Fuzzy Relation Inequalities Research Articles

Some Results of the Fuzzy Relation Inequalities With Addition–Min Composition

In this paper, some results of the fuzzy relation inequalities with addition–min composition are given. First, the uniqueness of minimal solutions of this system is discussed. Then, the relation be...

An interleaved depth-first search method for the linear optimization problem with disjunctive constraints

Being an extension of classical linear programming, disjunctive programming has the ability to express the problem constraints as combinations of linear equalities and inequalities linked with logic AND and OR operations. All the existing theories such as generalized disjunctive programming, optimization modulo theories, linear optimization over arithmetic constraint formula, and mixed logical linear programming pose one commonality of branching among different solving techniques. However, branching constructs a depth-first search which may traverse a whole bad subtree when the branching makes a mistake ordering a bad successor. In this paper, we propose the interleaved depth-first search with stochastic local optimal increasing (IDFS-SLOI) method for solving the linear optimization problem with disjunctive constraints. Our technique searches depth-first several subtrees in turn, accelerates the search by subtree splitting, and uses efficient backtracking and pruning among the subtrees. Additionally, the local optimal solution is improved iteratively by constructing and solving a stochastic linear programming problem. We evaluate our approach against existing counterparts on the rate-monotonic optimization problem (RM-OPT) and the linear optimization with fuzzy relation inequalities problem (LOFRI). Experimental results show that for the tested instances, the IDFS-SLOI method performs better from performance perspective, especially promising results have been obtained for the larger three groups where the execution time is reduced by 85.6 and 51.6% for RM-OPT and LOFRI, respectively.

Lexicographic Maximum Solution of Min-Product Fuzzy Relation Inequalities for Modeling the Optimal Pricing With Fixed Priority Grade in Supply Chain

Fuzzy relation inequalities composed by the min-product operation are established to model the pricing relation in a supply chain system. Basic properties of the min-product fuzzy relation inequalities are presented first, based on which the complete solution set could be characterized and obtained. In fact, each solution of the corresponding fuzzy relation inequalities is exactly a feasible price scheduling. Considering the fixed priority grade of the suppliers, the concept of lexicographic maximum solution is introduced and investigated, as an optimal price scheduling that maximizes the benefits. A detailed algorithm is developed to search the unique lexicographic maximum solution with a numerical illustrative example.

A New Characterisation of the Minimal Solution Set to Max-min Fuzzy Relation Inequalities

In this paper, max-min fuzzy relation inequalities are applied to describe a BitTorrent-like Peer-to-Peer file sharing system. In order to decrease the network congestion under some fixed priority grade of the terminals, we define concept of lexicographic solution in such system. It is found that each minimal solution of max-min fuzzy relation inequalities happen to be a lexicographic solution. Moreover, the lexicographic solution set is exactly the minimal solution set. Numerical example is given to illustrate our results. At last, for the addition-min fuzzy relation inequalities, we provide a guess, trying to describe the relation between its lexicographic solution set and minimal solution set.

Optimal-Vector-Based Algorithm for Solving Min-Max Programming Subject to Addition-Min Fuzzy Relation Inequality

Optimal management model in BitTorrent-like peer-to-peer (P2P) file sharing system can be reduced into a min–max programming problem subject to addition-min fuzzy relation inequality. To deal with such problem, we introduce the concept of optimal vector. Besides, a novel optimal-vector-based (OVB) algorithm is developed to find a minimal optimal solution. Based on practical application consideration, solution obtained by the OVB algorithm is theoretically demonstrated to be better than or equal to that obtained by the existing method. Advantage of our proposed OVB algorithm is illustrated by detailed numerical examples.

Computation of the greatest right and left invariant fuzzy quasi-orders and fuzzy equivalences

Right invariant fuzzy quasi-orders for fuzzy automata are broadly studied in the recent literature, as they arise as solutions to particular systems of fuzzy relation equations and inequalities. Some of their applications include determinization and state reduction procedures, as well as simulations and bisimulations for fuzzy automata. In this paper we provide a procedure for computing the greatest right invariant fuzzy quasi-order for a given fuzzy automaton over a complete residuated lattice. The proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite. When the previous condition is not satisfied, we show that the greatest right invariant fuzzy quasi-order can be obtained by taking the limit value of the convergent array of fuzzy quasi-orders for fuzzy automata over BL-algebras on the real unit interval [0,1]. Analogous procedures for computing the greatest left invariant fuzzy quasi-order, as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented. In addition, the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented.

