Margaret M. Wiecek

Quality utility : a Compromise Programming approach to robust design

In robust design, associated with each quality characteristic, the design objective often 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 robust 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 design. 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 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 robust design.

Dynamic programming approaches to the multiple criteria knapsack problem

We study the integer multiple criteria knapsack problem and propose dynamic-programming-based approaches to finding all the nondominated solutions. Different and more complex models are discussed, including the binary multiple criteria knapsack problem, problems with more than one constraint, and multiperiod as well as time-dependent models.

Lagrangian Coordination for Enhancing the Convergence of Analytical Target Cascading

DOI: 10.2514/1.15326 Analytical target cascading is a hierarchical multilevel multidisciplinary design methodology. In analytical target cascading, top-level design targets (i.e., specifications) are propagated to lower-level design problems in a consistent and efficient manner. In this paper, a modified Lagrangian dual formulation and coordination for analytical target cascadingaredevelopedtoenhanceaformulation andcoordinationproposedearlierintheliterature.Theproposed approach guarantees all the properties established earlier and additionally offers new significant advantages. As established previously for the convex case, the proposed analytical target cascading coordination converges to a global optimal solution with corresponding optimal Lagrange multipliers in the dual space. The Lagrange multipliers can be viewed as the weights for deviations in analytical target cascading formulations. Thus the proposed coordination algorithm finds the optimal solution and the optimal weights for the deviation terms simultaneously.Theenhancementallowsfortargetcascadingbetweenlevels,fortheuseofaugmentedLagrangianto improveconvergenceofthecoordinationalgorithm,andforpreventionofunboundedness.Aguidelinetosetthestep size for subgradient optimization when solving the Lagrangian dual problem is also proposed.

A review of different perspectives on uncertainty and risk and an alternative modeling paradigm

Abstract The literature in economics, finance, operations research, engineering and in general mathematics is first reviewed on the subject of defining uncertainty and risk. The review goes back to 1901. Different perspectives on uncertainty and risk are examined and a new paradigm to model uncertainty and risk is proposed using relevant ideas from this study. This new paradigm is used to represent, aggregate and propagate uncertainty and interpret the resulting variability in a challenge problem developed by Oberkampf et al. [2004, Challenge problems: uncertainty in system response given uncertain parameters. Reliab Eng Syst Safety 2004; 85(1): 11–9]. The challenge problem is further extended into a decision problem that is treated within a multicriteria decision making framework to illustrate how the new paradigm yields optimal decisions under uncertainty. The accompanying risk is defined as the probability of an unsatisfactory system response quantified by a random function of the uncertainty.

Unbiased approximation in multicriteria optimization

Abstract. Algorithms generating piecewise linear approximations of the nondominated set for general, convex and nonconvex, multicriteria programs are developed. Polyhedral distance functions are used to construct the approximation and evaluate its quality. The functions automatically adapt to the problem structure and scaling which makes the approximation process unbiased and self-driven. Decision makers preferences, if available, can be easily incorporated but are not required by the procedure.

An improved algorithm for solving biobjective integer programs

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented.

Generating Epsilon-Efficient Solutions in Multiobjective Programming

Scalarization approaches to purposely generating e-efficient solutions of multiobjective programs are investigated and a generic procedure for computing these solutions is proposed and illustrated with an example. Real-life decision making situations in which the solutions are of significance are described.

Norm-based approximation in multicriteria programming

Abstract Based on new theoretical results on norms, heuristic algorithms to approximate the nondominated set of multicriteria programs are proposed. By automatically adapting to the problems structure and scaling, the approximation is constructed objectively without interaction with the decision maker. As the algorithms extend the results obtained for bicriteria programs, difficulties encountered when dealing with more than two criteria are discussed.

Approximating Pareto curves using the hyper-ellipse

This paper proposes an approximation method to generate the set of Pareto solutions of bi-criteria convex optimization problems. The approximation is achieved by means of fitting a hyper-ellipse to a very small number of Pareto points and the equation of hyperellipse yields an explicit analytical description of the Pareto set. The method is applied to unconstrained and constrained problems, and illustrated on examples showing its efficiency. The paper identifies the limits of applicability of the approach and proposes further extensions.

Equivalence of balance points and Pareto solutions in multiple-objective programming

It is shown that the concept of balance points introduced by Galperin (Ref. 1) is equivalent to the concept of Pareto optimality.