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.     

Random fuzzy multi-objective linear programming: Optimization of possibilistic value at risk (pVaR)

This paper considers multiobjective linear programming problems (MOLPP) where random fuzzy variables are contained in objective functions and constraints. A new decision making model optimizing possibilistic value at risk (pVaR) is proposed by incorporating the concept of value at risk (VaR) into possibility theory. It is shown that the original MOLPPs involving random fuzzy variables are transformed into deterministic problems. An interactive algorithm is presented to derive a satisficing solution for a decision maker (DM) from among a set of Pareto optimal solutions. Each Pareto optimal solution that is a candidate of the satisficing solution is exactly obtained by using convex programming techniques. A simple numerical example is provided to show the applicability of the proposed methodology to real-world problems with multiple objectives in uncertain environments.     

Interactive multiobjective fuzzy random programming through the level set-based probability model

This paper considers a multiobjective linear programming problem involving fuzzy random variable coefficients. A new fuzzy random programming model is proposed by extending the ideas of level set-based optimality and a stochastic programming model. The original problem involving fuzzy random variables is transformed into a deterministic equivalent problem through the proposed model. An interactive algorithm is provided to obtain a satisficing solution for a decision maker from among a set of newly defined Pareto optimal solutions. It is shown that an optimal solution of the problem to be solved iteratively in the interactive algorithm is analytically obtained by a combination of the bisection method and the simplex method.     

COEVOLUTIONARY GENETIC ALGORITHMS FOR NONCONVEX NONLINEAR PROGRAMMING PROBLEMS: REVISED GENOCOP III

In this paper, we focus on general nonconvex nonlinear programming problems and consider an applicability of genetic algorithms. For such problems, Michalewicz et al. 1995 recently proposed the coevolutionary genetic algorithm, called GENOCOP III, by introducing the concepts of search points and reference points which, respectively, satisfy the linear constraints and all of the constraints. Unfortunately, however, in GENOCOP III, since an initial population is randomly generated, it is quite difficult to generate reference points. Furthermore, a new search point is randomly generated on the line segment between a search point and a reference point and effectiveness and speed of search may be quite low. Realizing such difficulties, in this paper we propose the revised GENOCOP III by introducing a method for generating a reference point by minimizing the sum of squares of violated nonlinear constraints and a bisection method for generating a new search point on the line segment between a search point and a reference point. Through a lot of numerical experiments, both feasibility and effectiveness of the proposed method are demonstrated.     

Satisficing solutions of multi-objective fuzzy optimization problems using genetic algorithm

In the present paper, a genetic algorithm for multi-objective optimization problems with max-product fuzzy relation equations as constraints is presented. Since the non-empty feasible domain of such problems is, in general, a non-convex set; the traditional optimization methods cannot be applied. Here, we are presenting a genetic algorithm (GA) to find “Pareto optimal solutions” for solving such problems observing the role of non-convexity of the feasible domain of decision problem. Solutions are kept within feasible region during the mutation as well as crossover operations. Test problems are developed to evaluate the performance of the proposed algorithm and to determine satisficing decisions. In case of two objectives, weighting method is also applied to find the locus of optimal solutions.     

Multiproduct aggregate production planning with fuzzy demands and fuzzy capacities

Given the uncertain market demands and capacities in production environment, this paper discusses some practical approaches to modeling multiproduct aggregate production planning problems with fuzzy demands, fuzzy capacities, and financial constraints. By formulating the fuzzy demand, fuzzy equation, and fuzzy capacities, a fuzzy production-inventory balance equation for single period and a dynamic balance equation are formulated as fuzzy/soft equations and they represent the possibility levels of meeting the market demands. Using this formulation and interpretation, a fuzzy multiproduct aggregate production planning model is developed, and its solutions using parametric programming, best balance and interactive techniques are introduced to cater to different scenarios under various decision making preferences. Using the proposed models and techniques, first, the decision maker can select a preferred production plan with a common satisfaction level or different combinations of preferred possibility level and satisfaction levels, according to the market demands and available production capacities, and second, the obtained structure of the optimal solution can help decision maker in aggregate production planning. The decision maker can also make a preferred and reasonable production plan corresponding to one's most concerned criteria. Hence, decision makers not only can come up with a reasonable aggregate production plan with minimum efforts, but also have more choices of making a preferred aggregate plan based on his most concerned criteria. These models can effectively enhance the capability of an aggregate plan to give feasible family disaggregation plans under different scenarios with fuzzy demands and capacities. Simulation and the results of analysis on the proposed techniques are also given in detail in this paper.     

