Fuzzy random chance-constrained programming

By fuzzy random programming, we mean the optimization theory dealing with fuzzy random decision problems. This paper presents a new concept of chance of fuzzy random events, and constructs a general framework of fuzzy random chance-constrained programming. We also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming. To speed up the process of handling uncertain functions, we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation. Finally, we integrate the fuzzy random simulation, neural network, and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples     

求解随机相关机会规划的有效算法

随机相关机会规划作为一类重要的随机规划,存在于许多领域中.为了寻找更为有效的求解随机相关机会规划的算法,采用随机仿真来逼近机会函数,在微粒群算法中利用随机仿真估计适应值,提出一种将随机仿真与微粒群算法相结合的随机相关机会规划算法.通过实例仿真测试该算法的性能,并与遗传算法进行比较,结果表明本算法具有一定的优势.     

Fuzzy random programming with equilibrium chance constraints

To model fuzzy random decision systems, this paper first defines three kinds of equilibrium chances via fuzzy integrals in the sense of Sugeno. Then a new class of fuzzy random programming problems is presented based on equilibrium chances. Also, some convex theorems about fuzzy random linear programming problems are proved, the results provide us methods to convert primal fuzzy random programming problems to their equivalent stochastic convex programming ones so that both the primal problems and their equivalent problems have the same optimal solutions and the techniques developed for stochastic convex programming can apply. After that, a solution approach, which integrates simulations, neural network and genetic algorithm, is suggested to solve general fuzzy random programming problems. At the end of this paper, three numerical examples are provided. Since the equivalent stochastic programming problems of the three examples are very complex and nonconvex, the techniques of stochastic programming cannot apply. In this paper, we solve them by the proposed hybrid intelligent algorithm. The results show that the algorithm is feasible and effectiveness.     

Fuzzy random programming models for location-allocation problem with applications

Three types of fuzzy random programming models based on the mean chance for the capacitated location-allocation problem with fuzzy random demands are proposed according to different criteria, including the expected cost minimization model, the α-cost minimization model, and the chance maximization model. In order to solve the proposed models, some hybrid intelligent algorithms are designed by integrating the network simplex algorithm, fuzzy random simulation, and genetic algorithm. Finally, some numerical examples about a container freight station problem are given to illustrate the effectiveness of the devised algorithms.     

Fuzzy mathematical programming

Fuzzy linear programming (FLP) was originally suggested to solve problems which could be formulated as LP-models, the parameters of which, however, were fuzzy rather than crisp numbers. It has turned out in the meantime that FLP is also well suited to solve LP-problems with several objective functions. FLP belongs to goal programming in the sense that implicitly or explicitly aspiration levels have to be defined at which the membership functions of the fuzzy sets reach their maximum or minimum. Main advantages of FLP are, that the models used are numerically very efficient and that they can in many ways be well adopted to different decision behaviors and contexts.     

Hybrid multilevel programming with uncertain random parameters

Multilevel programming is developed for modeling decentralized decision-making processes. For different management requirements and risk tolerances of different-level decision-makers, the decision-making criteria applied in different levels cannot be always the same. In this paper, a hybrid multilevel programming model with uncertain random parameters based on expected value model (EVM) and dependent-chance programming (DCP), named as EVM–DCP hybrid multilevel programming, is proposed. The corresponding concepts of Nash equilibrium and Stackelberg–Nash equilibrium are given. For some special case, an equivalent crisp mathematical programming is proposed. An approach integrating uncertain random simulations, Nash equilibrium searching approach and genetic algorithm is designed. Finally, a numerical experiment of uncertain random supply chain pricing decision problem is given.     

Fuzzy programming for mixed-integer optimization problems

Mixed-integer optimization problems belong to the group of NP-hard combinatorial problems. Therefore, they are difficult to search for global optimal solutions. Mixed-integer optimization problems are always described by precise mathematical programming models. However, many practical mixed-integer optimization problems have inherited a more or less imprecise nature. Under these circumstances, if we take into account the flexibility of the constraints and the fuzziness of the objectives, the original mixed-integer optimization problems can be formulated as fuzzy mixed-integer optimization problems. Mixed-integer hybrid differential evolution (MIHDE) is an evolutionary search algorithm which has been successfully applied to many complex mixed-integer optimization problems. In this article, a fuzzy mixed-integer mathematical programming model is developed to formulate the fuzzy mixed-integer optimization problem. In addition the MIHDE is introduced to solve the fuzzy mixed-integer programming problem. Finally, the illustrative example shows that satisfactory results can be obtained by the proposed method. This demonstrates that MIHDE can effectively handle fuzzy mixed-integer optimization problems.     

Interactive fuzzy programming for random fuzzy two-level programming problems through possibility-based fractile model

This paper focuses on interactive decision making methods for random fuzzy two-level linear programming problems. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level programming problems are reduced to deterministic ones. Interactive fuzzy nonlinear programming to obtain 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.     

