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

An equivalent condition to the Jensen inequality for the generalized Sugeno integral.

For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given

Lattice operations on fuzzy implications and the preservation of the exchange principle

In this work, we solve an open problem related to the preservation of the exchange principle (EP) of fuzzy implications under lattice operations ([3], Problem 3.1.). We show that generalizations of the commutativity of antecedents (CA) to a pair of fuzzy implications (I,J)(I,J), viz., the generalized exchange principle and the mutual exchangeability are sufficient conditions for the solution of the problem. Further, we determine conditions under which these become necessary too. Finally, we investigate the pairs of fuzzy implications from different families such that (EP) is preserved by the join and meet operations

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

Geometric Programming Subject to System of Fuzzy Relation Inequalities

In this paper, an optimization model with geometric objective function is presented. Geometric programming is widely used; many objective functions in optimization problems can be analyzed by geometric programming. We often encounter these in resource allocation and structure optimization and technology management, etc. On the other hand, fuzzy relation equalities and inequalities are also used in many areas. We here present a geometric programming model with a monomial objective function subject to the fuzzy relation inequality constraints with maxproduct composition. Simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. Also, an algorithm and two practical examples are presented to abbreviate and illustrate the steps of the problem resolution

Bandler–Kohout Subproduct With Yager’s Classes of Fuzzy Implications

The Bandler-Kohout subproduct (BKS) inference mechanism is one of the two established fuzzy relational inference (FRI) mechanisms; the other one being Zadeh's compositional rule of inference (CRI). Both these FRIs are known to possess many desirable properties. It can be seen that many of these desirable properties are due to the rich underlying structure, viz., the residuated algebra, from which the employed operations come. In this study, we discuss the BKS relational inference system, with the fuzzy implication interpreted as Yager's classes of implications, which do not form a residuated structure on [0,1] . We show that many of the desirable properties, viz., interpolativity, continuity, robustness, which are known for the BKS with residuated implications, are also available under this framework, thus expanding the choice of operations available to practitioners. Note that, to the best of the authors' knowledge, this is the first attempt at studying the suitability of an FRI where the operations come from a nonresiduated structure

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