Equality Constraints in Multiobjective Robust Design Optimization: Decision Making Problem

Robust design optimization (RDO) problems can generally be formulated by incorporating uncertainty into the corresponding deterministic problems. In this context, a careful formulation of deterministic equality constraints into the robust domain is necessary to avoid infeasible designs under uncertain conditions. The challenge of formulating equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is itself a complex task. A judicious formulation of equality constraints adds to this complexity because additional tradeoffs are introduced between constraint satisfaction under uncertainty and multiobjective performance. Equality constraints under uncertainty in multiobjective problems can therefore pose a complicated decision making problem. In this paper, we provide a new problem formulation that can be used as an effective multiobjective decision making tool, with emphasis on equality constraints. We present two numerical examples to illustrate our theoretical developments.     

多目标决策的中心方法之统一探讨

该文从统一的角度研究了多目标决策的中心方法的结构及其收敛性质．提出了形式一般的、可采用三种曲线搜索规则的中心方法之算法模型并在很弱的条件下证明了其全局收敛性以此为基础．讨论了模型中的搜索方向等参量的取法，给出了两类可实现的算法．该文结果统一和推广了已有的单（多）目标决策的中心方法．数值结果表明该算法是有效的．     

多目标多维模糊决策模型的模糊交叉算法

对多目标多维模糊决策模型做了进一步研究，提出一种模糊环境下带有目标权重主观监督因子的目标函数，提出了计算模糊决策识别矩阵与目标权重的模糊交叉计算公式。该算法既充分利用了模糊决策中的客观信息，又充分利用了决策专家的知识与经验，将客观定权与主观定权有机结合起来，为求解最优模糊决策识别矩阵和确定目标最优权重提供了一种有效途径，进一步丰富了多目标多维模糊决策理论模型。将本文提出的模糊决策方法应用于16家电炉炼钢企业的模糊综合评价决策，取得了较为满意的效果。     

Solving the multiobjective possibilistic linear programming problem

In conventional multiobjective decision making problems, the estimation of the parameters of the model is often a problematic task. Normally they are either given by the decision maker (DM), who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets for which a lot of fuzzy approaches to multiobjective programming have been developed. In this paper we propose a method to solve a multiobjective linear programming problem involving fuzzy parameters (FP-MOLP), whose possibility distributions are given by fuzzy numbers, estimated from the information provided by the DM. As the parameters, intervening in the model, are fuzzy the solutions will be also fuzzy. We propose a new Pareto Optimal Solution concept for fuzzy multiobjective programming problems. It is based on the extension principle and the joint possibility distribution of the fuzzy parameters of the problem. The method relies on α-cuts of the fuzzy solution to generate its possibility distributions. These ideas are illustrated with a numerical example.     

Gradient projection and local region search for multiobjective optimisation

This paper presents a new method for multiobjective optimisation based on gradient projection and local region search. The gradient projection is conducted through the identification of normal vectors of an efficient frontier. The projection of the gradient of a nonlinear utility function onto the tangent plane of the efficient frontier at a given efficient solution leads to the definition of a feasible local region in a neighbourhood of the solution. Within this local region, a better efficient solution may be sought. To implement such a gradient-based local region search scheme, a new auxiliary problem is developed. If the utility function is given explicitly, this search scheme results in an iterative optimisation algorithm capable of general nonseparable multiobjective optimisation. Otherwise, an interactive decision making algorithm is developed where the decision maker (DM) is expected to provide local preference information in order to determine trade-off directions and step sizes. Optimality conditions for the algorithms are established and the convergence of the algorithms is proven. A multiobjective linear programming (MOLP) problem is taken for example to demonstrate this method both graphically and analytically. A nonlinear multiobjective water quality management problem is finally examined to show the potential application of the method to real world decision problems.     

基于熵的多目标多维模糊决策模型

顾及多目标多维决策中存在的模糊性和随机性,基于Jaynes的信息熵最大原理,提出一种模糊环境下带有主观监督因子和信息熵的目标函数,导出了新的计算模糊决策识别矩阵与目标权重的模糊交叉计算公式.该模型将基于目标的客观决策与主观决策有机结合起来,为求解最优模糊决策识别矩阵和确定目标最优权重提供了一种有效途径,并把信息熵作为评价决策优劣的指标,进一步发展了多目标多维模糊决策理论模型.将本文提出的模糊决策方法应用于16家电炉炼钢企业的模糊综合评价,取得了较为满意的效果.     

