Quadratic and nonlinear programming problems solving and analysis in fully fuzzy environment

AbstractThis paper presents a comprehensive methodology for solving and analyzing quadratic and nonlinear programming problems in fully fuzzy environment. The solution approach is based on the Arithmetic Fuzzy Logic-based Representations, previously founded on normalized fuzzy matrices. The suggested approach is generalized for the fully fuzzy case of the general forms of quadratic and nonlinear modeling and optimization problems of both the unconstrained and constrained fuzzy optimization problems. The constrained problems are extended by incorporating the suggested fuzzy logic-based representations assuming complete fuzziness of all the optimization formulation parameters. The robustness of the optimal fuzzy solutions is then analyzed using the recently newly developed system consolidity index. Four examples of quadratic and nonlinear programming optimization problems are investigated to illustrate the efficacy of the developed formulations. Moreover, consolidity patterns for the illustrative examples are sketched to show the ability of the optimal solution to withstand any system and input parameters changes effects. It is demonstrated that the geometric analysis of the consolidity charts of each region can be carried out based on specifying the type of consolidity region shape (such as elliptical or circular), slope or angle in degrees of the centerline of the geometric, the location of the centroid of the geometric shape, area of the geometric shape, lengths of principals diagonals of the shape, and the diversity ratio of consolidity points. The overall results demonstrate the consistency and effectiveness of the developed approach for incorporation and implementation for fuzzy quadratic and nonlinear optimization problems. Finally, it is concluded that the presented concept could provide a comprehensive methodology for various quadratic and nonlinear systems’ modeling and optimization in fully fuzzy environments

Consolidity: Mystery of inner property of systems uncovered

AbstractThis paper uncovers the mystery of consolidity, an inner property of systems that was amazingly hidden. Consolidity also reveals the secrecy of why strong stable and highly controllable systems are not invulnerable of falling and collapsing. Consolidity is measured by its Consolidity Index, defined as the ratio of overall changes of output parameters over combined changes of input and system parameters, all operating in fully fuzzy environment. Under this notion, systems are classified into consolidated, quasi-consolidated, neutrally consolidated, unconsolidated, quasi-unconsolidated and mixed types. The strategy for the implementation of consolidity is elaborated for both natural and man-made existing systems as well as the new developed ones. An important critique arises that the by-product consolidity of natural or built-as-usual system could lead to trapping such systems into a completely undesired unconsolidity. This suggests that the ample number of conventional techniques that do not take system consolidity into account should gradually be changed, and adjusted with improved consolidity-based techniques. Four Golden Rules are highlighted for handling system consolidity, and applied to several illustrative case studies. These case studies cover the consolidity analysis of the Drug Concentration problem, Predator-Prey Population problem, Spread of Infectious Disease problem, AIDS Epidemic problem and Arm Race model. It is demonstrated that consolidity changes are contrary (opposite in sign) to changes of both stability and controllability. This is a very significant result showing that our present practice of stressing on building strong stable and highly controllable systems could have already jeopardized the consolidity behavior of an ample family of existing real life systems. It is strongly recommended that the four Golden Rules of consolidity should be enforced as future strict regulations of systems modeling, analysis, design and building of different disciplines of sciences. It can be stated that with the mystery of consolidity uncovered, the door is now wide open towards the launching of a new generation of systems with superior consolidity in various sciences and disciplines. Examples of these disciplines are basic sciences, evolutionary systems, engineering, astronautics, astronomy, biology, ecology, medicine, pharmacology, economics, finance, commerce, political and management sciences, humanities, social sciences, literature, psychology, philosophy, mass communication, and education

Consolidity analysis for fully fuzzy functions, matrices, probability and statistics

The paper presents a comprehensive review of the know-how for developing the systems consolidity theory for modeling, analysis, optimization and design in fully fuzzy environment. The solving of systems consolidity theory included its development for handling new functions of different dimensionalities, fuzzy analytic geometry, fuzzy vector analysis, functions of fuzzy complex variables, ordinary differentiation of fuzzy functions and partial fraction of fuzzy polynomials. On the other hand, the handling of fuzzy matrices covered determinants of fuzzy matrices, the eigenvalues of fuzzy matrices, and solving least-squares fuzzy linear equations. The approach demonstrated to be also applicable in a systematic way in handling new fuzzy probabilistic and statistical problems. This included extending the conventional probabilistic and statistical analysis for handling fuzzy random data. Application also covered the consolidity of fuzzy optimization problems. Various numerical examples solved have demonstrated that the new consolidity concept is highly effective in solving in a compact form the propagation of fuzziness in linear, nonlinear, multivariable and dynamic problems with different types of complexities. Finally, it is demonstrated that the implementation of the suggested fuzzy mathematics can be easily embedded within normal mathematics through building special fuzzy functions library inside the computational Matlab Toolbox or using other similar software languages