 11 well-known meta-heuristics which had been applied to many complex and practical engineering optimization problems. The obtained results confirm the high performance of the proposed algorithm in comparison to the other well-known algorithms

Some mathematical aspects of fuzzy systems

In this work, three topics which are important for the further development of fuzzy systems are chosen to be investigated. First, the mathematical aspects of fuzzy relational equations (FREs) are explored. Solving FREs is one of the most important problems in fuzzy systems. In order to identify the algebraic information of the fuzzy space, two new tools, called fuzzy multiplicative inversion and additive inversion, are proposed. Based on these tools, the relationship among fuzzy vectors in fuzzy space is studied. Analytical expressions of maximum and mean solutions for FREs, and an optimal algorithm for calculating minimum solutions are developed. Second, the possibility of applying functional analysis theory to Takagi-Sugeno (T-S) fuzzy systems design is investigated. Fuzzy transforms, which are based on the generalised Fourier transform in functional analysis, are proposed. It is demonstrated that, mathematically, a T-S fuzzy model is equivalent to a fuzzy transform. Hence the parameters of a T-S fuzzy system can be identified by solving equations constructed using the inner product between membership functions and a given target function. The functional point of view leads to an insight into the behaviour of a fuzzy system. It provides a theoretical basis for exploring improvements to the efficiency of T-S fuzzy modelling. Third, the mathematical aspects of model-based fuzzy control (MBFC) are investigated. MBFC theory is not suitable for general nonlinear systems, due to an implicit linearity assumption. This assumption limits fuzzy controller design to a special case of linear time-varying systems control. To apply MBFC in general nonlinear control, a new stability criterion for general nonlinear fuzzy system is proposed. The mathematical aspects investigated in this research, provide a systematic guidance on issues such as efficient fuzzy systems modelling, balanced 'soft' and 'hard' computing in fuzzy system design, and applicability of fuzzy control to general nonlinear systems. They serve as a theoretical basis for further development of fuzzy systems

Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems

Computer Science & Technology Series : XXI Argentine Congress of Computer Science. Selected papers

CACIC’15 was the 21thCongress in the CACIC series. It was organized by the School of Technology at the UNNOBA (North-West of Buenos Aires National University) in Junín, Buenos Aires. The Congress included 13 Workshops with 131 accepted papers, 4 Conferences, 2 invited tutorials, different meetings related with Computer Science Education (Professors, PhD students, Curricula) and an International School with 6 courses. CACIC 2015 was organized following the traditional Congress format, with 13 Workshops covering a diversity of dimensions of Computer Science Research. Each topic was supervised by a committee of 3-5 chairs of different Universities. The call for papers attracted a total of 202 submissions. An average of 2.5 review reports werecollected for each paper, for a grand total of 495 review reports that involved about 191 different reviewers. A total of 131 full papers, involving 404 authors and 75 Universities, were accepted and 24 of them were selected for this book.Red de Universidades con Carreras en Informática (RedUNCI

Computer Science & Technology Series : XXI Argentine Congress of Computer Science. Selected papers

CACIC’15 was the 21thCongress in the CACIC series. It was organized by the School of Technology at the UNNOBA (North-West of Buenos Aires National University) in Junín, Buenos Aires. The Congress included 13 Workshops with 131 accepted papers, 4 Conferences, 2 invited tutorials, different meetings related with Computer Science Education (Professors, PhD students, Curricula) and an International School with 6 courses. CACIC 2015 was organized following the traditional Congress format, with 13 Workshops covering a diversity of dimensions of Computer Science Research. Each topic was supervised by a committee of 3-5 chairs of different Universities. The call for papers attracted a total of 202 submissions. An average of 2.5 review reports werecollected for each paper, for a grand total of 495 review reports that involved about 191 different reviewers. A total of 131 full papers, involving 404 authors and 75 Universities, were accepted and 24 of them were selected for this book.Red de Universidades con Carreras en Informática (RedUNCI