Identifying and eliminating dominated alternatives in multi-attribute decision making with intuitionistic fuzzy information

Multi-attribute decision making under uncertainty is a usual task in our daily life. In the decision making process, the decision information provided by the decision maker (or expert) over alternatives may take the form of intuitionistic fuzzy numbers, and the weight information on attributes is usually incomplete. To this issue, we first transform the original decision matrix, whose elements are intuitionistic fuzzy numbers expressed by pairs of satisfaction degrees and dissatisfaction degrees, into its expected decision matrix, whose elements are composed of satisfaction degrees and hesitation degrees. We introduce the concept of dominated alternative, and give a method to identify the dominated alternatives. Then we develop an interactive method for eliminating any dominated alternatives by updating the decision maker's preferences gradually so as to find out the optimal one eventually. A further extension of the interactive method to interval-valued intuitionistic fuzzy situations is given, and the solution process of this interactive method is shown in detail through an illustrative example.     

An Interactive Satisfying Method Based on Alternative Tolerance for Multiple Objective Optimization With Fuzzy Parameters

An interactive satisfying method based on alternative tolerance is presented for the multiple objective optimization problem with fuzzy parameters. Using the $alpha $ -level sets of the fuzzy numbers, all the objectives are modeled as the fuzzy goals, and the tolerances of the objectives are iteratively changed according to a decision maker for a satisfying solution. Via a specific attainable point programming model, the membership functions can be modified, and then, a lexicographic two-phase programming procedure is constructed correspondingly to find the final solution. In a special case, the objective constraint is added instead of changing the membership functions; therefore, the dissatisfying objectives for the decision maker can be improved step by step. The presented method not only acquires the $alpha $ -Pareto optimal or weak $alpha $-Pareto optimal solution of the fuzzy multiple objective optimization, but also satisfies the progressive preference of the decision maker. A numerical example shows its power.      

An interactive fuzzy satisficing method based on variance minimization under expectation constraints for multiobjective stochastic linear programming problems

In this paper, we focus on multiobjective linear programming problems involving random variable coefficients in objective functions and constraints. Using the concept of chance constrained conditions, such multiobjective stochastic linear programming problems are transformed into deterministic ones based on the variance minimization model under expectation constraints. After introducing fuzzy goals to reflect the ambiguity of the decision maker??s judgements for objective functions, we propose an interactive fuzzy satisficing method to derive a satisficing solution for them as a fusion of the stochastic programming and the fuzzy one. The application of the proposed method to an illustrative numerical example shows its usefulness.     

An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making

This paper investigates the dynamic intuitionistic fuzzy multi-attribute group decision making (DIF-MAGDM) problems, in which all the attribute values provided by multiple decision makers (DMs) at different periods take the form of intuitionistic fuzzy numbers (IFNs), and develops an interactive method to solve the DIF-MAGDM problems. The developed method first aggregates the individual intuitionistic fuzzy decision matrices at different periods into an individual collective intuitionistic fuzzy decision matrix for each decision maker by using the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator, and then employs intuitionistic fuzzy TOPSIS method to calculate the individual relative closeness coefficient of each alternative for each decision maker and obtain the individual ranking of alternatives. After doing so, the method utilizes the hybrid weighted averaging (HWA) operator to aggregate all the individual relative closeness coefficients into the collective relative closeness coefficient of each alternative and obtain the aggregate ranking of alternatives, by which the optimal alternative can be selected. In addition, the spearman correlation coefficient for both the aggregate ranking and individual ranking of alternatives is calculated to measure the consensus level of the group preferences. Finally, a numerical example is used to illustrate the developed method.     

