A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Duality for a class of fuzzy nonlinear optimization problem under generalized convexity

In this paper, a Mond-Weir type dual program for a nonlinear primal problem under fuzzy environment is formulated. The solution concept of primal-dual problems is inspired by the nondominated solution. We have considered ordering among fuzzy numbers as a partial ordering and using the concept of Hukuhara difference between two fuzzy numbers and $H$ -differentiability, appropriate duality theorems are established under pseudo/quasi-convexity assumptions. We have also illustrated a numerical example which satisfies the duality relations discussed in the paper.     

Fuzzy Markov Chains and Decision-Making

Decision-making in an environment of uncertainty and imprecision for real-world problems is a complex task. In this paper it is introduced general finite state fuzzy Markov chains that have a finite convergence to a stationary (may be periodic) solution. The Cesaro average and the -potential for fuzzy Markov chains are defined, then it is shown that the relationship between them corresponds to the Blackwell formula in the classical theory of Markov decision processes. Furthermore, it is pointed out that recurrency does not necessarily imply ergodicity. However, if a fuzzy Markov chain is ergodic, then the rows of its ergodic projection equal the greatest eigen fuzzy set of the transition matrix. Then, the fuzzy Markov chain is shown to be a robust system with respect to small perturbations of the transition matrix, which is not the case for the classical probabilistic Markov chains. Fuzzy Markov decision processes are finally introduced and discussed.     

A Predictor-corrector algorithm with multiple corrections for convex quadratic programming

Recently an infeasible interior-point algorithm for linear programming (LP) was presented by Liu and Sun. By using similar predictor steps, we give a (feasible) predictor-corrector algorithm for convex quadratic programming (QP). We introduce a (scaled) proximity measure and a dynamical forcing factor (centering parameter). The latter is used to force the duality gap to decrease. The algorithm can decrease the duality gap monotonically. Polynomial complexity can be proved and the result coincides with the best one for LP, namely, $O(\sqrt{n}\log n\mu^{0}/\varepsilon)$ .     

Duality Theory in Fuzzy Optimization Problems

A solution concept of fuzzy optimization problems, which is essentially similar to the notion of Pareto optimal solution (nondominated solution) in multiobjective programming problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. We also introduce a concept of fuzzy scalar (inner) product based on the positive and negative parts of fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function for the fuzzy optimization problem are proposed via the concept of fuzzy scalar product. Under these settings, the weak and strong duality theorems for fuzzy optimization problems can be elicited. We show that there is no duality gap between the primal and dual fuzzy optimization problems under suitable assumptions for fuzzy-valued functions.     

Saddle Point Optimality Conditions in Fuzzy Optimization Problems

The fuzzy-valued Lagrangian function of fuzzy optimization problem via the concept of fuzzy scalar (inner) product is proposed. A solution concept of fuzzy optimization problem, which is essentially similar to the notion of Pareto solution in multiobjective optimization problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. Under these settings, the saddle point optimality conditions along with necessary and sufficient conditions for the absence of a duality gap are elicited.     

Total ordering for intuitionistic fuzzy numbers

Intuitionistic fuzzy set plays a vital role in decision making, data analysis, and artificial intelligence. Many decision‐making problems consist of different types of datum, where fuzzy set theoretical approaches may fail to obtain the optimal decision. Numerous approaches for intuitionistic fuzzy decision‐making problem have been introduced in the literature to overcome these short comings. But there is no single approach that can be used to solve all kinds of problems because of the partial ordering defined on the collection of intuitionistic fuzzy numbers (IFNs). Even though ranking of fuzzy numbers have been studied from early seventies in the last century, a total order on the entire class of fuzzy numbers has been introduced by Wang and Wang (Fuzzy Sets Syst 2014, 243, 131–141) only on 2014. A total order on the collection of all IFN is an open problem till today. In this article, a total order on the entire class of IFN using upper lower dense sequence in the interval [0, 1] is proposed and compared with existing techniques using illustrative examples, further an algorithm (which is problem independent) for solving any intuitionistic fuzzy multicriteria decision‐making problem (Intuitionistic fuzzy MCDM) is introduced. This new total ordering on IFNs generalizes the total ordering defined in Wang and Wang ( 22 ) for fuzzy numbers. © 2016 Wiley Periodicals, Inc. Complexity 21: 54–66, 2016     

模糊数的正向和反向表述及模糊数运算的拓展

模糊数运算的存在不可逆等问题,主要在于传统(正向)区间数严格限定所致.因此,提出了"反向区间数"的概念,利用该概念,能够给经典模糊分解定理、扩张原理新的表达形式.之后,分别以正(反)向区间为基础,分析模糊数的结构元表达形式,得到正(反)向区间对应结构元理论中单调增(减)函数.定义了反向区间数和反向区间数加、乘运算法则,利用结构元理论,证明了正、反向模糊数的加、乘运算解析表达式,得到了模糊方程解的判断定理.在保持传统运算法则不变的同时,对模糊数概念进行正(反)向的表述,并定义了二者的运算法则,这拓展了传统模糊数解的空间,进而解决模糊方程求解、不可逆等问题.通过算例看出,这两种表述在实际的计算过程中具有明显的意义.     

