Neural network technique for fuzzy multiobjective linear programming

Neural Network(NN) is well-known as one of powerful computing tools to solve optimization problems. Due to the massive computing unit-neurons and parallel mechanism of neural network approach we can solve the large-scale problem efficiently and optimal solution can be gotten. In this paper, we intoroduce improvement of the two-phase approach for solving fuzzy multiobjectve linear programming problem with both fuzzy objectives and constraints and we propose a new neural network technique for solving fuzzy multiobjective linear programming problems. The procedure and efficiency of this approach are shown with numerical simulations.     

Linear programming with fuzzy coefficients in constraints

This paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustrate the solution procedure.     

Decomposition's Dantzig–Wolfe applied to fuzzy multicommodity flow problems

We present, in this paper, a method for solving linear programming problems with fuzzy costs based on the classical method of decomposition's Dantzig–Wolfe. Methods using decomposition techniques address problems that have a special structure in the set of constraints. An example of such a problem that has this structure is the fuzzy multicommodity flow problem. This problem can be modeled by a graph whose nodes represent points of supply, demand and passage of commodities, which travel on the arcs of the network. The objective is to determine the flow of each commodity on the arcs, in order to meet demand at minimal cost while respecting the capacity constraints of the arcs and the flow conservation constraints of the nodes. Using the theory of fuzzy sets, the proposed method aims to find the optimal solution, working with the problem in the fuzzy form during the resolution procedure.     

Optimal design of trusses by approximate compatibility

Optimal design of elastic trusses is formulated as an approximate linear programming problem. Using the force method of analysis, the redundant forces are expressed in linearized terms of the design variables. The solution of the resulting linear programming problem can be viewed as an exact optimum for a truss with different displacements corresponding to the unknown redundants. The latter displacements, computed directly from the linear programming solution, indicate the degree of not satisfying the compatibility conditions. This information can be used to introduce imaginary displacements in subsequent iteration cycles.An iterative procedure of solution is proposed in which both the design and the imaginary displacements are modified until the compatible optimal solution is reached. Each iteration cycle requires the solution of a linear programming problem. The proposed procedure provides more flexibility in the solution process than the usual algorithms based on a sequence of linear programs and may improve the convergence. Numerical examples illustrate the application of this procedure in optimal design of simple trusses.     

Optimal design of elastic trusses by approximate equilibrium

Optimal design of elastic trusses is formulated as an approximate linear programming problem. Using the displacement method of analysis it is shown that the system equilibrium equations represent the only nonlinear functions of the variables. The linear programming formulation is obtained by ignoring temporarily the nonlinear terms in the latter equations. The solution of this approximate problem can be viewed as an exact optimum for a set of different loadings.An iterative procedure of solution, based on a sequence of linear programs, is proposed. In each iteration cycle both the design variables and a set of imaginary loadings are modified. The latter loadings can be introduced from those loadings corresponding to the exact optima at preceding iteration cycles. The proposed procedure provides more flexibility in the solution process compared with the usual algorithms based on a sequence of linear programs and may improve the convergence to the optimum.     

Interactive fuzzy programming for fuzzy random two-level linear programming problems through probability maximization with possibility

This paper considers two-level linear programming problems involving fuzzy random variables under cooperative behavior of the decision makers. Through the introduction of fuzzy goals together with possibility measures, the formulated fuzzy random two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. By adopting probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.     

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.     

Interactive fuzzy random cooperative two-level linear programming through level sets based probability maximization

In this paper, assuming cooperative behavior of the decision makers, two-level linear programming problems under fuzzy random environments are considered. To deal with the formulated fuzzy random two-level linear programming problems, α-level sets of fuzzy random variables are introduced and an α-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.     

Method for solving multiobjective aggregate production planning problem with fuzzy parameters

We propose an efficient method that transforms a fuzzy multiple objective linear programming (MOLP) problem model to crisp MOLP model, and an interactive solution procedure that suggest the best compromise aggregate production plans for the multi-period fuzzy multiple objective aggregate production planning (APP) problem.     

A fuzzy bi-criteria transportation problem

In this paper, a fuzzy bi-criteria transportation problem is studied. Here, the model concentrates on two criteria: total delivery time and total profit of transportation. The delivery times on links are fuzzy intervals with increasing linear membership functions, whereas the total delivery time on the network is a fuzzy interval with a decreasing linear membership function. On the other hand, the transporting profits on links are fuzzy intervals with decreasing linear membership functions and the total profit of transportation is a fuzzy number with an increasing linear membership function. Supplies and demands are deterministic numbers. A nonlinear programming model considers the problem using the max–min criterion suggested by Bellman and Zadeh. We show that the problem can be simplified into two bi-level programming problems, which are solved very conveniently. A proposed efficient algorithm based on parametric linear programming solves the bi-level problems. To explain the algorithm two illustrative examples are provided, systematically.     

Fast algorithm for a constrained infinite horizon LQ problem

This paper presents an efficient algorithmic solution to the infinite horizon linear quadratic optimal control problem for a discrete-time SISO plant subject to bound constraints on a scalar variable. The solution to the corresponding quadratic programming problem is based on the active set method and on dynamic programming. It is shown that the optimal solution can be updated after inclusion or removal of an active constraint by a simple procedure requiring in the order of kn operations, n being the system order and k the time at which the constraint is included or removed.     

A direct solution approach to fuzzy mathematical programs with fuzzy decision variables

In this paper, a direct solution approach is presented for solving fuzzy mathematical programming problems with fuzzy decision variables. In the proposed approach, a fuzzy ranking procedure for fuzzy numbers and a meta-heuristic algorithm is employed. A basic example is presented in the paper. It has been observed that fuzzy mathematical programs with fuzzy decision variables can be solved effectively by employing direct solution approaches which are based on fuzzy ranking procedures and meta-heuristics.     