An intuitionistic fuzzy goal programming approach for finding pareto-optimal solutions to multi-objective programming problems

Multi-objective optimization in the intuitionistic fuzzy environment is the process of finding a Pareto-optimal solution that simultaneously maximizes the degree of satisfaction and minimizes the degree of dissatisfaction of an intuitionistic fuzzy decision. In this paper, a new method for solving multi-objective programming problems is developed that unlike other methods in the literature, provides compromise solutions satisfying both the conditions of intuitionistic fuzzy efficiency and Pareto-optimality. This method combines the advantages of the intuitionistic fuzzy sets concept, goal programming, and interactive procedures, and supports the decision maker in the process of solving programming problems with crisp, fuzzy, or intuitionistic fuzzy objectives and constraints. A characteristic of the proposed method is that it provides a well-structured approach for determining satisfaction and the dissatisfaction degrees that efficiently uses the concepts of violation for both objective functions and constraints. Another feature of the proposed method comes from its continuous interaction with the decision maker. In this situation, through adjusting the problem's parameters, the decision maker would have the ability of revisiting the membership and non-membership functions. Therefore, despite the lack of information at the beginning of the solving process, a compromise solution that satisfies the decision maker's preferences can be obtained. A further feature of the proposed method is the introduction of a new two-step goal programming approach for determining the compromise solutions to multi-objective problems. This approach ensures that the compromise solution obtained during each iterative step satisfies both the conditions of intuitionistic fuzzy efficiency and Pareto-optimality. The application of the proposed model is also discussed in this paper.     

A multi-objective decision-making model with fuzzy rough coefficients and its application to the inventory problem

In this paper, we concentrate on developing a fuzzy rough multi-objective decision-making model according to uncertainty theory. We present some equivalent models and a traditional algorithm based on an interactive fuzzy satisfying method, which is similar to the interactive fuzzy rough satisfying method, in order to obtain a satisfying solution for the decision maker. In addition, the technique of fuzzy rough simulation is applied to deal with general fuzzy rough objective functions and fuzzy rough constraints which are usually difficult to convert into their equivalents. Furthermore, combined with the techniques of fuzzy rough simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy rough multi-objective programming problem. Finally, a model is applied to an inventory problem to illustrate the usefulness of the proposed model and algorithm, and then a sensitivity analysis is made.     

An interactive rectangle elimination method for biobjective decision making

This paper presents a new man-machine interactive method for biobjective decision making. It is specifically designed to cope with both the ill-defined nature of the decision problem and the high cost of computation points in the tradeoff (Pareto optimal) set. With this method, the decision maker may efficiently approximate the tradeoff set and/or estimate his preferred objective value. First, the notion of a rectangle representation of the tradeoff set by a set of points, called experiments, and a set of rectangles, defined by the experiments, is introduced. Next, a special class of decision makers is considered. For a decision maker in this special class, the rectangle representation of the tradeoff set defines a rectangle of uncertainty that contains the decision maker's preferred objective value. A measure of the worst ease uncertainity is formulated and minimized to yield an optimal strategy for interactively selecting experiments. Finally, this strategy is employed in a general interactive algorithm that works under minimal assumptions on the tradeoff set and on the decision maker.     

A hybrid fuzzy goal programming approach with different goal priorities to aggregate production planning

In this study a hybrid (including qualitative and quantitative objectives) fuzzy multi objective nonlinear programming (H-FMONLP) model with different goal priorities will be developed for aggregate production planning (APP) problem in a fuzzy environment. Using an interactive decision making process the proposed model tries to minimize total production costs, carrying and back ordering costs and costs of changes in workforce level (quantitative objectives) and maximize total customer satisfaction (qualitative objective) with regarding the inventory level, demand, labor level, machines capacity and warehouse space. A real-world industrial case study demonstrates applicability of proposed model to practical APP decision problems. GENOCOP III (Genetic Algorithm for Numerical Optimization of Constrained Problems) has been used to solve final crisp nonlinear programming problem.     

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.     

An interactive fuzzy multi-objective optimization method for engineering design

The coupling of performance functions due to common design variables and uncertainties in an engineering design process will result in difficulties in optimization design problems, such as poor collaboration among design objectives and poor resolution of design conflicts. To handle these problems, a fuzzy interactive multi-objective optimization model is developed based on Pareto solutions, where the metric function and some additional constraints are added to ensure the collaboration among design objectives. The trade-off matrix at the Pareto solutions was developed, and the method for selecting weighting coefficients of optimization objectives is also provided. The proposed method can generate a Pareto optimal set with the maximum satisfaction degree and the minimum distance from ideal solution. The favorable optimal solution can be then selected from the Pareto optimal set by analyzing the trade-off matrix and collaborative sensitivity. Two examples are presented to illustrate the proposed method.     