An interactive fuzzy satisficing method for multiobjective stochastic linear programming problems through an expectation model

For decision making problems involving uncertainty, both stochastic programming as an optimization method based on the theory of probability and fuzzy programming representing the ambiguity by fuzzy concept have been developing in various ways. In this paper, we focus on multiobjective linear programming problems with random variable coefficients in objective functions and/or constraints. For such problems, as a fusion of these two approaches, after incorporating fuzzy goals of the decision maker for the objective functions, we propose an interactive fuzzy satisficing method for the expectation model to derive a satisficing solution for the decision maker. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Priority based fuzzy goal programming technique to fractional fuzzy goals using dynamic programming

This paper describes the use of preemptive priority based fuzzy goal programming method to fuzzy multiobjective fractional decision making problems under the framework of multistage dynamic programming. In the proposed approach, the membership functions for the defined objective goals with fuzzy aspiration levels are determined first without linearizing the fractional objectives which may have linear or nonlinear forms. Then the problem is solved recursively for achievement of the highest membership value (unity) by using priority based goal programming methodology at each decision stages and thereby identifying the optimal decision in the present decision making arena. A numerical example is solved to represent potentiality of the proposed approach.     

A multiobjective decomposition-coordination framework with an application to vehicle layout and configuration design

This short paper addresses both researchers in multiobjective optimization as well as industrial practitioners and decision makers in need of solving optimization and decision problems with multiple criteria. To enhance the solution and decision process, a multiobjective decomposition-coordination framework is presented that initially decomposes the original problem into a collection of smaller-sized subproblems that can be solved for their individual solution sets. A common solution for all decomposed and, thus, the original problem is then achieved through a subsequent coordination mechanism that uses the concept of epsilon-efficiency to integrate decisions on the desired tradeoffs between these individual solutions. An application to a problem from vehicle configuration design is selected for further illustration of the results in this paper and suggests that the proposed method is an effective and promising new solution technique for multicriteria decision making and optimization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

SMAA - Stochastic multiobjective acceptability analysis

Stochastic multiobjective acceptability analysis (SMAA) is a multicriteria decision support technique for multiple decision makers based on exploring the weight space. Inaccurate or uncertain input data can be represented as probability distributions. In SMAA the decision makers need not express their preferences explicitly or implicitly; instead the technique analyses what kind of valuations would make each alternative the preferred one. The method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative, a central weight vector representing the typical valuations resulting in that decision, and a confidence factor measuring whether the input data is accurate enough for making an informed decision.     

具有稳定系数的多目标多维模糊决策算法

对多目标多维模糊决策模型的模糊交叉算法做了进一步研究,分析了多目标多维模糊决策模型中主观监督因子在三维以上模糊决策时对目标权重调节不灵敏的原因,提出一种新的模糊环境下带有目标权重主观监督因子和稳定系数的目标函数,给出了具有稳定系数和主观监督因子的目标权重计算公式,并给出了多目标多维模糊决策的算法,实例计算说明了算法的有效性。     

多目标多层次系统多维模糊决策理论模型

分析了模糊优选理论模型,指出了其对大系统决策的局限性,利用模糊模式识别理论与模糊关系优选理论,建立了多目标多层次系统多维模糊决策理论模型,进一步丰富了系统模糊决策理论。     

Dimension reduction in multiobjective optimization

In multiobjective optimization there is often the problem of the existence of a large number of objectives. For more than two objectives there is a difficulty with the representation and visualization of the solutions in the objective space. Therefore, it is not clear for the decision maker the trade-off between the different alternative solutions. Thus, this creates enormous difficulties when choosing a solution from the Pareto-optimal set and constitutes a central question in the process of decision making. Based on statistical methods as Principle Component Analysis and Cluster Analysis, the problem of reduction of the number of objectives is addressed. Several test examples with different number of objectives have been studied in order to evaluate the process of decision making through these methods. Preliminary results indicate that this statistical approach can be a valuable tool on decision making in multiobjective optimization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robustness to uncertain optimization using scalarization techniques and relations to multiobjective optimization

In this paper, we propose two kinds of robustness concepts by virtue of the scalarization techniques (Benson’s method and elastic constraint method) in multiobjective optimization, which can be characterized as special cases of a general non-linear scalarizing approach. Moreover, we introduce both constrained and unconstrained multiobjective optimization problems and discuss their relations to scalar robust optimization problems. Particularly, optimal solutions of scalar robust optimization problems are weakly efficient solutions for the unconstrained multiobjective optimization problem, and these solutions are efficient under uniqueness assumptions. Two examples are employed to illustrate those results. Finally, the connections between robustness concepts and risk measures in investment decision problems are also revealed.     

