Conditions of Slope-Rotatability for Third-Order Polynomial Regression Models

In experimental design for response surface analysis, it is sometimes of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of the response surface is important. In particular, Hader and Park (1978 Hader , R. J. , Park , S. H. ( 1978 ). Slope-rotatable central composite designs . Technometrics 20 : 413 – 417 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggested the concept of slope-rotatability and studied slope rotatable central composite designs. Until now, many response surface designs including central composite designs have been suggested for fitting second order response surface models. However, we often need to fit third-order polynomial regression models. In this article, we suggest extended central composite designs (ECCDs) to fit third-order models and find the necessary and sufficient conditions for slope-rotatability over all directions in the third-order polynomial models.     

Second-Order Response Surface Model with Neighbor Effects

This article considers the second-order response surface model in which the experimental units, i.e., plots experience the neighbor effects from immediate left and right neighboring plots assuming the plots to be placed adjacent linearly with no gaps. Conditions have been derived for the estimation of coefficients of second-order response surface model. A method of constructing designs for fitting second-order response surface in the presence of neighbor effects has been developed. The designs so obtained are found to be rotatable.     

An Algorithmic Construction of Four-Level Response Surface Designs

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. In this article, we present a method for constructing four-level design matrices in order to explore and optimize response surfaces where the predictor variables are each at four equally spaced levels, by utilizing a genetic algorithm. The produced designs achieve both properties of near-rotatability and estimation efficiency.     

A Compromise Experimental Design Method for Parametric Polynomial Response Surface Approximations

This study presents a compromise approach to augmentation of experimental designs, necessitated by the expense of performing each experiment (computational or physical), that yields higher quality parametric polynomial response surface approximations than traditional augmentation. Based on the D-optimality criterion as a measure of experimental design quality, the method simultaneously considers several polynomial models during the experimental design, resulting in good quality designs for all models under consideration, as opposed to good quality designs only for lower-order models, as in the case of traditional augmentation. Several numerical examples and an engineering example are presented to illustrate the efficacy of the approach.     

Construction of New Three-Level Response Surface Designs

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. In this article, we present a method for constructing three-level designs in order to explore and optimize response surfaces combining orthogonal arrays and covering arrays in a particular manner. The produced designs achieve the properties of rotatability, predictive performance and efficiency for the estimation of a second-order model.     

The augmentation of existing data for improving the path of steepest ascent

Response surface methodology is a collection of mathematical and statistical techniques that are useful for the modeling and analysis of problems in which a response of interest is influenced by several independent variables and the objective is to optimize this response. When we are at a point on the response surface that is remote from the optimum, such as the current operating conditions, there is little curvature in the system and the first-order model will be appropriate. In these circumstances, a preliminary procedure as the steepest ascent usually is employed to move sequentially in the direction of maximum increase in the response. To improve the estimation of parameters of the steepest ascent path, we present, in an efficient way, the augmentation of existing data such that the independent variables are made more orthogonal to each other. Additionally, when we estimate the true path using this method, the bias and magnitude of the covariance matrix of the estimated path is decreased, significantly.     

A reference prior for the analysis of a response surface

Standard response surface methodology employs a second order polynomial model to locate the stationary point ξ$\mathit{\xi }$ of the true response function. To make Bayesian analysis more direct and simpler, we refer to an alternative and equivalent parametrization, which contains ξ$\mathit{\xi }$ as parameter of interest. The marginal reference prior of ξ$\mathit{\xi }$ is derived in its general form and particular cases are also given in detail, showing the Bayesian role of rotatability.     

A Bayesian Approach for Multiple Response Surface Optimization in the Presence of Noise Variables

An approach for the multiple response robust parameter design problem based on a methodology by Peterson (2000) is presented. The approach is Bayesian, and consists of maximizing the posterior predictive probability that the process satisfies a set of constraints on the responses. In order to find a solution robust to variation in the noise variables, the predictive density is integrated not only with respect to the response variables but also with respect to the assumed distribution of the noise variables. The maximization problem involves repeated Monte Carlo integrations, and two different methods to solve it are evaluated. A Matlab code was written that rapidly finds an optimal (robust) solution in case it exists. Two examples taken from the literature are used to illustrate the proposed method.     

Evaluation of Response Surface Designs in Presence of Errors in Factor Levels

Experimenters are often confronted with the problem that errors in setting factor levels cannot be measured. In the robust design scenario, the goal is to determine the design that minimizes the variability transmitted to the response from the variables’ errors. The prediction variance performance of response surface designs with errors is investigated using design efficiency and the maximum and minimum scaled prediction variance. The evaluation and comparison of response surface designs with and without errors in variables are developed for second order designs on spherical regions. The prediction variance and design efficiency results and recommendations for their use are provided.     

On Some Two- and Three-dimensional D-minimax Designs for Estimating Slopes of a Third-order Response Surface

Design of experiments is considered for the situation where estimation of the slopes of a response surface is the main interest. Under the D-minimax criterion, the objective is to minimize the generalized variance of the estimated axial slopes at a point maximized over all points in the region of interest in the factor space. For the third-order model over spherical regions, the D-minimax designs are derived in two and three dimensions. The efficiencies of some two- and three-dimensional designs available in the literature are also investigated.     

