On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0–1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.     

On the unique solvability of fuzzy relational equations

The solution set of a consistent system of fuzzy relational equations with max-min composition can be characterized by one maximum solution and a finite number of minimal solutions. A polynomial-time method of O(mn) complexity is proposed to determine whether such a system has a unique minimal solution and/or a unique solution, where m, n are the dimensions of the input data. The proposed method can be extended to examining a system of fuzzy relational equations with max-T composition where T is a continuous triangular norm.     

A survey on fuzzy relational equations, part I: classification and solvability

Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of “composition of binary relations.” In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.     

Randomly generating test problems for fuzzy relational equations

Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems. To compare and evaluate the accuracy and efficiency of various solution methods proposed for solving systems of fuzzy relational equations as well as the associated optimization problems, a test problem random generator for systems of fuzzy relational equations is needed. In this paper, procedures for generating test problems of fuzzy relational equations with the sup-T{\mathcal{T}} composition are proposed for the cases of sup-T_M{\mathcal{T}_M}, sup-T_P{\mathcal{T}_P}, and sup-T_L{\mathcal{T}_L } compositions. It is shown that the test problems generated by the proposed procedures are consistent. Some properties are discussed to show that the proposed procedures randomly generate systems of fuzzy relational equations with various number of minimal solutions. Numerical examples are included to illustrate the proposed procedures.     

Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T_ω-based addition preserves the shape of L-R type fuzzy numbers. In this paper, we show T_ω-based multiplication also preserves the shape of L-R type fuzzy numbers. We then apply T_ω-based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L-R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.     

Application of possibility theory to investment decisions

Carlson and Fuller (2001, Fuzzy Sets and Systems, 122, 315–326) introduced the concept of possibilistic mean, variance and covariance of fuzzy numbers. In this paper, we extend some of these results to a nonlinear type of fuzzy numbers called adaptive fuzzy numbers (see Bodjanova (2005, Information Science, 172, 73–89) for detail). We then discuss the application of these results to decision making problems in which the parameters may involve uncertainty and vagueness. As an application, we develop expression for fuzzy net present value (FNPV) of future cash flows involving adaptive fuzzy numbers by using their possibilistic moments. An illustrative numerical example is given to illustrate the results.     

连续并既约元及其在刻画Fuzzy关系方程解集中的应用

本文首先引入连续并既约元(是并既约元但不是完全并既约元的元)的概念,并讨论了它的性质,然后应用连续并既约元的性质去刻画完备Brouwer格上无限Fuzzy关系方程A☉X=b的解集(其中A=(aj)j∈J和b已知,b为连续并既约元,X= (xj)j∈JT未知,“☉”表示“sup-inf”,J为无限集):给出了方程存在可达解与不可达解的充要条件及可达解与不可达解的一些性质,进一步刻画了方程的解集．     

Some remarks on the identification problem in fuzzy systems

Several identification problems in fuzzy systems are considered which are described by means of fuzzy relational equations. The determination of a family of fuzzy relations of the system is described in detail.     

The optimization problem over a distributive lattice

In this paper we give a necessary and sufficient condition for existence of minimal solution(s) of the linear system A * X ≥ b where A, b are fixed matrices and X is an unknown matrix over a lattice. Next, an algorithm which finds these minimal solutions over a distributive lattice is given. Finally, we find an optimal solution for the optimization problem min {Z = C * X | A * X ≥ b} where C is the given matrix of coefficients of objective function Z. This research was completed while the author was a visitor of the Center for Informatics and Applied Optimization, University of Ballarat, Ballarat, Australia.     

Probabilistic approach for granular media equations in the non-uniformly convex case

We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch. Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003) in the uniformly convex case. The main tool is an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T ₁ transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.      

Solution Sets of Interval-Valued Min-S-Norm Fuzzy Relational Equations

This paper introduces the concepts of tolerable solution set, united solution set, and controllable solution set for interval-valued fuzzy relational equations. Given a continuous s-norm, it is shown that each of the three types of the solution sets of interval-valued fuzzy relational equations with a min-s-norm composition, if nonempty, is composed of one minimum solution and a finite number of maximal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. An example is given to illustrate the proposed procedures. An erratum to this article is available at .     

Solution Sets of Interval-Valued Fuzzy Relational Equations

This paper introduces the concepts of tolerable solution set, united solution set, and controllable solution set of interval-valued fuzzy relational equations. Given a continuous t-norm, it is proved that each of the three types of the solution sets of interval-valued fuzzy relational equations with a max-t-norm composition, if nonempty, is composed of one maximum solution and a finite number of minimal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. Examples are given to illustrate the procedures. An erratum to this article is available at .     

Bivalent and other solutions of fuzzy relational equations via linguistic hedges

We show that the well-known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations. When the constraint is empty, constrained solutions are ordinary solutions. The generalization is obtained by employing intensifying and relaxing linguistic hedges, conceived in this paper as certain unary functions on the scale of truth degrees. One aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations. The other is to emphasize the role of linguistic hedges as constraints.     

Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

We establish conditions of the existence of solutions periodic in t with period T for a weakly nonlinear system of partial differential equations with pulse influence. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 601–605, April, 1997.     

On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains

Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R ³. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg in R ³ and Sohr-von Wahl in exterior domains to general domains.     

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f = f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x) = T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range − 1/2 < λ < 0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ. (Received: June 7, 2005)     

A New View on Fuzzy Hypermodules

We describe the relationship between the fuzzy sets and the algebraic hyperstructures. In fact, this paper is a continuation of the ideas presented by Davvaz in （Fuzzy Sets Syst., 117： 477- 484, 2001） and Bhakat and Das in （Fuzzy Sets Syst., 80： 359-368, 1996）. The concept of the quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set is introduced and this is a natural generalization of the quasi-coincidence of a fuzzy point in fuzzy sets. By using this new idea, the concept of interval-valued （α,β）-fuzzy sub-hypermodules of a hypermodule is defined. This newly defined interval-valued （α,β）-fuzzy sub-hypermodule is a We shall study such fuzzy sub-hypermodules and sub-hypermodules of a hypermodule. generalization of the usual fuzzy sub-hypermodule. consider the implication-based interval-valued fuzzy     

Optimal Three Cylinder Inequality at the Boundary for Solutions to Parabolic Equations and Unique Continuation Properties

Let Γ be a portion of a C ^1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous result we prove that if u (x, t) = O (|x|^k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically equal to zero. This work is partially supported by MURST, Grant No. MM01111258