Static output‐feedback control for interval type‐2 discrete‐time fuzzy systems

This article investigates the problem of reliable mixed control for discrete‐time interval type‐2 (IT2) fuzzy model‐based systems via static output‐feedback (SOF) control method. The number of fuzzy rules and the membership functions for the SOF controller are different from those for the plant. A sufficient criterion of reliable stability with mixed performance is derived for the closed‐loop system with sensor failure. The SOF controller is designed for two different cases (known sensor failure case and unknown sensor failure case) to guarantee the reliable stability with mixed performance. Moreover, novel criteria are presented to obtain the optical performance for the closed‐loop system. Finally, an example is used to verify the effectiveness of the proposed design scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 74–88, 2016     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

The Dual of a Logical Linear Programme

A Linear Programme (LP) involves a conjunction of linear constraints and has a well defined dual. It is shown that if we allow the full set of Boolean connectives {, , } applied to a set of linear constraints we get a model which we define as a Logical Linear Programme (LLP). This also has a well defined dual preserving most of the properties of LP duality. Generalisations of the connectives are also considered together with the relationship with Integer Programming formulation.     

A new linear ordering of fuzzy numbers on subsets of {{\mathcal F}({\pmb{\mathbb{R}}}})

We introduce a novel linear order on every family of fuzzy numbers which satisfies the assumption that their modal values must be all different and must form a compact subset of . A distinct new feature is that our linear determined procedure employs the corresponding order of a class interval associated with a confidence measure which seems intuitively anticipated. It is worthy noting that although we start from an entirely different rationale, we introduce a fuzzy ordering which initially coincides with the one established earlier by Ramik and Rimanek. However, this fuzzy ordering does not apply when the supports of the fuzzy numbers overlap. In order to cover such cases we extent this initial fuzzy ordering to the “extended fuzzy order” (XFO). This new XFO method includes a possibility and a necessity measure which are compared with the widely accepted PD and NSD indices of D. Dubois and H. Prade. The comparison shows that our possibility and necessity measures comply better with our intuition.     

Towards a new strategy for solving fuzzy optimization problems

Fuzzy Optimization models and methods has been one of the most and well studied topics inside the broad area of Soft Computing. Particularly relevant is the field of fuzzy linear programming (FLP). Its applications as well as practical realizations can be found in all the real world areas. As FLP problems constitute the basis for solving fuzzy optimization problems, in this paper a basic introduction to the main models and methods in FLP is presented and, as a whole, Linear Programming problems with fuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrix are analyzed. But fuzzy sets and systems based optimization methods do not end with FLP, and hence in order to solve more complex optimization problems, fuzzy sets based Meta-heuristics are considered, and two main operative approaches described. Provided that these techniques obtain efficient and/or effective solutions, we present a fuzzy rule based methodology for coordinating Meta-heuristics and in addition, to provide intelligence, we propose a process of extraction of the knowledge to conduct the coordination of the system.     

On stability in Fuzzy Linear Programming problems

This study focuses on the problem of stability (with respect to changes of centres of fuzzy parameters) of the solution in Fuzzy Linear Programming (FLP) problems with symmetrical triangular fuzzy numbers and extended operations and inequalities.     

Duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair

The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe’s primal and dual pair problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the Wolfe’s dual problem can be formulated by considering the gradients of differentiable fuzzy- valued functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.     

Dynamic Aggregation Operators Based on Intuitionistic Fuzzy Tools and Einstein Operations

The idea of combine aggregation and intuitionistic fuzzy information plays essential role in multi criteria decision making (MCDM) process. However, this new branch has attracted researchers that study in different fields recently. In this paper, we study MCDM problems with intuitionistic fuzzy environment. Firstly, we introduce some operations related with Einstein t-norm and t-conorm such as, Einstein sum, product and exponentiation. After that, we define dynamic intuitionistic fuzzy Einstein averaging (DIFWA${}^{?}$) operator and dynamic intuitionistic fuzzy Einstein geometric averaging (DIFWG${}^{?}$) operator. Their notable property is that collect and aggregate values in different period based on Einstein operations in intuitionistic fuzzy set (IFS)s. In addition, we compare the defined operators with the existing intuitionistic fuzzy dynamic operators and get the corresponding relations. We establish two methods using with DIFWA${}^{?}$ and DIFWG${}^{?}$ to solve MCDM problems with intuitionistic fuzzy tools. Finally, an illustrated example is presented to show the applicability of the introduced methods.     

On perturbations of certain nonconvex optimization problems

Using the concept of -conjugate functions, a wide class of nonconvex optimization problems can be investigated. By generalized Lagrangians, the problems indicated and a generalized notion of stability can be treated. It is possible to give duality theorems for these problems such that the approach given by Rockafellar (1967) for convex problems can be extended to the nonconvex case.     

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization*

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in ⁿ can be reformulated into certain smooth/convex unconstrained dual problems in ^m with m n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.