a compromise programming approach to robust designCHEN W 19991Word文档下载推荐.docx

1、Mechanical Engineering （M/C 251）842 W. Taylor St. University of Illinois at Chicago Chicago IL 60607-7022Phone: （312） 996-6072Fax: （312） 413-0447e-mail: weichen1uic.eduModified Manuscript to JMD, Oct. 14, 98ABSTRACTIn robust design, associated with each quality characteristic, the design objective o

2、ften involves multiple aspects such as “bringing the mean of performance on target” and “minimizing the variations”. Current ways of handling these multiple aspects using either the Taguchis signal-to-noise ratio or the weighted-sum method are not adequate. In this paper, we solve bi-objective robus

3、t design problems from a utility perspective by following upon the recent developments on relating utility function optimization to a Compromise Programming （CP） method. A robust design procedure is developed to allow a designer to express his/her preference structure of multiple aspects of robust d

4、esign. The CP approach, i.e., the Tchebycheff method, is then used to determine the robust design solution which is guaranteed to belong to the set of efficient solutions （Pareto points）. The quality utility at the candidate solution is represented by means of a quadratic function in a certain sense

5、 equivalent to the weighted Tchebycheff metric. The obtained utility function can be used to explore the set of efficient solutions in a neighborhood of the candidate solution. The iterative nature of our proposed procedure will assist decision making in quality engineering and the applications of r

6、obust design.Keywords: Robust Design, Multiobjective Optimization, Compromise Programming, Utility Function, Decision AnalysisMain Text Word Count: 5,658; Characters with space: 35,013.NOMENCLATURECP（ ,w） Weighted Tchebycheff Approach f（x） Objective FunctionF（x） Vector of Objective Functions F Candi

7、date Efficient Solution g（x） Constraint Functionkj Penalty FactorsK Loss Function CoefficientL1 Manhattan MetricL2 Euclidean MetricL Tchebycheff MetricLp Lp-metric S/N Signal/Noise w WeightsWS Weighted-sumWSP（w） Weighted-sum Problemx Vector of Design VariablesX Design SpacexL Lower Bound for Design

8、Variables xU Upper Bound for Design Variablesx0 Pareto Solution for Design Variablesx* Optimal Solution for Design Variablesx Candidate Solution in Design SpaceY Random Variable; Objective Space x Deviations of the Design Variables* Optimal Solution of -problemQuality Characteristic of S/Nu* Utopia

9、Point in CPf Mean of the Objective Function f（x）* Optimal Value of the Meanff Standard Deviation of the Objective Function f（x）* Optimal Value of the Standard Deviation1. INTRODUCTIONIn recent years, the Taguchi robust design method has been widely used to design quality into products and processes

10、（Phadke, 1989）. Using this method, the quality of a product is improved by minimizing the effect of the causes of variation without eliminating the causes （Taguchi, 1993）. While the majority of the early applications of robust design consider manufacturing as the cause for performance variations, re

11、cent developments in design methodology have produced approaches that utilize the same concept to improve the robustness of design decisions with respect to the variations associated with the design process （Chang et al., 1994; Chen et al., 1996b）.Although Taguchis robust design principle has been w

12、idely accepted, the methods Taguchi offers for robust design have received much criticism, in particular the two-part orthogonal array for experimental design and the signal-to-noise-ratio （S/N ratio） used as the robust optimization criterion （Box, 1988; Nair, 1992）. In the engineering design commun

13、ity, researchers are working on developing nonlinear programming methods that can be used for a variety of robust design applications （Otto and Antonsson, 1991; Parkinson et al., 1993; Yu and Ishii, 1994; and Cagan and Williams, 1993）, including probabilistic optimization methods for robust design （

14、Siddall, 1984; Eggert and Mayne,1993）. A comprehensive review of existing robust optimization methods is provided bySu and Renaud （1997）, and will not be repeated here.One issue that we find has not been adequately addressed in the previous investigations is the multiple aspects of the objective in