矿山产能分配的变量模糊线性规划模型

针对矿山资源开采过程中产能不确定的分配问题,引入了模糊结构元素理论。将产能用结构元表示,并利用结构元加权序将模糊数比较转化为单调函数比较,将含有模糊变量的线性规划问题等价转化为经典线性规划问题。以某矿山为例,建立矿山产能分配的变量模糊线性规划模型,并进行求解。结果表明：实现了将实际问题中的模糊事件进行精确表达,原问题的求解更简便。得到矿山产能取得最大可能利润时的可能分配。应用结构元加权序求解的线性规划模型优于结构元元序的。     

Multi-objective aggregate production planning with fuzzy parameters

In this paper, a direct solution method that is based on ranking methods of fuzzy numbers and tabu search is proposed to solve fuzzy multi-objective aggregate production planning problem. The parameters of the problem are defined as triangular fuzzy numbers. During problem solution four different fuzzy ranking methods are employed/tested. One of the primary objectives of this study is to show that how a multi-objective aggregate production planning problem which is stated as a fuzzy mathematical programming model can also be solved directly (without needing a transformation process) by employing fuzzy ranking methods and a metaheuristic algorithm. The results show that this can be easily achieved.     

Solving exclusionary side constrained transportation problem by using a hybrid spanning tree-based genetic algorithm

In real life applications we often have the following problem: How to find the reasonable assignment strategy to satisfy the source and destination requirement without shipping goods from any pairs of prohibited sources simultaneously to the same destination so that the total cost can be minimized. This kind of problem is known as the transportation problem with exclusionary side constraint (escTP). Since this problem is one of nonlinear programming models, it is impossible to solve this problem using a traditional linear programming software package (i.e., LINDO). In this paper, an evolutionary algorithm based on a genetic algorithm approach is proposed to solve it. We adopt a Prüfer number to represent the candidate solution to the problem and design the feasibility of the chromosome. Moreover, to handle the infeasible chromosome, here we also propose the repairing procedure. In order to improve the performance of the genetic algorithm, the fuzzy logic controller (FLC) is used to dynamically control the genetic operators. Comparisons with other conventional methods and the spanning tree-based genetic algorithm (st-GA) are presented and the results show the proposed approach to be better as a whole.     

Fuzzy based fast dynamic programming solution of unit commitment with ramp constraints

Abstract: A fast dynamic programming technique based on a fuzzy based unit selection procedure is proposed in this paper for the solution of the unit commitment problem with ramp constraints. The curse of dimensionality of the dynamic programming technique is eliminated by minimizing the number of prospective solution paths to be stored at each stage of the search procedure. Heuristics like priority ordering of the units, unit grouping, fast economic dispatch based on priority ordering, and avoidance of repeated economic dispatch through memory action have been employed to make the algorithm fast. The proposed method produced comparable results with the best performing methods found in the literature.     

New fuzzy control model and dynamic output feedback parallel distributed compensation

A new fuzzy modeling based on fuzzy linear fractional transformations model is introduced. This new representation is shown to be a flexible tool for handling complicated nonlinear models. Particularly, the new fuzzy model provides an efficient and tractable way to handle the output feedback parallel distributed compensation problem. We demonstrate that this problem can be given a linear matrix inequality characterization and hence is immediately solvable through available semidefinite programming codes. The capabilities of the new fuzzy modeling is illustrated through numerical examples.     

模糊调整在求解模糊规则中的应用*

提出一种求解模线性规划的非精确算法,它模拟人的调节过程,将模糊控制思想嵌入到遗传算法的变异与交叉算子中求解出一个模糊糊优解集,取代了以往利用单纯形注解模糊糊线性规划问题的一个最优解。     

Application of fuzzy sets to multi-objective project management decisions

In real-world project management (PM) decision problems, input data and/or related parameters are frequently imprecise/fuzzy over the planning horizon owing to incomplete or unavailable information, and the decision maker (DM) generally faces a fuzzy multi-objective PM decision problem in uncertain environments. This work focuses on the application of fuzzy sets to solve fuzzy multi-objective PM decision problems. The proposed possibilistic linear programming (PLP) approach attempts to simultaneously minimise total project costs and completion time with reference to direct costs, indirect costs, relevant activities times and costs, and budget constraints. An industrial case illustrates the feasibility of applying the proposed PLP approach to practical PM decisions. The main advantage of the proposed approach is that the DM may adjust the search direction during the solution procedure, until the efficient solution satisfies the DM's preferences and is considered to be the preferred satisfactory solution. In particular, computational methodology developed in this work can easily be extended to any other situations and can handle the realistic PM decision problems with simplified triangular possibility distributions.     

INTERACTIVE DECISION MAKING FOR MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMINGPROBLEMS WITH FUZZY PARAMETERS

Abstract

In this paper, we focus on multiobjective linear fractional programming problems with fuzzy parameters and present a new interactive decision making method for obtaining the satisficing solution of the decision maker (DM) on the basis of the linear programming method. The fuzzy parameters in the objective functions and the constraints are characterized by fuzzy numbers. The concept of a-Pareto optimality is introduced in which the ordinary Pareto optimality is extended based on the α-level sets of the fuzzy numbers. In our interactive decision making method, in order to generate a candidate for the satisficing solution which is also a-Pareto optimal, if the DM specifies the degree α of the a-level sets and the reference objective values, the minimax problem is solved by combined use of the bisection method and the linear programming method and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree a. Then by considering the current values of the objective functions and a as well as the trade-off rates, the DM acts on this solution by updating his/her reference objective values and/or degree a. In this way the satisficing solution for the DM can be derived efficiently from among an a-Pareto optimal solution set. A numerical example illustrates various aspects of the results developed in this paper.