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

Minimizing and maximizing a linear objective function under a fuzzy max⁡−∗\max -\ast relational equation and an inequality constraint

summary:This paper provides an extension of results connected with the problem of the optimization of a linear objective function subject to $\max-\ast$ fuzzy relational equations and an inequality constraint, where $\ast$ is an operation. This research is important because the knowledge and the algorithms presented in the paper can be used in various optimization processes. Previous articles describe an important problem of minimizing a linear objective function under a fuzzy $\max-\ast$ relational equation and an inequality constraint, where $\ast$ is the $t$-norm or mean. The authors present results that generalize this outcome, so the linear optimization problem can be used with any continuous increasing operation with a zero element where $\ast$ includes in particular the previously studied operations. Moreover, operation $\ast$ does not need to be a t-norm nor a pseudo-$t$-norm. Due to the fact that optimal solutions are constructed from the greatest and minimal solutions of a $\max-\ast$ relational equation or inequalities, this article presents a method to compute them. We note that the linear optimization problem is valid for both minimization and maximization problems. Therefore, for the optimization problem, we present results to find the largest and the smallest value of the objective function. To illustrate this problem a numerical example is provided

IEEE Transactions on Fuzzy Systems : Vol. 21, No. 4, August 2013

1. Knowledge-Leverage-Based Fuzzy System and Its Modeling / Zhaohong Deng, et al. 2. Trilateral Teleoperation of Adaptive Fuzzy Force / Motion Control for Nonlinear Teleoperations with Comunication Random Delay / Zhijun Li, et al. 3. Reliable All-Pairs Evolving Fuzzy Classifiers / Edwin Lughofer, Oliver Buchtala 4. Priority Wight Intervals Derived From Intuitionistic Multiplicative Preference Relations / Zeshui Xu 5. A Novel Control Design on Discrete-Time Takagi-Sugeno Fuzzy Systems with Time-Varying Delay / Xiaojie Su, et al. 6. Looking for Like-Minded Individuals in Social Networks Using Tagging and E Fuzzy Sets / Ronald R. Yager, Marek Z. Reformat 7. Adaptive Moving-Target Tracking Contor of a Vision-Based Mobile Robot via a Dynamic Petri Recurrent Fuzzy Neural Network / Rong-Jong Wai, You-Wei Lin 8. A Fuzzy Logic-Based System for Indoor Localization Using WiFi in Ambient Intelligent Environments / Teresa Garcia-Valverde, et al. 9. Extending the Contraposition Property of Propositional Logic for Fuzzy Abduction / Aruna Chakraborty, et al. 10. A Behavioral Distance for Fuzzy-Transition Systems / Yongzhi Cao, et al. 11. New Stability Condition Based on Piecewise Fuzzy Lyapunov Functions and Tensor Product Transformations / Victor C.S. Campos, et al. 12. Short Papers: Relaxation Technique for a T-S Fuzzy Control Design Based on a Continuous-Time Fuzzy Weighting-Dependent Lyapunov Function / Sung Hyun Kim 13. Paralel Algorithms for Fuzzy Ontology Reasoning / Fernando Bonillo, Miguel Delgado, Jose Cesar Sanchez-Sanchez 14. A Matrix Approach to Latticized Linear Programming with Fuzzy-Relation Inequality Constraints / Haitao Li, Yuzhen Wan