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Response surface methodology for advanced manufacturing technology optimization: theoretical fundamentals, practical guidelines, and survey literature review

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Abstract

Process optimization normally involves the combination of mathematical and statistical techniques which can be approached by distinct ways. Despite the fact that different methods can be found in the literature, the response surface methodology raised as one of the most effective ways for performing process optimization, by combining design and analysis of experiments, modeling techniques, and optimization methods. However, practical guidelines for response surface methodology and critical analysis of its applications are quite scarce. Thus, this paper aims to present the theoretical principles and practical guidelines for carrying response surface methodology as well as to provide empirical evidence on its critical aspects for manufacturing optimization. In order to accomplish with this objective, 49 papers published in the International Journal of Advanced Manufacturing Technology (IJAMT) from 2014 to 2017 were investigated and reproduced, allowing the analysis of 123 response surfaces. Surprisingly, more than 75.29% of the models have presented a saddle shape. The practical meaning of this finding is that the stationary point is not a suitable solution for the optimization of those surfaces. Besides, multiple response surfaces are more commonly found in the literature than individual ones. From this amount of papers, 71.88% of the works investigated have presented significant correlation with their peers and 87.61% have convexity incompatible with the optimization direction. Most of the optimization solutions have found outside of experimental region which reveals a preponderant neglect of the nonlinear constraints involving the definition of the experimental region. It was also verified that the proportion of functions in saddle format corresponds to 96.86% of the models estimated in flat regions. Moreover, it was found that the number of center points is commonly changed and all the manufacturing processes investigated are driven by at most five control parameters. Finally, considering the theoretical principles, the practical guidelines, and the obtained results, a follow-along example involving the optimization of AISI H13-hardened steel turning with PCBN wiper, previously published by the authors in IJAMT, was revisited by using response surface methodology. The results corroborated the proposed framework suitability.

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Abbreviations

f :

Feed rate

V c :

Cutting speed

d :

Depth of cut

y :

Response variable, function value, or estimated model

β :

Model coefficient

x :

Independent variable

ε :

Experimental error or residual

N :

Total number of experiments

k :

Number of factors or input variables

n C :

Number of center points

α :

Axial distance

n F :

Number of factorial points

SS :

Squared sum

MS :

Squared mean

F 0 :

Fisher-Snedecor statistics

∅:

Number of degrees of freedon of F0

Γ:

Gama function

δi:

Spherical moments

m :

Index of the space of the model

M :

Matrix of moments

x s :

Stationary point coordinate vector

B :

Matrix of coefficients of second-order terms

b :

Vector of coefficients of first-order terms

ρ :

Design radius or the Pearson correlation

y s :

Estimative of the response variable in the stationary point

λ :

Eigenvalues of \( \hat{\boldsymbol{B}} \)

w :

Canonical variables

I :

Identity matrix

μ :

Mean vector

c :

Chi-squared variable

\( \boldsymbol{\sum}^{\sim } \) :

Variance-covariance matrix

∇:

Gradient vector

P :

Matrix of the eigenvectors of \( {\tilde{\boldsymbol{\Sigma}}}^{-\mathbf{1}} \)

X :

Vector of experimental factors

f (X):

Objective function

g j(X):

Generic inequality constraint

l j(X):

Generic equality constraint

α 1, α 2 :

Relative weights of linearly combined functions

n :

Total number of data

\( {\chi}_0^2 \) :

Qui-squared statistic

ABNT:

Brazilian Association of Technical Standards

ANOVA:

Analysis of variance

BBD:

Box-Behnken design

CCD:

Central composite design

DOE:

Design of experiments

FCCD:

Face-centered central composite design

FFD:

Full factorial design

EWR:

Electrode wear rate

GRG:

Generalized reduced gradient

IQR:

Interquartile range

NBI:

Normal boundary intersection

NOMATI:

Manufacturing Optimization and Innovation Technology Center

Ortho.:

Orthogonal

MRR:

Material removal rate

MRV:

Multiple response variables

RSM:

Response surface methodology

OLS:

Ordinary least squares

PC:

Principal component score

PCA:

Principal component analysis

SAM:

Steepest ascent method

SPV:

Scaled prediction variance

UP:

Uniform precision

WPCA:

Weighted principal component analysis

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Acknowledgements

The authors also would like to acknowledge Prof. Ph.D. João Paulo Davim Tavares da Silva from the University of Aveiro, Portugal, who allowed and sponsored the consecution of experimental data set, the Foundation of Support Research of the State of Minas Gerais (FAPEMIG), and the anonymous reviewer for the careful reading of our paper and for the positive comments and suggestions.

Funding

This paper is supported by the Coordination for the Improvement of Higher Education Personnel (CAPES), through doctoral scholarships and project CAPES 9801-12.0 and the National Council for Scientific and Technological Development (CNPq) through projects CNPq 303586/2015-0 and CNPq 409318/2017-5.

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  1. Institute of Industrial Engineering and Management, Federal University of Itajuba, Itajuba, Minas Gerais, Brazil

    Lucas Guedes de Oliveira, Anderson Paulo de Paiva, Pedro Paulo Balestrassi, João Roberto Ferreira, Sebastião Carlos da Costa & Paulo Henrique da Silva Campos

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  1. Lucas Guedes de Oliveira
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  2. Anderson Paulo de Paiva
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Correspondence to Pedro Paulo Balestrassi.

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de Oliveira, L.G., de Paiva, A.P., Balestrassi, P.P. et al. Response surface methodology for advanced manufacturing technology optimization: theoretical fundamentals, practical guidelines, and survey literature review. Int J Adv Manuf Technol 104, 1785–1837 (2019). https://doi.org/10.1007/s00170-019-03809-9

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