Extended scaled prediction variance optimality for modified central composite design

Robust parameter designs (RPDs) enable the experimenter to discover how to modify the design of the product to minimize the effect due to variation from noise sources. The aim of this article is to show how this amount of work can be reduced under modified central composite design (MCCD). We propose a measure of extended scaled prediction variance (ESPV) for evaluation of RPDs on MCCD. Using these measures, we show that we can check the error or bias associated with estimating the model parameters and suggest the values of α recommended for MCCS under minimum ESPV.     

Efficient Mixture Design Fitting Quadratic Surface with Quantile Responses Using First-degree Polynomial

This study considers efficient mixture designs for the approximation of the response surface of a quantile regression model, which is a second degree polynomial, by a first degree polynomial in the proportions of q components. Instead of least squares estimation in the traditional regression analysis, the objective function in quantile regression models is a weighted sum of absolute deviations and the least absolute deviations (LAD) estimation technique should be used (Bassett and Koenker, 1982 Bassett, G., Koenker, R. (1982). An empirical quantile function for linear models with i.i.d. errors. Journal of the American Statistical Association 77:407–415.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]; Koenker and Bassett, 1978 Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica 46(1):33–50.[Crossref], [Web of Science ®] , [Google Scholar]). Therefore, the standard optimal mixture designs like the D-optimal or A-optimal mixture designs for the least squared estimation are not appropriate. This study explores mixture designs that minimize the bias between the approximated 1st-degree polynomial and a 2nd-degree polynomial response surfaces by the LAD estimation. In contrast to the standard optimal mixture designs for the least squared estimation, the efficient designs might contain elementary centroid design points of degrees higher than two. An example of a portfolio with five assets is given to illustrate the proposed efficient mixture designs in determining the marginal contribution of risks by individual assets in the portfolio.     

Design and analysis of mixture experiments with process variable

A mixture experiment is an experiment in which the response is assumed to depend on the relative proportions of the ingredients present in the mixture and not on the total amount of the mixture. In such experiment process, variables do not form any portion of the mixture but the levels changed could affect the blending properties of the ingredients. Sometimes, the mixture experiments are costly and the experiments are to be conducted in less number of runs. Here, a general method for construction of efficient mixture experiments in a minimum number of runs by the method for projection of efficient response surface design onto the constrained region is obtained. The efficient designs with a less number of runs have been constructed for 3rd, 4th, and 5th component of mixture experiments with one process variable.     

Response Surface Designs Where Levels of Some Factors are Difficult to Change

This article considers response surface designs in which the number of levels of some of the factors are constrained. Two general types of designs are examined: CUBE designs and STAR designs. The specific factor levels are chosen to give variance contours with a high level of sphericity, thus providing designs that are close to rotatable.     

An Adaptive Exploration-Exploitation Algorithm for Constructing Metamodels in Random Simulation Using a Novel Sequential Experimental Design

In this article, we propose a novel algorithm for sequential design of metamodels in random simulation, which combines the exploration capability of most one-shot space-filling designs with the exploitation feature of common sequential designs. The algorithm continuously maintains a balance between the exploration and the exploitation search throughout the search process in a sequential and adaptive manner. The numerical results indicate that the proposed approach is superior to one of the existing well-known sequential designs in terms of both the computational efficiency and speed in generating efficient experimental designs.     

Note on reduction of dimensionality for second-order response surface design model

To reduce the dimensionality of the second-order response surface design model, variance component indices under imposing and non imposing restrictions on the moment matrix toward the orthogonality are derived and presented and the same is illustrated with suitable examples in this article.     

A new model and class of designs for mixture experiments with process variables

Experiments that involve the blending of several components are known as mixture experiments. In some mixture experiments, the response depends not only on the proportion of the mixture components, but also on the processing conditions, A new combined model is proposed which is based on Taylor series approximation and is intended to be a compromise between standard mixture models and standard response surface models. Cost and/or time constraints often limit the size of industrial experiments. With this in mind, we present a new class of designs that will accommodate the fitting of the new combined model.     

Sequential Designs for Binary Data with the Purpose to Maximize the Probability of Response

Two kinds of sequential designs are proposed for finding the point that maximizes the probability of response assuming a binary response variable and a quadratic logistic regression model. One is a parametric optimal design approach, and the other one is a nonparametric stochastic approximation approach. The suggested sequential designs are evaluated and compared in a simulation study. In summary, the parametric approach performed very well whereas its competitor failed in some cases.     

A Parameter-Tuned Genetic Algorithm for Economic-Statistical Design of Variable Sampling Interval X-Bar Control Charts for Non-Normal Correlated Samples

Among innovations and improvements that occurred in the past two decades on the techniques and tools used for statistical process control (SPC), adaptive control charts have shown to substantially improve the statistical and/or economical performances. Variable sampling intervals (VSI) control charts are one of the most applied types of the adaptive control charts and have shown to be faster than traditional Shewhart control charts in identifying small changes of concerned quality characteristics. While in the designing procedure of the VSI control charts the data or measurements are assumed independent normal observations, in real situations the validity of these assumptions is under question in many processes. This article develops an economic-statistical design of a VSI X-bar control chart under non-normality and correlation. Since the proposed design consists of a complex nonlinear cost model that cannot be solved using a classical optimization method, a genetic algorithm (GA) is employed to solve it. Moreover, to improve the performances, response surface methodology (RSM) is employed to calibrate GA parameters. The solution procedure, efficiency, and sensitivity analysis of the proposed design are demonstrated through a numerical illustration at the end.