Document Type : Article

**Authors**

Department of Industrial Engineering, K.N. Toosi University, Tehran, 1999143344, Iran

**Abstract**

By rapid advancements in technologies, studying and simulating a complex system with uncertain parameters is too demanding. Based on the literature, there are three approaches to identify and simulate different systems: engineering, statistical, and engineering-statistical approaches. The purpose of this study is to apply the engineering-statistical approach to calibration and adjustment of a Laser Assisted Micro-Machining (LAMM) process with two correlated outputs which are basically known as cutting and thrust forces. This paper contributes to the existing literature by extending the most relevant approach of the calibration of single-output complicated processes to multi-output settings where discrepancy function is modeled by a multivariate Gaussian process and multivariate analysis of variance is used to identify variables whose adjustment benefits the most. For the best case reported in previous studies, Mean Squared Prediction Error (MSPE), as the comparison index, was reported around 1.48 for thrust force whereas the proposed approach resulted in a better value of 1.9425×10-4. Moreover, for cutting force output, the index was obtained as 0.21 by the Kennedy and O’Hagan, 1.41 by Roshan and Yan, and 1.6×10-8 by the presented model. These values demonstrate reasonable and comparable results for the MSPE, in comparison with the models considering the outputs individually.

**Keywords**

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42. Diaz-Garcia, J.A. "On generalized multivariate analysis of variance", Brazilian Journal of Probability and Statistics, 25(1), pp. 1-13 (2011).

2. Kennedy, M.C. and O'Hagan, A. "Bayesian calibration of computer models", Journal of Royal Statistical Society - Series B, 63, pp. 425-464 (2001).

3. Roshan, V.J. and Melkote, S.N., Statistical Adjustments to Engineering Models, Georgia Institute of Technology, pp. 1-24 (2008).

4. Yan, H. "Statistical adjustment, calibration, and uncertainty quantification of complex computer models", Ph.D. Thesis, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology (2014).

5. Roshan, V.J. and Yan, H. "Engineering-driven statistical adjustment and calibration", Technometrics, 57(2), pp. 257-267 (2015).

6. Sheikhi, H. and Saghaie, A. "Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage", Chinese Journal of Aeronautics, 30(1), pp. 175-185 (2017).

7. Rahimi, M., Shafieezadeh, A., Wood, D., et al. "Bayesian calibration of multi-response systems via multivariate Kriging: Methodology and geological and geotechnical case studies", Engineering Geology, 260, 105248, ISSN 0013-7952 (2019). https://doi.org/10.1016/j.enggeo.2019.105248.

8. Daniel, W., David, M., James, W.J., and Francois Working with Dynamic Crop Models Methods, Tools and Examples for Agriculture and Environment, Elsevier, 3rd Edition (2019).

9. Yan, G., Sun, H., and Waisman, H. "A guided Bayesian inference approach for detection of multiple flaws in structures using the extended finite element method", Computers and Structures, 152, pp. 27-44 (2015).

10. Singh, R.K., Josef, V.R., and Melkote, S.N. "A statistical approach to the optimization of a laserassisted micromachining process", International Journal of Advanced Manufacturing Technology, 53, pp. 221-230 (2011).

11. Allaire, G., Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation, Oxford, New York (2007).

12. Nocedal, J. and Wright, S.J., Numerical optimization, Springer, New York (2006).

13. Szabo, B.A. "The use of a priori estimates in engineering computations", Computer Methods in Applied Mechanics and Engineering, 82(1-3), pp. 139-154 (1990).

14. Mathelin, L. and Hussaini, M.Y., A Stochastic Collocation Algorithm for Uncertainty Analysis, NASA Center for AeroSpace Information (CASI), pp. 1-16 (2003).

15. Vepsa, A., Haapaniemi, H., Luukkanen, P., et al. "Application of finite element model updating to a feed water pipeline of a nuclear power plant", Nuclear Engineering and Design, 235(17-19), pp. 1849-1865 (2005).

16. Palumbo, G., Piccininni, A., Piglionico, V., et al. "Modelling residual stresses in sand-cast superduplex stainless steel", Journal of Materials Processing Technology, 217, pp. 253-261 (2015).

17. Stickler, B. and Schachinger, E., Basic Concepts in Computational Physics, Springer, New York (2014).

18. Sena, P.K. and Silvapulle, M.J. "An appraisal of some aspects of statistical inference under inequality constraints", Journal of Statistical Planning and Inference, 107, pp. 3-43 (2002).

19. Kumar, R., Tewari, P.C., and Khanduja, D. "Parameters optimization of fabric finishing system of a textile industry using teaching-learning-based optimization algorithm", International Journal of Industrial Engineering Computations, 6(2), pp. 221-234 (2018).