Lexicographic optimal solution of the multi-objective programming problem subject to max-product fuzzy relation inequalities

It is shown in this paper that the emission base stations in wireless communication can be reduced into a system of fuzzy relation inequalities with max-product composition. For optimal management in such system, we introduce the fuzzy relation multi-objective programming. Concept of feasible index set (FIS) is defined, based on which a novel algorithm, named FIS algorithm, is developed to find the unique lexicographic optimal solution of the proposed problem with polynomial computational complexity. Applying this method, we needn't to find out all the minimal solutions of the constraint. A numerical application example is provided to illustrate the feasibility and efficiency of the FIS algorithm.

A modified PSO algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints (FRI-FC)

In this paper, optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t-norm is considered as fuzzy composition, and fuzzy inequality replaces ordinary inequality in the constraints. Unlike most optimization algorithms, in fuzzy relational inequalities with fuzzy constraints (FRI-FC) we find a near-feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp (ordinary) inequality constraints. Such solutions are called super-optima in this paper. Subsequently, an algorithm is proposed to find a super-optimum with pre-specified desirable infeasibility. For this purpose, we firstly study some structural properties of the FRI-FC problem and present a new formulization that is independent of t-norms used in the constraints of the problem. This new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters. However, it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super-optimum under a certain sufficient condition. Finally, a modified PSO is presented in which the ability of PSO to solve unconstrained problems with continuous domain, the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized FRI-FC problems. The modified PSO algorithm has been applied to the generalized FRI-FC problem defined with ten well-known continuous t-norms. Additionally, an idea of the FRI-FC problems as outer approximators for FRI problems with ordinary inequalities is also investigated.

A Linear Programming Approach for Minimizing a Linear Function Subject to Fuzzy Relational Inequalities With Addition–Min Composition

In this paper, we study an optimization problem of minimizing a linear function subject to fuzzy relational inequalities with the addition–min composition. This optimization setting has recently been proposed to model the network cogestion issue when a BitTorrent-like peer-to-peer file-sharing system is used for data transmission. In a 2014 paper, a pseudominimal index (PMI)-based approach was proposed to search for an optimal solution. It turns out that the PMI-based approach may require to solve several to many linear programming problems in order to get an optimal solution. In this paper, we point out that the feasible domain is indeed convex. And we only need to solve a single linear programming problem to generate an optimal solution for the original optimization problem. Furthermore, our approach could be extended to the case with a nonlinear continuous objective function.

Fuzzy multi-objective optimization of linear functions subject to max-arithmetic mean relational inequality constraints

The main goal of this work is to nd a better solution to a kind of multiobjective optimization problem subject to a system of fuzzy relational inequalities with max-arithmetic mean composition. First, this problem is solved and, then, in the case that the decision maker is not satis ed with any of the solutions, by assigning linear membership functions to the inequalities in the constraints and objective functions and using Bellman-Zadeh decision, a new solution is found. This new solution does not belong to the feasible domain but is considered acceptable based on the decision maker's view. In order to fou nd this solution easier, some simpli cation processes are given. Afterwards, an algorithm is presented to generate the new solution. Finally, an example is given to illustrate the steps of the algorithm.

Addition-min fuzzy relation inequalities with application in BitTorrent-like Peer-to-Peer file sharing system

The data transmission mechanism in BitTorrent-like Peer-to-Peer (P2P) file sharing systems may be reduced to some addition-min fuzzy relation inequalities. The solution set of addition-min fuzzy relation inequalities plays an important role in the corresponding optimization problem. In this paper, we study some properties of the solutions to such system. Convexity of the solution set and number of minimal solutions are discussed, with comparison to those of the classical max-T fuzzy relation equations or inequalities. Besides, vertex solution and variable-ordering minimal solution are also investigated, with application in BitTorrent-like P2P file sharing system. Two numerical examples are given to illustrate the feasibility and efficiency of the algorithm for solving the variable-ordering solution.

Fuzzy relation equations and inequalities with two unknowns and their applications

In this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets. Solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise. We show that solutions to each of these systems form a complete lattice, and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations. We also demonstrate the application of solutions to these systems in data reduction.

