Integrated Robust Design Using Response Surface Methodology and Constrained Optimization

System design, parameter design, and tolerance design are the three stages of product or process development advocated by Genichi Taguchi. Parameter design, or robust parameter design (RPD), is the method to determine nominal parameter values of controllable variables such that the quality characteristics can meet the specifications and the variability transmitted from uncontrollable or noise variables is minimized for the process or product. Tolerance design is used to determine the best limits for the parameters to meet the variation and economical requirements of the design. In this thesis, response surface methodology (RSM) and nonlinear programming methods are adopted to integrate the parameter and tolerance design. The joint optimization method that conducts parameter design and tolerance design simultaneously is more effective than the traditional sequential process. While Taguchi proposed the crossed array design, the combined array design approach is more flexible and efficient since it combines controllable factors, internal noise factors, and external noise factors in a single array design. A combined array design and the dual response surface method can provide detailed information of the process through process mean and process variance obtained from the response model. Among a variety of cuboidal designs and spherical designs, standard or modified central composite designs (CCD) or face-centered cube (FCC) designs are ideal for fitting second-order response surface models, which are widely applied in manufacturing processes. Box-Behnken design (BBD), mixed resolution design (MRD), and small composite design (SCD) are also discussed as alternatives. After modeling the system, nonlinear programming can be used to solve the constrained optimization problem. Dual RSM, mean square error (MSE) loss criterion, generalized linear model, and desirability function approach can be selected to work with quality loss function and production cost function to formulate the object function for optimization. This research also extends robust design and RSM from single response to the study of multiple responses. It was shown that the RSM is superior to Taguchi approach and is a natural fit for robust design problems. Based on our study, we can conclude that dual RSM can work very well with ordinary least squares method or generalized linear model (GLM) to solve robust parameter design problems. In addition, desirability function approach is a good selection for multiple-response parameter design problems. It was confirmed that considering the internal noise factors (standard deviations of the control factors) will improve the regression model and have a more appropriate optimal solution. In addition, simulating the internal noise factors as control variables in the combined array design is an attractive alternative to the traditional method that models the internal noise factors as part of the noise variables. The purpose of this research is to develop the framework for robust design and the strategies for RSM. The practical objective is to obtain the optimal parameters and tolerances of the design variables in a system with single or multiple quality characteristics, and thereby achieve the goal of improving the quality of products and processes in a cost effective manner. It was demonstrated that the proposed methodology is appropriate for solving complex design problems in industry applications

Constructing model robust mixture designs via weighted G-optimality criterion

We propose and develop a new G-optimality criterion using the concept of weighted optimality criteria and certain additional generalizations. The goal of the weighted G-optimality is to minimize a weighted average of the maximum scaled prediction variance in the design region over a set of reduced models. A genetic algorithm (GA) is used for generating the weighted G-optimal exact designs in an experimental region for mixtures. The performance of the proposed GA designs is evaluated and compared to the performance of the designs produced by our genetic algorithm and the PROC OPTEX exchange algorithm of SAS/QC. The evaluation demonstrates the advantages of GA designs over the designs generated using exchange algorithm, showing that the proposed GA designs have better model-robust properties and perform better than the designs generated by the PROC OPTEX exchange algorithm

Development of the D-Optimality-Based Coordinate-Exchange Algorithm for an Irregular Design Space and the Mixed-Integer Nonlinear Robust Parameter Design Optimization

