# Algorithms for Solving Inverse Problems of Simulation Modeling

## Authors

• Ekaterina Gribanova

## Keywords:

inverse problems, simulation modeling, inverse calculations, mathematical optimization, function approximation, inventory control

## Abstract

This paper is devoted to solving inverse problems of simulation modeling, which are presented in the form of an optimization problem. The article discusses the use of direct search methods taking into account the specifics of the problem under consideration. Due to the fact that these methods require a lot of computational experiments, two algorithms based on approximation were proposed for solving the problem. The first algorithm consists in determining and evaluating the parameters of the function of dependence (which can be linear or non-linear) of the output variable on the input variables and solving the inverse problem by minimizing increments of the arguments. In the second algorithm a linear function of dependence is iteratively constructed using the data set generated by changing the input variables in given increments, and the inverse problem is solved by minimizing increments of the arguments. The classical inventory management model with a threshold strategy is considered as an example. The inverse problem was solved using direct search and approximation-based methods.

## References

R.C. Aster, B. Borchers, C.H. Thurber, Parameter Estimation and Inverse Problems, Elsevier, 2019, 404 p. https://doi.org/10.1016/B978-0-12-804651-7.00015-8.

D.L. Colton, Surveys on Solution Methods for Inverse Problems, Springer, Wien, 2000, 275 p. https://doi.org/10.1007/978-3-7091-6296-5.

A.A. Shananin, “Inverse problems in economic measurements,” Computational Mathematics and Mathematical Physics, vol. 58, issue 2, pp. 170-179, 2018. https://doi.org/10.1134/S0965542518020161.

I. Ekeland, N. Djitte, “An inverse problem in the economy theory of demand,” Annales de l'Institut Henri Poincare Non Linear Analysis, vol. 23, issue 2, pp. 269-281, 2006. https://doi.org/10.1016/j.anihpc.2005.10.001.

S. I. Kabanihin, “Inverse problems of natural science and computer modeling,” First-hand Science, vol. 49, issue 1, pp. 32-43, 2013. (in Russian)

B.E. Odincov, Inverse Calculations in the Formation of Economic Decisions, Moscow: Finance and Statistics, 2004, 256 p. (in Russian)

E. B. Gribanova, “Development of iterative algorithms for solving the inverse problem using inverse calculations,” Eastern-European Journal of Enterprise Technologies, vol. 3, issue 4, pp. 27-34, 2020. https://doi.org/10.15587/1729-4061.2020.205048.

F. Masoud, “A new efficient conjugate gradient method for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 300, pp. 207-216, 2016. https://doi.org/10.1016/j.cam.2015.12.035.

V. Cerda, J. L. Cerda, A. M. Idris, “Optimization using the gradient and simplex methods,” Talanta, vol. 148, pp. 641-648, 2016. https://doi.org/10.1016/j.talanta.2015.05.061.

S. Afandizadeh, M. Ameri, M. H. M. Moghaddam, “Introducing a modified gradient vector method for optimization of accident prediction non-linear functions,” Applied Mathematical Modelling, vol. 190, pp. 5500-5506, 2011. https://doi.org/10.1016/j.apm.2011.03.053.

M. Li, J. Bai, L. Li, X.Meng, Q. Liu, B. Chen, “A gradient-based aero-stealth optimization design method for flying wing aircraft,” Aerospace Science and Technology, vol. 92, pp. 156-169, 2019. https://doi.org/10.1016/j.ast.2019.05.067.

C. Audet, “A survey on direct search methods for blackbox optimization and their applications,” in: P.M. Pardalos, T.M. Rassias (Eds.), Mathematics without Boundaries, Springer, New York, 2014, pp. 31-56. https://doi.org/10.1007/978-1-4939-1124-0_2.

T. G. Kolda, R. M. Lewis, V. T. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM review, vol. 45, issue 3, pp. 385-482, 2003. https://doi.org/10.1137/S003614450242889.

F. Peng, K. Tang, G. Chen, X. Yao, “Population-based algorithm portfolios for numerical optimization,” IEEE Transaction on Evolutionary Computation, issue 5, pp. 782-800, 2010. https://doi.org/10.1109/TEVC.2010.2040183.

P. Vikhar, “Evolutionary algorithms: A critical review and its future prospects,” Proceedings of the International Conference on Global Trends in Signal Processing, Information Computing and Communication, Jalgaon, India, December 22-24, 2016, pp. 261-265. https://doi.org/10.1109/ICGTSPICC.2016.7955308.

J. Saldivar, C. Vairetti, C. Rodriguez, F. Daniel, F. Casati, R. Alarcon, “Analysis and improvement of business process models using spreadsheets,” Information Systems, vol. 5, pp. 1-19, 2016. https://doi.org/10.1016/j.is.2015.10.012.

A. Tiger, J. Loucks, C. Burns, “Spreadsheet-based supply chain simulation for teaching risk pooling combined with facility location,” Independent Journal of Management & Production, vol. 10, issue 6, pp. 1932-1951, 2019. https://doi.org/10.14807/ijmp.v10i6.960.

B. Mileva Boshkoska, T. Damij, F. Jelenc, N. Damija, “Abdominal surgery process modeling framework for simulation using spreadsheets,” Computer Methods and Programs in Biomedicine, vol. 121, issue 1, pp. 1-13, 2015. https://doi.org/10.1016/j.cmpb.2015.05.001.

R. Lewis, V. Torczon, M. Trosset, “Direct search methods: then and now,” Journal of Computational and Applied Mathematics, vol. 124, issue 1–2, pp. 191-207, 2000. https://doi.org/10.1016/S0377-0427(00)00423-4.

M. Gilli, D. Maringer, E. Schumann, Numerical Methods and Optimization in Finance, Academic Press, 2011, 256 p. https://doi.org/10.1016/B978-0-12-375662-6.00011-0.

A. A. Mitsel, E. B. Gribanova, “Development of a system for simulating economic objects based on an object-oriented approach,” Proceedings of Tomsk Polytechnic University, vol. 311, issue 6, pp. 11-15, 2007. (in Russian)

2021-09-30

## How to Cite

Gribanova, E. (2021). Algorithms for Solving Inverse Problems of Simulation Modeling. International Journal of Computing, 20(3), 433-439. https://doi.org/10.47839/ijc.20.3.2290

Articles