Response surface optimization for a nonlinearly constrained irregular experimental design space

Finding optimum conditions for process factors in an engineering optimization problem with response surface functions requires structured data collection using experimental design. When the experimental design space is constrained owing to external factors, its design space may form an asymmetrical and irregular shape and thus standard experimental design methods become ineffective. Computer-generated optimal designs, such as D-optimal designs, provide alternatives. While several iterative exchange algorithms for D-optimal designs are available for a linearly constrained irregular design space, it has not been clearly understood how D-optimal design points need to be generated when the design space is nonlinearly constrained and how response surface models are optimized. This article proposes an algorithm for generating the D-optimal design points that satisfy both feasibility and optimality conditions by using piecewise linear functions on the design space. The D-optimality-based response surface design models are proposed and optimization procedures are then analysed.

[1]  Wei Chen,et al.  An Efficient Algorithm for Constructing Optimal Design of Computer Experiments , 2005, DAC 2003.

[2]  A. Wald.  On the Efficient Design of Statistical Investigations , 1943 .

[3]  V. Fedorov,et al.  Invited Discussion Paper Constrained Optimization of Experimental Design , 1995 .

[4]  O. Dykstra.  The Augmentation of Experimental Data to Maximize [X′X] , 1971 .

[5]  Alan J. Miller,et al.  A Fedorov Exchange Algorithm for D-optimal Design , 1994 .

[6]  Valerii V. Fedorov,et al.  Design of experiments with unknown parameters in variance , 2002 .

[7]  R. D. Cook,et al.  A Comparison of Algorithms for Constructing Exact D-Optimal Designs , 1980 .

[8]  Fred Spiring,et al.  Introduction to Statistical Quality Control , 2007, Technometrics.

[9]  James M. Lucas,et al.  Which Response Surface Design is Best: A Performance Comparison of Several Types of Quadratic Response Surface Designs in Symmetric Regions , 1976 .

[10]  W. Näther.  Optimum experimental designs , 1994 .

[11]  W. DuMouchel,et al.  A simple Bayesian modification of D-optimal designs to reduce dependence on an assumed model , 1994 .

[12]  Toby J. Mitchell,et al.  An Algorithm for the Construction of “D-Optimal” Experimental Designs , 2000, Technometrics.

[13]  Matteo Bruggi,et al.  Optimization of sensor placement to detect damage in flexible plates , 2013 .

[14]  N. N. Chan.  A-Optimality for Regression Designs. , 1982 .

[15]  Luc Pronzato,et al.  Improvements on removing nonoptimal support points in D-optimum design algorithms , 2007, 0706.4394.

[16]  Alan J. Miller,et al.  A review of some exchange algorithms for constructing discrete D-optimal designs , 1992 .

[17]  G. Geoffrey Vining,et al.  Combining Taguchi and Response Surface Philosophies: A Dual Response Approach , 1990 .

[18]  André I. Khuri,et al.  Response surface methodology , 2010 .

[19]  Margaret J. Robertson,et al.  Design and Analysis of Experiments , 2006, Handbook of statistics.

[20]  Byung Rae Cho,et al.  A Nonlinear Integer Programming Approach to Solving the Robust Parameter Design Optimization Problem , 2016, Qual. Reliab. Eng. Int..

[21]  J. Kiefer,et al.  Optimum Designs in Regression Problems , 1959 .

[22]  Douglas C. Montgomery,et al.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[23]  Wanzhu Tu,et al.  Dual response surface optimization , 1995 .

[24]  G. Gary Wang,et al.  ADAPTIVE RESPONSE SURFACE METHOD - A GLOBAL OPTIMIZATION SCHEME FOR APPROXIMATION-BASED DESIGN PROBLEMS , 2001 .

[25]  W. J. Studden,et al.  Theory Of Optimal Experiments , 1972 .

[26]  Kalyanmoy Deb,et al.  Towards automating the discovery of certain innovative design principles through a clustering-based optimization technique , 2011 .

[27]  R. C. St. John,et al.  D-Optimality for Regression Designs: A Review , 1975 .

[28]  K. Smith.  ON THE STANDARD DEVIATIONS OF ADJUSTED AND INTERPOLATED VALUES OF AN OBSERVED POLYNOMIAL FUNCTION AND ITS CONSTANTS AND THE GUIDANCE THEY GIVE TOWARDS A PROPER CHOICE OF THE DISTRIBUTION OF OBSERVATIONS , 1918 .

[29]  B. Cho,et al.  Response surface-based robust parameter design optimization with both qualitative and quantitative variables , 2017 .

[30]  C McCollin.  Applied stochastic models in business and industry , 2011 .

[31]  Jennifer Seberry,et al.  D-optimal designs , 2011 .

[32]  Larsgunnar Nilsson,et al.  Robust optimization of front members in a full frontal car impact , 2013 .

[33]  Luc Pronzato,et al.  Removing non-optimal support points in D-optimum design algorithms , 2003 .

[34]  Akin Ozdemir,et al.  Development of the D-Optimality-Based Coordinate-Exchange Algorithm for an Irregular Design Space and the Mixed-Integer Nonlinear Robust Parameter Design Optimization , 2017 .

[35]  Edward J. Wegman,et al.  D-optimal design methods for robust estimation of multivariate location and scatter , 1998 .

[36]  Srinivas Kodiyalam,et al.  TUNING PID CONTROLLERS USING ERROR-INTEGRAL CRITERIA AND RESPONSE SURFACES BASED OPTIMIZATION , 2000 .