Response-Surface Methods in R, Using rsm

This article describes the recent package rsm, which was designed to provide R support for standard response-surface methods. Functions are provided to generate central-composite and Box-Behnken designs. For analysis of the resulting data, the package provides for estimating the response surface, testing its lack of fit, displaying an ensemble of contour plots of the fitted surface, and doing follow-up analyses such as steepest ascent, canonical analysis, and ridge analysis. It also implements a coded-data structure to aid in this essential aspect of the methodology. The functions are designed in hopes of providing an intuitive and effective user interface. Potential exists for expanding the package in a variety of ways.

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

Application of flexible recipes for model building, batch process optimization and control

Unlike the traditionally fixed recipes in batch process operation, flexible recipes allow the adjustment of some of its relevant recipe items. These adjustments can either be predefined in cases of planned experimentation, or suggested by a formal process optimization or control algorithm on the basis of actual information. In both the response surface methodology and the simplex evolutionary operation (EVOP), some well-known methods for empirical model building and process optimization, flexible recipes are involved. Another application of flexible recipes arises in a feedforward quality control strategy of batch processes when variations in market or process conditions are known a priori. The experimental results of these strategies are presented for the batchwise production of benzylalcohol on a pilotplant scale. Experiments have been performed to obtain a reliable model of the yield. On the basis of this model, better process conditions have been suggested, which substantially deviate from the final simplex resulted from experiments within simplex EVOP. Finally, an adaptive feedforward control strategy has been applied for a priori known disturbances in the process inputs

Response Surface Methodology and Its Application in Optimizing the Efficiency of Organic Solar Cells

Response surface methodology (RSM) is a ubiquitous optimization approach used in a wide variety of scientific research studies. The philosophy behind a response surface method is to sequentially run relatively simple experiments or models in order to optimize a response variable of interest. In other words, we run a small number of experiments sequentially that can provide a large amount of information upon augmentation. In this dissertation, the RSM technique is utilized in order to find the optimum fabrication condition of a polymer solar cell that maximizes the cell efficiency. The optimal device performance was achieved using 10.25 mg/ml polymer concentration, 0.42 polymer-fullerene ratio, and 1624 rpm of active layer spinning speed. The cell efficiency at the optimum stationary point was found to be 5.23% for the Poly(diketopyrrolopyrrole-terthiophene) (PDPP3T)/PC60BM solar cells. Secondly, we explored methods for constructing a confidence region for the stationary point in RSM. In particular, we developed methods for constructing simultaneous confidence intervals for the coordinates of a stationary point in a quadratic response surface model. The methods include Bonferroni adjustment, a plug-in approach based on the asymptotic distribution of maximum likelihood estimators, and bootstrapping. The simultaneous coverage probabilities of the proposed methods are assessed via simulation. The coverage probabilities for the Bonferroni and plug-in approaches are pretty close to the nominal levels of 0.95 for large sample sizes. The metaheuristic method is also considered in order to search for an alternative solution to the design matrix that may be near to the optimal solution. Finally, we explored recent developments in RSM including generalized linear models and the case of multivariate response variables

REVIEW OF EXPERIMENTAL DESIGN IN ANALYTICAL CHEMISTRY

The ability of a chromatographic method to successfully separate, identify and quantitative species is determined by many factors, many of which are in the control of the experimenter. When attempting to discover the important factors and then optimize a response by turning these factors by using multivariate statistical techniques for the optimization of chromatographic system. The surface response methodologies and experimental design give a powerful suite of statistical methodology. Advantage includes modeling by empherical function, a defined number of experiments to be performed and available software to accomplish the task of two uses of experimental design in chromatography for showing lack of significant factors and then optimizing a response within their method development. Plackett - Burman design (Screening) widely used in validation studies and fraction factorial designs and their extensions such as (response surface) central composite designs are most popular optimizers. Box-Behnken and Doehlert designs are becoming more used as efficient alternatives. The use of mixture designs for optimization of mobile phase is also related. A discussion about model validation is presented. Then simultaneously the multiple responses are optimized, the desirability function is used and discussed the criteria for judging the quality of a chromatogram by using multi criteria decision making studies. Some applications of multivariate techniques for optimization of chromatographic methods are also summarized

Effect of design selection on response surface performance

Artificial neural nets and polynomial approximations were used to develop response surfaces for several test problems. Based on the number of functional evaluations required to build the approximations and the number of undetermined parameters associated with the approximations, the performance of the two types of approximations was found to be comparable. A rule of thumb is developed for determining the number of nodes to be used on a hidden layer of an artificial neural net and the number of designs needed to train an approximation is discussed

