METHODS FOR OPTIMAL EXPERIMENTAL DESIGN UNDER DIFFERENT COST STRUCTURES: OFF-LINE AND SEQUENTIAL APPROACHES

Open Access
Author:
Santiago Duran, Eduardo
Graduate Program:
Industrial Engineering
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
October 04, 2010
Committee Members:
  • Enrique Del Castillo, Dissertation Advisor
  • Enrique Del Castillo, Committee Chair
  • M Jeya Chandra, Committee Member
  • Dennis Kon Jin Lin, Committee Member
  • Patrick M Reed, Committee Member
Keywords:
  • optimal designs
  • sequential optimization
  • genetic algorithms
  • optimization
Abstract:
An experimental design is defined by the values of the factors to be tried in a set of experimental runs. Optimal experimental design methods aim at finding a set of runs, or points in the factor space, that optimize some property of the fitted responses, a property usually related to the variance of the predictions of the response models. To get good experimental designs in practice, a large percentage of existing optimal design methods rely on a candidate list of potential experimental points from where an exchange algorithm selects a subset which is hoped to be close to the true variance optimal design. The quality of the final design depends therefore on the quality of the candidate list, and selecting the latter is not an easy task, particularly if the experimental region is irregular due to the existence of constraints. A smaller number of optimal design algorithms have been proposed based on evolutionary or genetic optimization methods in an attempt to get better local optimal designs. These methods do not rely on a candidate list of points, but have a representation of a design that is too simplistic, with the levels of the controllable factors forming a string of “genes'” in this biological metaphor which does not allow to explore efficiently the very large design space. The themes of this dissertation consider the optimal design problems mentioned above. A recently proposed evolutionary algorithm, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), is used to find experimental designs that are both D- and G-efficient. The ability of this algorithm to consider any non-convex, non-differentiable function composed of continuous variables makes this method an ideal global optimizer for finding experimental designs combining multiple alphabetic optimality criteria. The algorithm is applied to finding experimental designs for controllable and noise factors in the so-called Robust Parameter Design problem popularized by G. Taguchi. For its specific modeling flexibility and its powerful search of the design space, CMA-ES is an excellent choice to find designs that perform optimally with respect to multiple criteria. The resulting evolutionary designs, obtained with this existing algorithm outperform the existing designs available in the literature. A second topic of this dissertation addresses the lack of richness of the design representation, or genotype, in traditional genetic optimization algorithms applied to experimental design. Instead of representing a design chromosome by a string of numbers that simply give the levels of the factors, a new representation based on Lindenmayer systems (L-systems) is presented. In this representation, the instructions on how the experimental design is constructed are stored as a sequence of symbols. The symbols themselves are subject to the usual genetic optimization operations of mutation, crossover, and selection. The main advantage of this approach is that the design space is searched more efficiently, and the resulting designs contain an overall structure which can result in intuitive geometrical properties (such as symmetries) that may not be obtainable under other methods. In this genetic optimization metaphor, the L-system based algorithm defines the “genotype” as a candidate list of points, subject to evolutionary optimization pressure. The phenotype is selected, or filtered, from the genotype by a D-optimal or other similar exchange algorithm. With this, there is redundancy in the genotype, making the metaphor closer to what is observed in the natural world. The final theme this dissertation addresses is the problem of minimizing the costs incurred while the experimental design is conducted. This leads to a sequential optimization strategy, where the sequence of experimental points must balance the exploration of the space of the controllable factors with the “exploitation” of the fitted models such that good response values are obtained. The problem has been studied in the Control Theory literature, and is commonly referred to as the Dual Control problem. We present a multiple response Dual Control algorithm that minimizes the sum of quadratic off-target and controllable factor adjustment costs over the duration of the experiment. The algorithm allows a faster optimization of an industrial process compared with the usual two-stage approach followed in Response Surface Methods where a model is fitted first based on an experimental design and then the optimal settings are found from such fitted model. New features of the algorithm are allowance for multiple responses and the ability to deal with the unknown covariance matrix case, not previously addressed in the literature. Sequential optimization methods are important also in the field of Engineering Design, where an experimental design is used to determine the set of computer runs or simulations, which may take several hours or even days to complete, thus metamodels are fitted and optimized in as few runs as possible. For this reason, the new Dual Control algorithm presented in this thesis is compared not only with off-line experimental designs but also with the EGO (Efficient Global Optimization) algorithm, a popular Kriging-based method for the sequential optimization of computer models in Engineering Design.