20. Roussas, G., An Introduction to Measure-Theoretic Probability, 2nd Ed., Elsevier, USA (2014).

21. Zio, E., The Monte Carlo Simulation Method for System Reliability and Risk Analysis, Springer, London (2013).

22. Dong, L., Xiaojing, L., and Yanhua, Y. "Investigation of uncertainty quantification method for be models using MCMC approach and application to assessment with feba data", Annals of Nuclear Energy, 107, pp. 62-70 (2017).

23. Parnianifard, A., Azfanizam, A.S., Ariffin, M.K.A., et al. "An overview on robust design hybrid metamodeling: Advanced methodology in process optimization under uncertainty", International Journal of Industrial Engineering Computations, 9(1), pp. 1-32 (2018).

24. Park, I. and Grandhi, R.V. "A Bayesian statistical method for quantifying model form uncertainty and two model combination methods", Reliability Engineering and System Safety, 129, pp. 46-56 (2014).

25. Caiado, C.C.S. and Goldstein, M. "Bayesian uncertainty analysis for complex physical systems modelled by computer simulators with applications to tipping points", Communications in Nonlinear Science and Numerical Simulation, 26(1-3), pp. 123-136 (2015).

26. Wong, S.W.K., Lum, C., Wu, L., et al. "Quantifying uncertainty in lumber grading and strength prediction: a Bayesian approach", Technometrics, 58(2), pp. 236- 243 (2016).

27. Sankararaman, S. and Mahadevan, S. "Integration of model verification, validation, and calibration for uncertainty quantification in engineering systems", Reliability Engineering and System Safety, 138, pp. 194-209 (2015).

28. Duru, O., Bulut, E., and Yoshida, Sh. "A fuzzy extended delphi method for adjustment of statistical time series prediction: an empirical study on dry bulk freight market case", Expert Systems with Applications, 39(1), pp. 840-848 (2012).

29. Azzimonti, D., Bect, J., Chevalier, C., et al., Quantifying Uncertainties on Excursion Sets under a Gaussian Random Field Prior, Cornell University (arXiv), (2015).

30. Reid, N. "Statistical sufficiency", International Encyclopedia of the Social & Behavioral Sciences, 2, pp. 418-422 (2015).

31. Junaid, M.M. andWani, M.F. "Modelling and analysis of tool wear and surface roughness in hard turning of AISI D2 steel using response surface methodology", International Journal of Industrial Engineering Computations, 9(1), pp. 63-74 (2018).

32. Brandt, S., Data Analysis: Statistical and Computational Methods for Scientists and Engineers, Springer, 4, New York (2014).

33. Korunovic, N., Madic, M., Trajanovic, M., and Radovanovic, M. "A procedure for multi-objective optimization of tire design parameters", International Journal of Industrial Engineering Computations, 6(2), pp. 199-210 (2015).

34. Lin, H.D. and Lin, W.T. "Automated process adjustments of chip cutting operations using neural network and statistical approaches", Expert Systems with Applications, 36, pp. 4338-4345 (2009).

35. Lunardon, N. and Ronchetti, E. "Composite likelihood inference by nonparametric saddle point tests", Computational Statistics and Data Analysis, 79, pp. 80-90 (2014).

36. Pratola, M.T. and Higdon, D.M. "Bayesian additive regression tree calibration of complex high-dimensional computer models", Technometrics, 58(2), pp. 166-179 (2016).

37. Recep, M.G., Seung-Kyum, C., and Christopher, J.S. "Uncertainty quantification and validation of 3D lattice scaffolds for computer-aided biomedical applications", Journal of the Mechanical Behavior of Biomedical Materials, 71, pp. 428-440 (2017).

38. Saikumar, R.Y., Michael, G.G., Christos, A., et al. "Bayesian uncertainty quantification and propagation for validation of a microstructure sensitive model for prediction of fatigue crack initiation", Reliability Engineering & System Safety, 164, pp. 110-123 (2017).

39. Chen, R.B., Wang, W., and Wu, C.F.J. "Sequential designs based on bayesian uncertainty quantification in sparse representation surrogate modeling", Technometrics, 59(2), pp. 139-152 (2017).

40. Neal, R.M. "Regression and classification using gaussian process priors", Bayesian Statistics, 6, pp. 475- 501 (1998).

41. Mondal, A., Mallick, B., Efendiev, Y., and Datta- Gupta, A. "Bayesian uncertainty quantification for subsurface inversion using a multiscale hierarchical model", Technometrics, 56(3), pp. 381-392 (2014).

42. Diaz-Garcia, J.A. "On generalized multivariate analysis of variance", Brazilian Journal of Probability and Statistics, 25(1), pp. 1-13 (2011).

Transactions on Industrial Engineering (E)

November and December 2022Pages 3394-3403