Fuzzy Relation Inequality Constraints Research Articles

Tri-composed fuzzy relation inequality with weighted-max-min composition and the relevant min-max optimization problem

In the existing works, the classical fuzzy relation inequality (FRI) with double-composed operators has been introduced to describe the P2P network system. In this work, considering the successful rate of the line connection within the P2P network system, we establish the so-called tri-composed FRI, with a weighted-max-min composition, for the first time. We study the solutions induced through paths and further construct the complete solution set of the tri-composed FRI using the path-induced solutions. Moreover, the whole solution set could be fully determined by all the path-induced solutions and its unique maximum solution. Furthermore, considering a specific optimal management objective in the P2P network, we further study the min-max optimization problem with the tri-composed FRI constraints. A path-based algorithm is proposed for handling the problem we studied, accompanied by an illustrative example.

Optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by Dubois-Prade family of t-norms

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated In doing so, the feasible region is formed by the intersection of two inequality fuzzy systems and Dubois-Prade family of t-norms which are considered as fuzzy composition. The most well-known continuous t-norms are Archimedean such as Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family. An interesting family of t-norms that is not Archimedean has been introduced by Dubois and Prade. In this paper, the resolution of the feasible region of the problem is initialy investigated when it is defined with max-Duboise-Prade composition. A necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility. of the problem. Moreover, two procedures have also been presented with the aim of simplifying the current linear problems. A method is proposed to generate random feasible max-Dubois-Prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem. Finally, an example is described to illustrate this algorithm.

A single-variable method for solving min–max programming problem with addition-min fuzzy relational inequalities

In this paper, we study the min–max programming problem with n addition-min fuzzy relational inequality constraints. We prove that when the problem is feasible, an optimal solution always exists with all variables being of the same value. Based on this result, the min–max programming problem can be simplified as a single-variable optimization problem with the same optimal objective value. To solve the corresponding single-variable optimization problem, we propose an analytical method and an iterative method by successively approximating the lower bound of the optimal value. Numerical examples are given to illustrate our methods.

Optimal Strong Solution of the Minimax Problem With Two-Sided Fuzzy Relation Inequality Constraints

To avoid unoccupied base stations, we introduce the concept of a strong solution to max-product fuzzy relation inequalities in this paper. Such a strong solution enables all base stations take part in wireless communication activities. The structure of the set of all strong solutions is discussed. The strong solution set is composed of a finite number of closed intervals. To decrease the damage caused by electromagnetic radiation, one always aims to find an optimal strong solution, in which each component reaches its minimum value. However, this is generally impossible. Hence, finding an optimal strong solution with a specific objective function is more feasible. In this work, we investigate the optimization, minimizing the largest component of a strong solution. A detailed algorithm is developed to find an optimal strong solution. The experimental results show that our proposed algorithm is efficient. In addition, we further discuss the structure of the complete optimal strong solution set.

Optimizing of Linear Problems Subjected to Sugeno - Weber FRI

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno-Weber family of t-norms is considered as fuzzy composition. SugenoWeber family of t-norms and t-conorms is one of the most applied one in various fuzzy modeling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for socalled λ –fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Sugeno-Weber composition. A necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Sugeno-Weber fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.

Optimizing of Linear Problems Subjected to SugenoWeber FRI

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno-Weber family of t-norms is considered as fuzzy composition. SugenoWeber family of t-norms and t-conorms is one of the most applied one in various fuzzy modeling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for socalled λ –fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Sugeno-Weber composition. A necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Sugeno-Weber fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.

Comment on “An algorithm for solving optimization problems with fuzzy relational inequality constraints”

A counterexample is provided to illustrate the incorrectness of I0 in Rule 1 presented by Guo et al. [F.F. Guo, L.P. Pang, D. Meng, Z.Q. Xia. An algorithm for solving optimization problems with fuzzy relational inequality constraints, Information Sciences 252 (2013) 20-31]. In other words, (7b) and (7b*) in the referenced paper above are not equivalent unless index set I0 is revised, since the definition of I0 in Rule 1 can only be correct when the constraints are fuzzy relational equations. And finally the correct of I0 is also presented in this note.