Domination, discretization, and scalarization

This is an overview of a few possibilities that are open by model theory in applied mathematics. The most attention is paid to the present state and frontiers of the Cauchy method of majorants, approximation of operator equations with finite-dimensional analogs, and the Lagrange multiplier principle in multiobjective decision making.     

一类大系统模糊随机多目标决策的建模与求解方法

本文提出和建立了一类大系统模糊随机多目标决策（ＦＳＭＯＤＭ）模型，讨论了模糊事件的模糊概率的若干基本性质．在此基础上，将ＦＳＭＯＤＭ模型转化为清晰的多目标决策模型，并给出了递阶分解协调求解方法．把该模型应用到大连市水资源与经济发展规划研究中取得了令人满意的结果．     

Robust multiobjective optimization with application to Internet routing

Robust optimization addressing decision making under uncertainty has been very well developed for problems with a single objective function and applied to areas of human activity such as portfolio selection, investment decisions, signal processing, and telecommunication-network planning. As these decision problems typically have several decisions or goals, we extend robust single objective optimization to the multiobjective case. The column-wise uncertainty model can be carried over to the multiobjective case without any additional assumptions. For the row-wise uncertainty model, we show under additional assumptions that robust efficient solutions are efficient to specific instance problems and can be found as the efficient solutions of another deterministic problem. Being motivated by the fact that Internet traffic must be maintained in a reliable yet affordable manner in situations of complex and dynamic usage, we apply the row-wise model to an intradomain multiobjective routing problem with polyhedral traffic uncertainty. We consider traditional objective functions corresponding to link utilizations and implement the biobjective case using the parametric simplex algorithm to compute robust efficient routings. We also present computational results for the Abilene network and analyze their meaning in the context of the application.     

Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation

The purpose of the present work is to implement well-known statistical decision and game theory strategies into multiobjective stochastic control problems of fluid dynamics. Such goal is first justified by the fact that deterministic (either singleobjective or multiobjective) control problems that are obtained without taking into account the uncertainty of the model are usually unreliable. Second, in most real-world problems, several goals must be satisfied simultaneously to obtain an optimal solution and, as a consequence, a multiobjective control approach is more appropriate. Therefore, we develop a multiobjective stochastic control algorithm for general fluid dynamics applications, based on the Bayes decision, adjoint formulation and the Nash equilibrium strategies. The algorithm is exemplified by the multiobjective stochastic control of a periodic Burgers equation.     

ISTMO: An interval reference point-based method for stochastic multiobjective programming problems

In this paper, we present an interactive algorithm (ISTMO) for stochastic multiobjective problems with continuous random variables. This method combines the concept of probability efficiency for stochastic problems with the reference point philosophy for deterministic multiobjective problems. The decision maker expresses her/his references by dividing the variation range of each objective into intervals, and by setting the desired probability for each objective to achieve values belonging to each interval. These intervals may also be redefined during the process. This interactive procedure helps the decision maker to understand the stochastic nature of the problem, to discover the risk level (s)he is willing to assume for each objective, and to learn about the trade-offs among the objectives.     

Solution of a possibilistic multiobjective linear programming problem

The estimate of the parameters which define a conventional multiobjective decision making model is a difficult task. Normally they are either given by the Decision Maker who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from the past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets and several fuzzy approaches to multiobjective programming have been developed 1, 9, 10, 11. The fuzziness of the parameters gives rise to a problem whose solution will also be fuzzy, see 2, 3, and which is defined by its possibility distribution. Once the possibility distribution of the solution has been obtained, if the decision maker wants more precise information with respect to the decision vector, then we can pose and solve a new problem. In this case we try to find a decision vector, which approximates as much as possible the fuzzy objectives to the fuzzy solution previously obtained. In order to solve this problem we shall develop two different models from the initial solution and based on Goal Programming: an Interval Goal Programming Problem if we define the relation “as accurate as possible” based on the expected intervals of fuzzy numbers, as we showed in [4], and an ordinary Goal Programming based on the expected values of the fuzzy numbers that defined the goals. Finally, we construct algorithms that implement the above mentioned solution method. Our approach will be illustrated by means of a numerical example.