J Intell Manuf DOI 10.1007/s10845-014-0967-7 Multi-objective genetic programming approach for robust modeling of complex manufacturing processes having probabilistic uncertainty in experimental data A. Jamali · E. Khaleghi · I. Gholaminezhad · N. Nariman-Zadeh · B. Gholaminia · A. Jamal-Omidi Received: 17 May 2014 / Accepted: 11 September 2014 © Springer Science+Business Media New York 2014 Abstract In this paper, a multi-objective uniform-diversity genetic programming (MUGP) algorithm deployed for robust Pareto modeling and prediction of complex nonlinear processes using some input-output data table. The uncertain- ties included in measured data are considered to obtain more robust models. The considered benchmarks are an explosive cutting and forming processes, in which the nonlinear behav- ior between the input and output of processes are detected using MUGP. For both case studies, a multi-objective mod- eling and prediction procedure firstly performed using deter- ministic data. Secondly, the same identification procedure carried out using probabilistic uncertainty in the experi- mental input-output data. The objective functions consid- ered are namely, training error, prediction error and num- ber of tree nodes (complexity of models) in the deterministic approach. Accordingly, the mean and standard deviation of training error and prediction error are considered in robust Pareto modeling and prediction of such processes. In this way, Pareto front of such modeling and prediction is first obtained for both explosive cutting and forming processes with deterministic data. Such Pareto front is then obtained using experimental input-output-data having probabilistic uncertainty in input parameters through a Monte Carlo sim- ulation (MCS) approach. In addition, it has been shown that for both cases, the trade-off models obtained from determin- istic data have significant biases when tested on data with A. Jamali (B ) · E. Khaleghi · I. Gholaminezhad · N. Nariman-Zadeh · B. Gholaminia · A. Jamal-Omidi Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, P.O. Box: 3756, Rasht, Iran e-mail: ali.jamali@guilan.ac.ir N. Nariman-Zadeh Intelligent-based Experimental Mechanics Center of Excellence, School of Mechanical Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran probabilistic uncertainty. Finally, the obtained results of such multi-objective robust model identification show promising results in terms of compensating uncertainty in the experi- mental input-output-data. Keywords Robust modeling · Genetic programming · Multi-objective · MCS Introduction System identification and modeling of complex processes using input-output data have always attracted many research efforts. In fact, system identification techniques are applied in many fields in order to model and predict the behaviors of unknown and/or very complex systems based on given input-output data (Jamali et al. 2013a, b). It is clear that, in most experimental input-output-data there are a variety of typical sources of uncertainty because of inaccuracy of sen- sory measurements, human fault, and etc. In order to obtain more robust models, it is required to consider all uncertainties included in measured data. It has been shown that designing without considering uncertainties generally leads to poten- tially insecure solution (Jamali et al. 2013b). Therefore, it is very desirable to use methods, which have considered uncertainties in order to reduce performance variation of the obtained mathematical models in a noisy environment. There are several methods which have been used by researchers for uncertainty consideration for different purposes such as Kalman filtering (Oakes et al. 2009), Latin hypercube sam- pling (Kweon et al. 2007), generalized likelihood uncertainty estimation (Zhang et al. 2014) and Monte Carlo Simulation (Bi et al. 2013; Paris et al. 2012; Bieda 2011; Zanjani et al. 2011; Biwer et al. 2005). 123