A comment on “The quadratic programming problem with fuzzy relation inequality constraints”

This note provides counterexamples to illustrate that Definition 1 and Theorem 2 which are proposed by Molai (2012) are incorrect. We first point out some deficiencies and errors found in the above referenced paper. The incorrect of Theorem 2 is due to the inaccuracy of Definition 1. Then we present a counterexample to illustrate the incorrect of FRI path. At last, we propose a new definition of FRI path of max-product fuzzy relation inequality. Some examples are presented to illustrate that the new definition of FRI path is correct.

Monomial geometric programming with an arbitrary fuzzy relational inequality

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 an arbitrary function. The feasible solution set is determined and compared with some common results in the literature. A necessary and sufficient condition and three other necessary conditions are presented to conceptualize the feasibility of the problem. In general a lower bound is always attainable for the optimal objective value by removing the components having no effect on the solution process. By separating problem to non-decreasing and non-increasing function to prove the optimal solution, we simplify operations to accelerate the resolution of the problem.

Linear optimization with mixed fuzzy relation inequality constraints using the pseudo-t-norms and its application

This paper studies the minimization problem of a linear objective function subject to mixed fuzzy relation inequalities (MFRIs) over finite support with regard to max-$$T_1$$T1 and max-$$T_2$$T2 composition operators, where $$T_1$$T1 and $$T_2$$T2 are two pseudo-t-norms. We first determine the structure of its feasible domain and then show that the solution set of a MFRI system is determined by a maximum solution and a finite number of minimal solutions. Moreover, sufficient and necessary conditions are proposed to check whether the feasible domain of the problem is empty or not. The MFRI path is defined to determine the minimal solutions of its feasible domain. The resolution process of the optimization problem is also designed based on the structure of its feasible domain. Procedures are proposed to reduce the size of the problem. With regard to the above points and the procedures, an algorithm is designed to solve the problem. Its application is expressed in the area of investing and covering. Finally, the algorithm is compared with other approaches.

A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints

The minimization problem of a quadratic objective function with the max-product fuzzy relation inequality constraints is studied in this paper. In this problem, its objective function is not necessarily convex. Hence, its Hessian matrix is not necessarily positive semi-definite. Therefore, we cannot apply the modified simplex method to solve this problem, in a general case. In this paper, we firstly study the structure of its feasible domain. We then use some properties of n×n real symmetric indefinite matrices, Cholesky’s decomposition, and the least square technique, and convert the problem to a separable programming problem. Furthermore, a relation in terms of a closed form is presented to solve it. Finally, an algorithm is proposed to solve the original problem. An application example in the economic area is given to illustrate the problem. Of course, there are other application examples in the area of digital data service and reliability engineering.

A Matrix Approach to Latticized Linear Programming With Fuzzy-Relation Inequality Constraints

This paper investigates the latticized linear programming that is subject to the fuzzy-relation inequality (FRI) constraints with the max-min composition by using the semi-tensor product method, and proposes a matrix approach to this problem. First, the resolution of the FRI is studied, and it is proved that all the minimal solutions and the unique maximum solution are within the finite parameter set solutions. Based on this and using the semi-tensor product, solving FRIs is converted to solving a set of algebraic inequalities, and some new results on the resolution of FRIs are presented. Second, the latticized linear programming that is subject to the FRI constraints is solved by taking the following two key steps: 1) the optimal value is obtained by calculating the minimum value of the objective function among all the minimal solutions to the FRI constraints; and 2) the optimal solution set is obtained by solving the fuzzy-relation equation that is generated by letting the objective function equal to the optimal value. The study of illustrative examples shows that the new results that are obtained in this paper are very effective in solving the latticized linear programming subject to the FRI constraints.

Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions

We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely "?", "?" and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.

Linear optimization with an arbitrary fuzzy relational inequality

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated whereby an arbitrary function is considered as fuzzy composition. The feasible solution set is determined and compared with some common results in the literature. A necessary and sufficient condition and three other necessary conditions are presented to conceptualize the feasibility of the problem. It is shown that in general a lower bound is always attainable for the optimal objective value. Moreover, we prove that the optimal solution of the problem is obtained if the problem is defined by a non-decreasing or non-increasing function. An algorithm is presented to summarize the process and an example is described to illustrate the algorithm.

An algorithm for solving optimization problems with fuzzy relational inequality constraints

An algorithm for solving a kind of optimization problems with fuzzy relational inequalities was proposed by Guo and Xia [6]. However, it is too expensive to verify the optimal condition. In this paper, some rules for reducing these problems are proposed and the relationship between minimal solutions and FRI paths is also given. These lead to a new algorithm for solving this kind of problems. Numerical experiments are presented for illustrating the efficiency of the proposed algorithm.

The quadratic programming problem with fuzzy relation inequality constraints

The quadratic programming has been widely applied to solve real world problems. The quadratic functions are often applied in the inventory management, portfolio selection, engineering design, molecular study, and economics, etc. Fuzzy relation inequalities (FRI) are important elements of fuzzy mathematics, and they have recently been widely applied in the fuzzy comprehensive evaluation and cybernetics. In view of the importance of quadratic functions and FRI, we present a fuzzy relation quadratic programming model with a quadratic objective function subject to the max-product fuzzy relation inequality constraints. Some sufficient conditions are presented to determine its optimal solution in terms of the maximum solution or the minimal solutions of its feasible domain. Also, some simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. The simplified problem can be converted into a traditional quadratic programming problem. An algorithm is also proposed to solve it. Finally, some numerical examples are given to illustrate the steps of the algorithm.

Duality of optimization problems with generalized fuzzy relation equation and inequality constraints

A generalization of fuzzy relation equations and inequalities is introduced. An explicit method for solving optimization problems, the feasible set of which is described by a system of generalized fuzzy relation equations and inequalities, is considered. The objective function of the problems is equal to the maximum of continuous increasing functions of one variable. A duality concept for such problems is proposed. The problems are considered in a finitely dimensional space.

Fuzzy linear objective function optimization with fuzzy-valued max-product fuzzy relation inequality constraints

In this paper, we firstly consider an optimization problem with a linear objective function subject to a system of fuzzy relation inequalities using the max-product composition. Since its feasible domain is non-convex, traditional linear programming methods cannot be applied to solve it. An algorithm is proposed to solve this problem using fuzzy relation inequality paths. Then, a more general case of the problem, i.e., an optimization model with one fuzzy linear objective function subject to fuzzy-valued max-product fuzzy relation inequality constraints, is investigated in this paper. A new approach is proposed to solve this problem based on Zadeh’s extension principle and the algorithm. This paper develops a procedure to derive the fuzzy objective value of the recent problem. A pair of mathematical program is formulated to compute the lower and upper bounds of the problem at the possibility level α . From different values of α , the membership function of the objective value is constructed. Since the objective value is expressed by a membership function rather than by a crisp value, more information is provided to make decisions.

Monomial geometric programming with fuzzy relation inequality constraints with max-product composition

Monomials function has always been considered as a significant and most extensively used function in real living. Resource allocation, structure optimization and technology management can often apply these functions. In optimization problems the objective functions can be considered by monomials. In this paper, we present monomials geometric programming with fuzzy relation inequalities constraint with max-product 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 a few practical examples are presented to abbreviate and illustrate the steps of the problem resolution.

Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max–min composition

In this paper, we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max–min composition in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with Ordinary Inequality constraints (denoted by x OI in this paper). We have presented an algorithm to generate such optimal solutions. Also two examples are given to illustrate the algorithm presented and some real application of this work.