Maximizing a monomial geometric objective function subject to bipolar max-product fuzzy relation constraints

In this paper, the problem of maximizing a monomial geometric objective function subject to bipolar max-product fuzzy relation constraints is studied. First of all, it is shown that the bipolar max-product Fuzzy Relation Inequality (FRI) system can equivalently be converted to a bipolar max-product Fuzzy Relation Equation (FRE) system. Hence, the structure of feasible domain of the problem is determined in the case of the bipolar max-product FRE system. It is shown that its solution set is non-convex, in a general case. Some sufficient conditions are proposed for solution existence of its feasible domain. An algorithm is designed to solve the optimization problem with regard to the structure of its feasible domain and the properties of the objective function. Its importance is also illustrated by an application example in the area of economics and covering problem. Some numerical examples are given to illustrate the above points.

SOLUTION-SET INVARIANT MATRICES AND VECTORS IN FUZZY RELATION INEQUALITIES BASED ON MAX-AGGREGATION FUNCTION COMPOSITION

Fuzzy relation inequalities based on max-F composition are discussed, where F is a binary aggregation on [0,1]. For a fixed fuzzy relation inequalities system $ A circ^{F}textbf{x}leqtextbf{b}$, we characterize all matrices $ A^{'} $ For which the solution set of the system $ A^{' } circ^{F}textbf{x}leqtextbf{b}$ is the same as the original solution set. Similarly, for a fixed matrix $ A $, the possible perturbations $ b^{'} $ of the right-hand side vector $ b $ not modifying the original solution set are determined. Several illustrative examples are included to clarify the results of the paper.

Linear fractional programming problem with max-Hamacher FRI

In this paper, the linear fractional programming problem subject to a system of fuzzy relation inequalities (FRI) with the max-Hamacher composition operator is studied. First, the structure of its feasible domain is investigated, and its feasible solution set determined. Then, some sufficient conditions are given that under them, some of the optimal components of the problem are directly determined. The optimal solution of the linear fractional programming problem might not be any of the minimal solutions. However, in the process of obtaining the optimal solution, we need to compute the minimal solutions. Therefore, some reductions are presented that under them, we can compute the minimal solutions fast. The original problem can be transformed into some traditional linear fractional programming subproblems and eventually optimized in a small search space. Finally, an algorithm is designed to solve the problem based on the above reductions. An application and some numerical examples are provided to illustrate the procedure.

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.

Latticized linear programming subject to max-product fuzzy relation inequalities with application in wireless communication

In this paper we introduce the latticized linear programming problem subject to max-product fuzzy relation inequalities with application in the optimization management model of wireless communication emission base stations. Resolution of max-product fuzzy relation inequalities is studied by comparing with that of the corresponding max-product fuzzy relation equations. A solution matrix approach is developed for solving the proposed problem without finding all the (quasi-) minimal solutions of the constraint. For carrying out the solution matrix approach, we provide a step-by-step algorithm illustrated by a numerical example.

Lexicography minimum solution of fuzzy relation inequalities: applied to optimal control in P2P file sharing system

A peer-to-peer (P2P) file sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Concept of lexicography minimum solution is introduced and applied to such system. It is found that the unique lexicography minimum solution can be selected from the minimal solution set of the corresponding fuzzy relation inequalities. However it is difficult to find the minimal solution set which is probably infinite. In order to avoid such difficulty, we propose a so-called Circulation Algorithm to find the unique lexicography minimum solution. The algorithm is developed step by step and illustrated by a numerical application example.

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.

Min-Max Programming Problem Subject to Addition-Min Fuzzy Relation Inequalities

A BitTorrent-like peer-to-peer file-sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Addition-min is a new composition, and the solution set of addition-min fuzzy relation inequalities differs much from that of general max-t-norm fuzzy relation inequalities or equations. In order to avoid network congestion and improve the stability of data transmission, a min-max programming problem is proposed subject to addition-min fuzzy relation inequalities in this paper. Based on some relevant theorems on resolution, a novel algorithm is developed step by step to find an optimal solution of the proposed problem. Moreover, an application example is given to illustrate the feasibility and efficiency of the algorithm. Finally, some further discussions are made concerning the optimal solution of the proposed problem.