Abstract
The majority of engineering optimization problems (design, identification, design of controlled systems, optimization of large-scale systems, operational development of prototypes, and so on) are essentially multicriteria. The correct determination of the feasible solution set is a major challenge in engineering optimization problems. In order to construct the feasible solution set, a method called PSI (Parameter Space Investigation) has been created and successfully integrated into various fields of industry, science, and technology. Owing to the PSI method, it has become possible to formulate and solve a wide range of multicriteria optimization problems. In addition to giving an overview of the PSI method, this paper also describes the methods for approximation of the feasible and Pareto optimal solution sets, identification, decomposition, and aggregation of the large-scale systems.
Similar content being viewed by others
References
Statnikov, R., Statnikov, A.: The Parameter Space Investigation Method Toolkit. Artech House, Boston (2011)
Statnikov, R., Matusov, J.: Multicriteria Analysis in Engineering. Kluwer Academic Publishers, Dordrecht (2002)
Statnikov, R., Matusov, J.: Multicriteria Optimization and Engineering. Chapman & Hall, New York (1995)
Statnikov, R.: Multicriteria Design: Optimization and Identification. Kluwer Academic Publishers, Dordrecht (1999)
Sobol’, I.M., Statnikov, R.B.: Selecting Optimal Parameters Multicriteria Problems, 2nd edn. Drofa, Moscow (2006) (in Russian)
Statnikov, R.B., Matusov, J.B.: Use of P τ nets for the approximation of the Edgeworth-Pareto set in multicriteria optimization. J. Optim. Theory Appl. 91(3), 543–560 (1996)
Statnikov, A., Astashev, V., Matusov, J., Toporkov, M., P’iankov, K., Statnikov, A., Yakovlev, E., Yanushkevich, I.: MOVI 1.4 (multicriteria optimization and vector identification) software package, certificate of registration, register of copyrights. USA, Registration Number TXU 1-698-418, Date of Registration: May 22, 2010
Dempe, S.: Foundations of Bilevel Programming. Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht (2010)
Statnikov, R., Anil, K., Bordetsky, A., Statnikov, A.: Visualization approaches for the prototype improvement problem. J. Multi-Criteria Decis. Anal. 15, 45–61 (2008)
Lichtenstein, S., Slovic, P.: The Construction of Preference. Cambridge University Press, New York (2006)
Weyl, H.: Uber die gleichverteilung von zahlen mod. Eins. Math. Ann. 77(3), 313–352 (1916)
Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)
Hammersley, J.M.: Monte Carlo methods for solving multivariable problems. Ann. N.Y. Acad. Sci. 86, 844–874 (1960)
Hlawka, E., Taschner, R.: Geometric and Analytic Number Theory. Springer, Berlin (1991)
Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). In: Acta Arith, vol. 41, pp. 337–351 (1982) (in French)
Niederreiter, H.: Statistical independence properties of pseudorandom vectors produced by matrix generators. J. Comput. Appl. Math. 31, 139–151 (1990)
Dick, J., Niederreiter, H.: On the exact t-value of Niederreiter and Sobol’ sequences. J. Complex. 24(5–6), 572–581 (2008)
Anil, K.A.: Multi-Criteria Analysis in Naval Ship Design. Master’s Thesis, Naval Postgraduate School, Monterey, CA, USA (2005)
Statnikov, R., Anil, K., Bordetsky, A., Statnikov, A.: Visualization tools for multicriteria analysis of the prototype improvement problem. In: Proceedings of the First IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making (MCDM 2007), Honolulu, Hawaii, USA (2007)
Statnikov, R., Bordetsky, A., Matusov, J., Sobol’, I., Statnikov, A.: Definition of the feasible solution set in multicriteria optimization problems with continuous, discrete, and mixed design variables. Nonlinear Anal. 71(12), e109–e117 (2009)
Matusov, J., Statnikov, R.: Approximation and regularization in vector optimization problems. In: Problems and Methods of Decision Making in Managerial Systems, pp. 56–62. The Institute for Systems Science (VNIISI), Moscow (1985) (in Russian)
Sobol’, I.: Multidimensional Quadrature Formulas and Haar Functions. Nauka, Moscow (1969) (in Russian)
Sobol’, I.: On functions satisfying the Lipschitz condition in multidimensional problems of computational mathematics. Dokl. Akad. Nauk SSSR 293(6), 1314–1319 (1987) (in Russian)
Statnikov, R., Matusov, J.: Multicriteria Machine Design. Znaniye, Moscow (1989) (in Russian)
Lieberman, E.: Multi-Objective Programming in the USSR. Academic Press, New York (1991)
Ozernoy, V.: Multiple criteria decision making in the USSR: A survey. Nav. Res. Logist. 35, 543–566 (1988)
White, D.: A bibliography on the applications of mathematical programming multiple-objective methods. J. Oper. Res. Soc. 41(8), 669–691 (1990)
Vasil’ev, F.: Methods of Solving Extremum Problems. Nauka, Moscow (1981) (in Russian)
Kelley, J.L.: General Topology. Van Nostrand Reinhold, New York (1957)
Matusov, J., Statnikov, R.: Approximation and vector optimization of large systems. Dokl. Akad. Nauk SSSR 296(3), 532–536 (1987) (in Russian)
Graupe, D.: Identification of Systems. Robert E. Krieger Publishing Company, Huntington (1976)
Ljung, L.: System Identification: Theory for the User. Prentice-Hall, Englewood Cliffs (1987)
Yakimenko, O.A., Statnikov, R.B.: On multicriteria parametric identification of the cargo parafoil model with the of PSI method. In: Proceedings of the 18th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminary (AIAA 2005), Munich, Germany (2005)
Dobrokhodov, V., Statnikov, R.: Multi-criteria identification of a controllable descending system. In: Proceedings of the First IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making (MCDM 2007), Honolulu, Hawaii, USA (2007)
Statnikov, R., Bordetsky, A., Statnikov, A.: Multicriteria analysis tools in real-life problems. Comput. Math. Appl. 52(1–2), 1–32 (2006)
Dobrokhodov, V., Statnikov, R., Statnikov, A., Yanushkevich, I.: Modeling and simulation framework for multiple objective identification of a controllable descending system. In: Proceedings of the International Conference on Adaptive Modeling and Simulation (ADMOS), Geteborg, Sweden (2003)
Statnikov, R., Bordetsky, A., Statnikov, A.: Multicriteria analysis of real-life engineering optimization problems: statement and solution. Nonlinear Anal. 63, e685–e696 (2005)
Zverev, I.A.: Vector identification of the parameters of spindle units of metal-cutting machines. J. Mach. Manuf. Reliab. 6, 58–63 (1997)
Bondarenko, M.I., Nazemkin, A.Y., Pozhalostin, A.A., Statnikov, R.B., Shenfel’d, V.S.: Construction of consistent solutions in multicriteria problems of optimization of large systems. Phys.-Dokl. 39(4), 274–279 (1994)
Xargay, E., Hovakimyan, N., Dobrokhodov, V., Statnikov, R.B., Kaminer, I., Cao, C., Gregory, I.M.: L 1 adaptive flight control system: systematic design and V&V of control metrics. In: AIAA Guidance, Navigation, and Control Conference, Toronto, Ontario, Canada (2010)
Acknowledgement
The authors would like to thank Dr. N.N. Bolotnik for his feedback and discussion of the results.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Statnikov, R., Matusov, J. & Statnikov, A. Multicriteria Engineering Optimization Problems: Statement, Solution and Applications. J Optim Theory Appl 155, 355–375 (2012). https://doi.org/10.1007/s10957-012-0083-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-012-0083-9