An epsilon-constraint method for fully fuzzy multiobjective linear programming

Linear ranking functions are often used to transform fuzzy multiobjective linear programming (MOLP) problems into crisp ones. The crisp MOLP problems are then solved by using classical methods (eg, weighted sum, epsilon-constraint, etc), or fuzzy ones based on Bellman and Zadeh's decision-making model. In this paper, we show that this transformation does not guarantee Pareto optimal fuzzy solutions for the original fuzzy problems. By using lexicographic ranking criteria, we propose a fuzzy epsilon-constraint method that yields Pareto optimal fuzzy solutions of fuzzy variable and fully fuzzy MOLP problems, in which all parameters and decision variables take on LR fuzzy numbers. The proposed method is illustrated by means of three numerical examples, including a fully fuzzy multiobjective project crashing problem.     

基于模糊逻辑NSGA-Ⅲ的开关磁阻发电机多目标优化算法

低资源型的汉越神经机器翻译中,数据稀疏问题是影响翻译性能的主要原因,目前缓解该问题的途径之一是通过语料扩充方法生成伪平行数据,并用于机器翻译模型的训练,伪平行数据生成方法主要有基于词的替换、单语数据回译和枢轴翻译3种.目前的研究集中于3种方法的单独使用,缺少方法间融合利用方面的研究工作,针对此问题,提出了融入双语词典的正反向枢轴方法,利用英语作为枢轴语言,在汉到英到越正向枢轴的基础上,融入利用稀有词构建的汉-英和英-越双语词典,将汉语单语数据通过模型翻译成英语数据,再利用英-越模型将其翻译成越南语数据,其次进行越到英到汉反向枢轴翻译将越南语单语数据翻译为汉语,以此在2个方向上生成汉越伪平行数据,并利用语言模型对生成的伪平行数据进行筛选.汉-越翻译任务上的实验结果表明,提出的融入双语词典的正反向枢轴方法,能够产生更优的伪平行语料,进而显著提升汉越神经机器翻译任务的性能.     

基于平行多种群与冗余基因策略的置信规则库优化方法

置信规则库(Belief rule base, BRB)的参数学习和结构学习共同影响着置信规则库的建模精度和复杂度. 为了提高BRB结构学习和参数学习的优化效率, 本文提出了一种基于平行多种群(Parallel multi-population)策略和冗余基因(Redundant genes)策略的置信规则库优化方法. 该方法采用平行多种群策略以实现对具有不同数量规则BRB同时进行优化的目的, 采用冗余基因策略以确保具有不同数量规则的BRB能够顺利进行(交叉, 变异等)相关优化操作. 最终自动生成具有不同数量规则BRB的最优解, 并得出帕累托前沿(Pareto frontier), 决策者可以根据自身偏好和实际问题需求, 综合权衡并在帕累托前沿中筛选最优解. 最后以某输油管道泄漏检测问题作为示例验证本文提出方法的有效性, 示例分析结果表明本文提出的方法可以一次生成具有多条规则BRB的最优解, 并且可以准确绘制出帕累托前沿, 为综合决策提供较强的决策支持.     

MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment

This paper proposes an intuitionistic fuzzy decision method based on prospect theory and the evidential reasoning approach, aiming at analyzing multi-attribute decision making problems in which the criteria values are intuitionistic fuzzy numbers and the information of attributes weights is unknown. Firstly, the measures of entropy and cross entropy are defined for intuitionistic fuzzy sets by taking into consideration the preference of decision maker towards hesitancy degree. Secondly, combined with bounded rationality, the prospect decision matrix is calculated in the light of prospect theory and intuitionistic fuzzy distance. Thirdly, the correlational analyses are conducted between the attribute weights and three indicators which are entropy, cross entropy and prospect value, and optimization models for identifying attribute weights are built under the circumstances that the weights are incomplete and unknown. Finally, in order to avoid the loss of decision making information, the evidential reasoning approach is applied to the calculation of comprehensive prospective values for all alternatives. Following the value calculation, the ranking and the optimal alternative are determined based on the comprehensive prospective values. Illustrating examples demonstrate that the proposed method is reasonable and feasible.