Contents lists available at ScienceDirect Precision Engineering journal homepage: www.elsevier.com/locate/precision Prediction capability of Pareto optimal solutions: A multi-criteria optimization strategy based on model capability ratios Lucas Guedes de Oliveira * , Anderson Paulo de Paiva, Paulo Henrique da Silva Campos, Emerson José de Paiva, Pedro Paulo Balestrassi Institute of Industrial Engineering and Management, Federal University of Itajuba, Itajuba, Minas Gerais, Brazil ARTICLE INFO Keywords: Prediction variance Model capability ratios Design of experiments Response surface methodology Normal boundary intersection Multiobjective optimization ABSTRACT Response Surface Methodology is an efective framework for performing modelling and optimization of in- dustrial processes. The Central Composite Design is the most popular experimental design for response surface analyses given its good statistical properties, such as decreasing prediction variance in the design center, where it is expected to fnd the stationary points of the regression models. However, the common practice of reducing center points in response surface studies may damage this property. Moreover, stationary and optimum points are rarely the same in manufacturing processes, for several reasons, such as saddle-shaped models, convexity incompatible with optimization direction, conficting responses, and distinct convexities. This means that even when the number of center points is appropriate, the optimal solutions will lie in regions with larger prediction variance. Considering that, in this paper, we advocate that the prediction variance should also be considered into multiobjective optimization problems. To do this, we propose a multi-criteria optimization strategy based on capability ratios, wherein (1) the prediction variance is taken as the natural variability of the model and (2) the diferences of expected values to nadir solutions are taken as the allowed variability. Normal Boundary Intersection method is formulated for performing the optimization of capability ratios and obtaining the Pareto frontiers. To illustrate the feasibility of the proposed approach, we present a case study of the turning without cutting fuids of AISI H13 steel with wiper CC650 tool. The results have supported that the proposed approach was able to fnd a set of optimal solutions with satisfactory prediction capabilities for both responses of interest (tool life T and surface roughness Ra), for a case with reduced number of center points, a saddle-shaped function for T and a convex function for Ra, with conficting objectives. Although it was a response more difcult to control, the optimization benefted more Ra, which was a desired result. Finally, we also provide the sample sizes to detect diferences between Pareto optimal solutions, allowing the decision maker to fnd distinguishable solutions at given levels of risk. 1. Introduction Response Surface Methodology (RSM) is a framework widely used in the modelling and optimization of industrial processes [1–3]. In es- sence, RSM employs statistical and mathematical techniques and methods to estimate response variables as a function of explanatory factors by using planned experiments [4]. Often, second-order models are employed to estimate a given region of interest of a response with some explanatory factors. The core idea is that these models are used as objective functions in optimization problems, providing the real values of the factors that efectively improve the processes. The Central Composite Design (CCD) is recognized in the literature as the most popular second-order design for RSM experimental studies because of its good statistical properties [5]. The CCD combines three types of points to allow the estimation of the main efects and their interactions (factorial points), the quadratic efects (axial points) and the error component (center points). Additionally, the CCD ofers the minimum prediction variance in the design center [6], since the sta- tionary is expected to be the optimal point and lie in the central region of the design. Nevertheless, these assumptions are not always true in real cases. First, when using the CCD, the number of center points re- commended [7] is often reduced [8–10] which leads to profound modifcations in the prediction variance functions, thus impacting on https://doi.org/10.1016/j.precisioneng.2019.06.008 Received 27 March 2019; Received in revised form 25 May 2019; Accepted 19 June 2019 * Corresponding author. E-mail addresses: lucasguedesdeoliveira@gmail.com (L.G. de Oliveira), andersonppaiva@unifei.edu.br (A.P. de Paiva), paulohcamposs@unifei.edu.br (P.H. da Silva Campos), emersonpaiva@unifei.edu.br (E.J. de Paiva), ppbalestrassi@gmail.com (P.P. Balestrassi). Precision Engineering 59 (2019) 185–210 Available online 27 July 2019 0141-6359/ © 2019 Elsevier Inc. All rights reserved. T