15、robust design. It was illustrated by one of the authors （Chen et al., 1996b） that associated with each quality （performance） characteristic, the robust design objective could be generalized into two aspects, namely,“optimizing the mean of performance” and “minimizing the variation of performance”. A

16、brief mathematical background that supports the above statement is provided in Section2.1. Through our previous applications （Chen et al., 1996a and Chen et al., 1997）, we observe that the performance variation is often minimized at the cost of sacrificing the best performance, and therefore the tra

17、deoff between the aforementioned two aspects cannot be avoided. In the literature, though the multiple aspects of the objective in robust design is acknowledged （Sundaresan et al., 1993）, single robust design objective function is often utilized. Ramakrishnan and Rao, 1991, formulate the robust desi

18、gn problem as a nonlinear optimization problem with Taguchis loss function as the objective. Sundaresan et al. （1993） employ a single objective function that utilizes weighting factors for target performance and variance represented by the Sensitivity Index （SI）. Bras and Mistree （1995） and Chen et

19、al. （1996b） introduce the compromise Decision Support Problem （DSP） （Mistree et al., 1993）, a goal programming approach, to model the multiple aspects of robust design objective as separate goals. We assert that the use of weighted sums of objectives is a very simplistic approach to multiobjective o

20、ptimization problems. A closer look at the drawbacks of minimizing weighted sums of objectives in multicriteria optimization is provided by Das and Dennis （1997）. More rigorous methods need to be considered for representing the preference structure of multiple objectives in robust design.For modelin

21、g designers preference structure, one of the commonly used methods is based on the utility theory （von Neumann and Morgenstern, 1947; Keeney and Raifa,1976; Hazelrigg, 1996; Thurston, 1991）. Under the notion of utility theory, the ultimate overall worth of a design is represented by a multiattribute

22、 utility function which incorporates consideration of attributes that cannot be directly converted to a commonmetric. Ideally, when the preference of the multiple aspects of the objective in robust design could be captured by the multiattribute utility analysis, robust design could be solved as a si

23、ngle objective optimization problem. However, one difficulty associated with using the utility function approach is that, in practice, it is often impossible to obtain a reliable mathematical representation of the decision-makers actual utility function. In the literature, approaches that take diffe

24、rent paradigms for solving multicriteria optimization problems are proposed. For instance, Messac （1996） develops the method of physical programming which eliminates the need for weight setting or utility function building in multicriteria optimization. In this work, we propose to use Compromise Pro

25、gramming （CP） （Yu, 1973 and Zeleny, 1973） to address the multiple aspects of robust design.CP is one of the approaches that take a paradigm different from the utility theory. The basic idea in CP is the identification of an ideal solution as a point where each attribute under consideration achieves

26、its optimum value and seek a solution that is as close as possible to the ideal point （Zelenys axiom of choice）. Though the weights representing relative importance are used as the preference structure when applying CP, it has been mathematically proven that CP is superior to the weighted-sum （WS） m

27、ethod in locating the efficient solutions, or the so called Pareto points （Steuer, 1986）. However, there are few applications of CP to mechanical engineering design problems. Miura and Chargin （1996） develop a variation of CP and apply it to optimal structural design. Athan and Papalambros （1996） do

28、 not refer to CP but propose to minimize the sum of the exponentially weighted objective functions and illustrate their approach also on some structural design problems.Though utility theory and CP are considered very different paradigms and methodologies to measure preferences as well as to determi

29、ne decision makers optima onthe efficient frontier, researchers have illustrated a linkage between the two approaches （Ballestero and Romero, 1991）. One of the authors established a relationship between a CP approach and a quadratic weighted-sums scalarization of multiobjective problems （Tind and Wi

30、ecek, 1997）. In this paper, we apply CP, specifically the Tchebycheff method, to multiobjective robust design problems from a utility perspective by following upon the recent developments. An interactive robust design procedure is developed to support decision making in robust design applications.2. TECHNOLOGICAL BASIS OF OUR APPROACH2.1 Multiple Quality Aspects of Robust DesignThe quality loss function is used by Taguchi as a metric for robust optimization. The relationship between quality loss and the amount of deviation from the target value is expressed by the loss functions for diffe