A-Optimal Designs And R-Optimal Designs For Second-order Response Surface Models With Qualitative Factors

Response surface methodology(RSM)is a kind of experimental design method considering statistics,which can be used to analyze the problems of response affected by variables.To construct a second-order response surface model to simulate the actual problems,central composite design(CCD)is widely used in biology,pharmacology,environmental science,food technology and many other fields.At present,most RSM researches only consider the case with quantitative factors,and the discussion on the model with qualitative factors is not so much.When there are qualitative factors in the designs,The qualitative factors are often treated as dummy variables or quantified factors when experiments have qualitative factors?To some extent,this method is effective,but it is not suitable in real situation.Based on the D-optimal criterion,Lee and Huang(2011)discussed the problem of experimental design with one qualitative factor.Considering that the qualitative factor affects the linear effect,interaction effect and quadratic effect of the quantitative factors respectively.They discussed that the D-optimal design at each level of qualitative factor can be composed of three kinds of set points in central composite design.The problem of seeking optimal design can be transformed into finding the weight of three kinds of design points in CCD.In addition to the Doptimal criterion,the A-optimal criterion which minimizes the sum of variance of unknown parameter components and the R-optimal criterion which minimizes the rectangular confidence space also have great statistical properties.A-optimal designs and R-optimal designs are widely used in reality.This paper extends Lee and Huang's research to A-optimal criterion and R-optimal criterion.Firstly,we introduce the research background and development history,and list several common experimental design methods.We also introduce the current research results related and the basic concept of RSM,optimal criterion,information matrix.Then,we introduce the concept of secondorder RSM response surface model with qualitative factors and central composite design.Three kinds of models under different effects combination of qualitative factors and quantitative factors are listed.We give the weight calculation formulas of three kinds of design points under Aoptimal criterion design and R-optimal criterion in central composite design,and compare the efficiency of each model.Finally,by means of numerical simulation,we compare the statistical properties of A-optimal designs,R-optimal designs and the common equal weight designs of the second-order response surface model with qualitative factors.