Robust parameter design (RPD), originally conceptualized by Taguchi, is an effective statistical design method for continuous quality improvement by incorporating product quality into the design of processes. The primary goal of RPD is to identify optimal input variable level settings with minimum process bias and variation. Because of its practicality in reducing inherent uncertainties associated with system performance across key product and process dimensions, the widespread application of RPD techniques to many engineering and science fields has resulted in significant improvements in product quality and process enhancement. There is little disagreement among researchers about Taguchi\u27s basic philosophy. In response to apparent mathematical flaws surrounding his original version of RPD, researchers have closely examined alternative approaches by incorporating well-established statistical methods, particularly the response surface methodology (RSM), while accepting the main philosophy of his RPD concepts. This particular RSM-based RPD method predominantly employs the central composite design technique with the assumption that input variables are quantitative on a continuous scale. There is a large number of practical situations in which a combination of input variables is of real-valued quantitative variables on a continuous scale and qualitative variables such as integer- and binary-valued variables. Despite the practicality of such cases in real-world engineering problems, there has been little research attempt, if any, perhaps due to mathematical hurdles in terms of inconsistencies between a design space in the experimental phase and a solution space in the optimization phase. For instance, the design space associated with the central composite design, which is perhaps known as the most effective response surface design for a second-order prediction model, is typically a bounded convex feasible set involving real numbers due to its inherent real-valued axial design points; however, its solution space may consist of integer and real values. Along the lines, this dissertation proposes RPD optimization models under three different scenarios. Given integer-valued constraints, this dissertation discusses why the Box-Behnken design is preferred over the central composite design and other three-level designs, while maintaining constant or nearly constant prediction variance, called the design rotatability, associated with a second-order model. Box-Behnken design embedded mixed integer nonlinear programming models are then proposed. As a solution method, the Karush-Kuhn-Tucker conditions are developed and the sequential quadratic integer programming technique is also used. Further, given binary-valued constraints, this dissertation investigates why neither the central composite design nor the Box-Behnken design is effective. To remedy this potential problem, several 0-1 mixed integer nonlinear programming models are proposed by laying out the foundation of a three-level factorial design with pseudo center points. For these particular models, we use standard optimization methods such as the branch-and-bound technique, the outer approximation method, and the hybrid nonlinear based branch-and-cut algorithm. Finally, there exist some special situations during the experimental phase where the situation may call for reducing the number of experimental runs or using a reduced regression model in fitting the data. Furthermore, there are special situations where the experimental design space is constrained, and therefore optimal design points should be generated. In these particular situations, traditional experimental designs may not be appropriate. D-optimal experimental designs are investigated and incorporated into nonlinear programming models, as the design region is typically irregular which may end up being a convex problem. It is believed that the research work contained in this dissertation is the initial examination in the related literature and makes a considerable contribution to an existing body of knowledge by filling research gaps

Two-Stage Stochastic Linear Programming with Recourse: A Characterization of Local Regions using Response Surface Methodology

The LP recourse problem applies to two-stage optimization problems where uncertainty in resource availability of the second stage hinders informed decision making. The recourse function affords a way to compensate later for an error in prediction now. The literature provides a rich body of work on the optimization of such problems, but little research has been accomplished regarding the characterization of the surface in the local region of optimality, in particular sensitivity analysis. A decision maker faced with considerations other than the modeled objective function must be presented with a way to estimate the impact of operating at non-optimal decision variable values. This work develops and demonstrates a technique for characterizing the surface using response surface methodology. Specifically, the flexibility and utility of RSM techniques applied to this class of problems is demonstrated, and a methodology for characterizing the surface in the local region using a low-order polynomial is developed

Comparison of the Optimal Design: Split-Plot Experiments

Over the past decade, there have been rapid advances in the development of methods for the design and analysis of optimal split-plot experiment. Industrial experimentation involving optimality criteria has not been fully exhausted. In this study, we present a literature review on the development of optimal design of split-plot experiments. Split-plot designs, optimality criteria are discussed. Recent developments of optimal split-plot designs is evaluated and compared. Keywords: Optimality, Designs, Experiment, Randomization, Split-plot, Algorithm, Blocks.

Modeling and Optimization of Stochastic Process Parameters in Complex Engineering Systems

For quality engineering researchers and practitioners, a wide number of statistical tools and techniques are available for use in the manufacturing industry. The objective or goal in applying these tools has always been to improve or optimize a product or process in terms of efficiency, production cost, or product quality. While tremendous progress has been made in the design of quality optimization models, there remains a significant gap between existing research and the needs of the industrial community. Contemporary manufacturing processes are inherently more complex - they may involve multiple stages of production or require the assessment of multiple quality characteristics. New and emerging fields, such as nanoelectronics and molecular biometrics, demand increased degrees of precision and estimation, that which is not attainable with current tools and measures. And since most researchers will focus on a specific type of characteristic or a given set of conditions, there are many critical industrial processes for which models are not applicable. Thus, the objective of this research is to improve existing techniques by not only expanding their range of applicability, but also their ability to more realistically model a given process. Several quality models are proposed that seek greater precision in the estimation of the process parameters and the removal of assumptions that limit their breadth and scope. An extension is made to examine the effectiveness of these models in both non-standard conditions and in areas that have not been previously investigated. Upon the completion of an in-depth literature review, various quality models are proposed, and numerical examples are used to validate the use of these methodologies