应急资源调度的模糊规划模型

研究了出救点资源可用量及出救所需时间为三角模糊数的应急资源调度问题。以应急开始时间满意度和资源需求满意度最大为第一目标,出救点最少为第二目标构建资源调度模糊规划模型。设计了将此模型化为确定性规划模型,进而用分层序列法求解模型的方法。以算例展示了模型中各目标间的协调及模型的实用性、算法的合理性和有效性。     

Fuzzy compromise programming for Group decision making

A multicriteria technique named fuzzy compromise programming is combined with a methodology known as group decision making under fuzziness to come up with a new technique that supports decision making with multiple criteria and multiple participants (or experts). All criteria (qualitative and quantitative) are modeled by way of fuzzy sets, utilizing the fact that criteria values in most water resources problems are vague, imprecise and/or ill defined. The involvement of multiple experts in the decision process is achieved by incorporating each participant's perception of criteria weights, best and worst criteria values, relative degrees of risk acceptance, as well as other parameters into the problem. The proposed methodology is illustrated with a case study taken from the literature, combined with the input of four expert individuals with diverse backgrounds. After processing the input from the experts, a group compromise decision is formulated.     

基于AHP的模糊线性规划求解方法

对于基于AHP的多准则分析过程,存在不一致区间判断的复杂评估问题.通过有下限和上限的区间数表示元素之间的比较比率,构造模糊约束集合矩阵,引入模糊集的隶属度函数表示对各种优先权矢量的满意程度,利用线性规划求解具有最大满意度的优先权矢量,得出候选者的总体优先顺序,并举例说明了应用该方法的计算过程.     

An interval programming approach for the bilevel linear programming problem under fuzzy random environments

In this paper, we address a class of bilevel linear programming problems with fuzzy random variable coefficients in objective functions. To deal with such problems, we apply an interval programming approach based on the $\alpha $ -level set to construct a pair of bilevel mathematical programming models called the best and worst optimal models. Through expectation optimization model, the best and worst optimal problems are transformed into the deterministic problems. By means of the Kth best algorithm, we obtain the best and worst optimal solutions as well as the corresponding range of the objective function values. In this way, more information can be provided to the decision makers under fuzzy random circumstances. Finally, experiments on two examples are carried out, and the comparisons with two existing approaches are made. The results indicate the proposed approaches can get not only the best optimal solution (ideal solution) but also the worst optimal solution, and is more reasonable than the existing approaches which can only get a single solution (ideal solution).     

考虑缺货的模糊随机缺陷生产系统

针对不可靠的生产过程,研究了生产故障时间为模糊随机变量且允许缺货的缺陷生产系统.建立含缺货费和模糊随机重修费的经济生产批量模型.基于可信性理论,建立其期望费用模型,揭示了费用函数的性质,并证明了使费用最小的最优生产时间的存在性和唯一性,从而确定了最优生产时间的上下界.基于此,设计了最优生产时间的二分法求解过程.最后通过算例验证了所提出模型的有效性,并分析了缺货费用、重修费用和缺陷产品比例对最优生产策略的影响.     

模糊随机信息系统及其属性约简

属性约简是粗糙集理论研究的核心内容之一。在信息系统中引入模糊随机变量,给出了模糊随机信息系统的概念,定义了期望相关关系。基于该关系,讨论了模糊随机信息系统属性约简的判定和方法,并通过实例分析了相应方法的具体计算步骤。     

系数为对称梯形模糊数的模糊线性规划

针对模糊系数的线性规划, 提出了一种系数为对称梯形模糊数的线性规划的方法, 同时得出一些定理、命题以及相应的算法, 并通过实例验证了该方法的有效性. 该方法与常规方法的不同之处在于无须将模糊线性规划转化为经典线性规划就能得到满意的模糊优化解, 因此提出的方法所取得的规划结果更加满足决策者的需要.     

Genetic based fuzzy goal programming for multiobjective chance constrained programming problems with continuous random variables

Solution procedure consisting of fuzzy goal programming and stochastic simulation-based genetic algorithm is presented, in this article, to solve multiobjective chance constrained programming problems with continuous random variables in the objective functions and in chance constraints. The fuzzy goal programming formulation of the problem is developed first using the stochastic simulation-based genetic algorithm. Without deriving the deterministic equivalent, chance constraints are used within the genetic process and their feasibilities are checked by the stochastic simulation technique. The problem is then reduced to an ordinary chance constrained programming problem. Again using the stochastic simulation-based genetic algorithm, the highest membership value of each of the membership goal is achieved and thereby the most satisfactory solution is obtained. The proposed procedure is illustrated by a numerical example.     

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.     

Fuzzy multilevel programming with a hybrid intelligent algorithm

In order to model fuzzy decentralized decision-making problem, fuzzy expected value multilevel programming and chance-constrained multilevel programming are introduced. Furthermore, fuzzy simulation, neural network, and genetic algorithm are integrated to produce a hybrid intelligent algorithm for finding the Stackelberg-Nash equilibrium. Finally, two numerical examples are provided to illustrate the effectiveness of the hybrid intelligent algorithm.     

Fuzzy stochastic dynamic programming for marketing decision support

One of the most important goals in marketing is to realize the highest profit by applying appropriate means to optimize the process of acquiring customers. To assist the marketer in making marketing decisions, this paper introduces a stochastic dynamic programming model for the process of acquiring customers. It is actually a stochastic multistage decision process, whose state space consists of granularized information on customers and whose transitions are controlled by marketing actions. Then it shows how to control this process using fuzzy constraints and how to characterize the goal of maximizing profit by a fuzzy set. After an overview of approaches in dynamic programming under fuzziness given by Bellman and Zadeh, this paper further presents a new model of fuzzy stochastic dynamic programming to solve the decision problem for a stochastic system with implicitly defined termination time. It is argued that this study can facilitate research and development of both financial engineering and e‐commerce. © 2000 John Wiley & Sons, Inc.