Vol.:(0123456789) 1 3 Engineering with Computers https://doi.org/10.1007/s00366-018-0660-0 ORIGINAL ARTICLE Fuzzy multivariate mean square error in equispaced pareto frontiers considering manufacturing process optimization problems Juliana Helena Daroz Gaudêncio 1  · Fabrício Alves de Almeida 1  · Rachel Campos Sabioni 2  · João Batista Turrioni 1  · Anderson Paulo de Paiva 1  · Paulo Henrique da Silva Campos 1 Received: 20 June 2018 / Accepted: 3 November 2018 © Springer-Verlag London Ltd., part of Springer Nature 2018 Abstract This paper proposes a combined approach using the normal boundary intersection (NBI) and multivariate mean square error (MMSE) that is an alternative approach to outperform the traditional NBI driving to an equispaced Pareto Frontier in a low- dimension space with a considerable reduction in the number of iterations. The method participating in the evolutionary stage of creating a uniformly spread Pareto Frontier for a nonlinear multi-objective problem is the NBI using normalized objective functions allied to MMSE. In sequence, the fuzzy MMSE approach is utilized to determine the optimal point of the multi-objective optimization. For sake of comparison, the performance of arc homotopy length, global criterion method, and weighted sums were explored. To illustrate this proposal, a multivariate case of AISI H13 hardened steel-turning process is used. Experimental results indicate that the solution found by NBI-MMSE approach is a more appropriate Pareto frontier that surpassed all the competitors and also provides the best-compromised solution to set the machine input parameters. Further, this algorithm was also tested in benchmark functions to confrm the NBI-MMSE efciency. Keywords Principal component analysis · Multivariate mean square error · Normal boundary intersection · Fuzzy decision maker · Hardened steel turing 1 Introduction Engineering problems have been precisely solved with the advancement of mathematical modeling coupled with com- puter methods, once industrial processes have several input variables that infuence the output variables. This way, it is possible to fnd in the literature studies, the mathematical and computational strategies applied to engineering, such as genetic algorithm [1, 2], particle swarm [3], ant colony [4], grey wolf optimizer [5], multilevel cross entropy optimizer [6], and sunfower optimization [7]. These applications are widely used in manufacturing processes, such as the machin- ing process, where it is possible to fnd jobs that aim at the efciency of the process, using techniques such as mean square error (MSE) [8], Taguchi method [9], and response surface methodology (RSM) [10]. According to Almeida et al. [ 8], RSM is a kind of design of experiment (DOE), a statistical technique capa- ble of modeling, optimizing, and reducing experimental costs in manufacturing processes. Usually, these processes present several responses of interest that may present a cer- tain level of correlation. In applications and mathematical models of correlated characteristics, the variance–covari- ance structure of these characteristics should be consid- ered [11]. Thus, the modeling of correlated characteristics for application in heuristic strategies should be seen as a multivariate strategy to optimize all variables simultane- ously. In a multi-objective optimization, eforts must be made to fnd the set of optimal solutions, by considering all objectives to be important. Within this approach, this work presents an algorithm focused in the engineering process optimization that combines the normal boundary intersection (NBI) method with multivariate mean square Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00366-018-0660-0) contains supplementary material, which is available to authorized users. * Fabrício Alves de Almeida fabricio.alvesdealmeida@gmail.com 1 Institute of Industrial Engineering and Management, Federal University of Itajubá, Itajubá, Brazil 2 Department of Mechanical Engineering, Sorbonne University, University of Technology of Compiègne, Compiègne, France