Effect of design selection on response surface performance

The mathematical formulation of the engineering optimization problem is given. Evaluation of the objective function and constraint equations can be very expensive in a computational sense. Thus, it is desirable to use as few evaluations as possible in obtaining its solution. In solving the equation, one approach is to develop approximations to the objective function and/or restraint equations and then to solve the equation using the approximations in place of the original functions. These approximations are referred to as response surfaces. The desirability of using response surfaces depends upon the number of functional evaluations required to build the response surfaces compared to the number required in the direct solution of the equation without approximations. The present study is concerned with evaluating the performance of response surfaces so that a decision can be made as to their effectiveness in optimization applications. In particular, this study focuses on how the quality of approximations is effected by design selection. Polynomial approximations and neural net approximations are considered

HIGH PERFORMANCE AND ULTRA HIGH PERFORMANCE CONCRETE WITH LOCALLY AVAILABLE MATERIALS FROM SASKATCHEWAN

Reinforced concrete structures exhibit various durability problems, such as the corrosion of reinforcing steel, sulfate attack, etc., when exposed to harsh environments. This type of damage often leads to very serious technical and economic problems, such as a short lifetime of infrastructure and high costs associated with their long term maintenance and repair. High performance concrete (HPC) and ultra-high performance concrete (UHPC) could play key roles in solving or in mitigating these problems. The main research goal of this thesis was to determine whether it is possible to produce high performance concrete (HPC), very-high performance concrete (VHPC) and ultra-high performance concrete (UHPC) that have unique combinations of strength, freeze-thaw durability and self-placeability at competitive costs using materials locally available in Saskatchewan. To develop HPC and VHPC/UHPC, a statistical experimental design was used to perform experimental designs, analyze the fitting models and optimize multiple responses. The procedure was implemented using the Design-Expert Version 9.0 software. Seven materials were researched in this project to make concrete, namely: water, cement, silica fume, silica flour, fine sand, steel fiber, and superplasticizer (SP). Four different properties were measured, including the compressive strength, splitting tensile strength, air content of hardened concrete and flow cone test. After analyzing the results of these tests, it was found that the goal of developing a HPC material with the specified properties was achieved (flow cone spread value = 274 mm and, after 28 days, the obtained properties were: compressive strength = 82 MPa, splitting tensile strength = 23 MPa and air content = 6%.). The goal of making VHPC with the specified properties was obtained (flow cone spread value = 274 mm and, after 28 days, the obtained properties were: compressive strength iv = 102.4 MPa and splitting tensile strength = 23 MPa) regardless of air content. Nevertheless, the results of the analysis clearly showed that it would be impossible to produce a UHPC with a 28 day compressive strength greater than 150 MPa using the mix ingredients and fabrication processes adopted in this study

A REVIEW ON OPTIMIZATION OF DRUG DELIVERY SYSTEM WITH EXPERIMENTAL DESIGNS

The present review article aims at determining the various possible techniques available to enhance the quality, safety and efficacy of pharmaceutical formulations by exploring most suitable and practically applicable experimental designs and optimization techniques. As we know that pharmaceutical industries are constantly in search of novel ideas to improve quality by various optimization techniques, hence in present review article we shall discuss latest optimization techniques and experimental designs to achieve the best combination of product and process characteristics under the given set of conditions. Experimental designs and optimization techniques are the tools that are simultaneously and systematically used to identify various types of problems that may influence research, development and production of pharmaceutical formulations. These are organized an approach to determine the relationship between the factors affecting a process and the output of that process. The screening methods discussed here include factorial design, fractional factorial designs, full factorial design, mixture designs etc. Recently, different software has been used in implementing optimization techniques in pharmaceutical products to enhance product quality by using most suitable available facilities

A MODIFIED CLASS OF COMPOSITE DESIGNS FOR THE RESPONSE MODEL APPROACH WITH NOISE FACTORS

A class of composite designs involves factorial, axial, and center points. Factorial points are with a variance-optimal design for a first-order or interaction model, and axial points provide information about the existence of curvature. The center points allow for efficient estimation of the pure quadratic terms. From these properties, a class of composite designs is recommended if resources are readily available and a high degree of precision of parameter estimate is expected and evolves from their use in sequential experimentation. However, there are often cost constraints imposed on experiments. Previous studies show that resolution, orthogonal quadratic effect property, and saturated or near-saturated design reduce the number of experiments. This study extends the response model approach with noise factors to composite designs satisfying these properties. These modified composite designs are further discussed and examined in terms of scaled prediction error variance and extended scaled prediction variance, which provides a good distribution of the prediction variance of the response. Based on these criteria, the best performance design is suggested according to the number of control and noise factors. As a result, we show that the modified designs showing robustness to noise factors and stability of predictive variance are a class of modified small composite designs and modified augmented-pair designs