Multi-criteria Optimization of Multi-step Matrix Game in Collision Avoidance of Ships

ABSTRACT: This research article formulates a mathematical model of the matrix game of the safe ship control process containing: state variables and control, collision risk definition and the form of a collision risk matrix. Multicriteria optimization of the matrix game was introduced, leading to non-cooperative and cooperative game control algorithms and non-game control. Simulation safe trajectories of own ship for various types of control were compared to the example of the real situation at sea.

KEYWORDS: Colregs, ARPA, Mathematical Model, Collision Avoidance, Multicriteria Optimisation, Collision Risk, Matrix Game, Safe Trajectories

REFERENCES

Basar, T. & Bernhard, P. 2008. H-Infinity optimal control and related mini-max design problems: A dynamic game approach. Berlin: Springer. - doi:10.1007/978-0-8176-4757-5

Bist, D.S. 2000. Safety and security at sea. Oxford-New Delhi: Butter Heinemann.

Breton, M., & Szajowski, K. 2010. Advances in dynamic games: theory, applications, and numerical methods for differential and stochastic games. Boston: Birkhauser. - doi:10.1007/978-0-8176-8089-3

Ehrgott, M. 2005. Multicriterial optimization. Berlin: Springer.

Ehrgott, M. & Gandibleux, X. 2002. Multiple criteria optimization: state of the art annotated bibliographic surveys. New York: Kluwer Academic Press. - doi:10.1007/b101915

Engwerda, J.C. 2005. LQ dynamic optimization and differential games. New York: John Wiley & Sons.

Eshenauer, H., Koski, J., Osyczka, A. 1990. Multicriteria design optimization: procedures and application. Berlin: Springer-Verlag. - doi:10.1007/978-3-642-48697-5

Isaacs, R. 1965. Differential games. New York: John Wiley & Sons.

Kazimierski, W. & Stateczny. A. 2013. Fusion of Data from AIS and Tracking Radar for the Needs of ECDIS, Proc. of the Signal Processing Symp., Jachranka, Poland. - doi:10.1109/SPS.2013.6623592

Kouemou, G. 2009. Radar technology. Chapter 4 by Józef Lisowski: Sensitivity of safe game ship control on base information from ARPA radar. Croatia, In-tech, 61-86.

Kun, G. 2001. Stabilizability, controllability, and optimal strategies of linear and nonlinear dynamical games. PhD. Thesis. Aachen: RWTH.

Lazarowska, A. 2017. A new deterministic approach in a decision support system for ship’s trajectory planning. Expert Systems with Applications, Vol. 71, Issue C: 469–478. - doi:10.1016/j.eswa.2016.11.005

Łebkowski A.: Design of an Autonomous Transport System for Coastal Areas. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, Vol. 12, No. 1, doi:10.12716/1001.12.01.13, pp. 117-124, 2018

Lisowski, J. 2014. Optimization-supported decision-making in the marine game environment. Solid State Phenomena, Vol. 210: 215-222. - doi:10.4028/www.scientific.net/SSP.210.215

Lisowski J.: Analysis of Methods of Determining the Safe Ship Trajectory. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, Vol. 10, No. 2, doi:10.12716/1001.10.02.05, pp. 223-228, 2016

Mesterton-Gibbons, M. 2001. An introduction to game theoretic modeling. Providence: American Mathematical Society. - doi:10.1090/stml/011

Millington, I. & Funge, J. 2009. Artificial intelligence for games. Amsterdam-Tokyo: Elsevier. - doi:10.1016/B978-0-12-374731-0.00008-6

Miloh, T. 1974. Determination of critical manoeuvres for collision avoidance using the theory of differential games. Hamburg: Inst. Fur Schiffbau.

Modarres, M. 2006. Risk analysis in engineering. Boca Raton: Taylor & Francis Group.

Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. 2007. Algorithmic game theory. New York: Cambridge University Press. - doi:10.1017/CBO9780511800481

Olsder, G.J. & Walter, J.L. 1977. A differential game approach to collision avoidance of ships. Proc. of the 8th IFIP Symp. on Optimization Techniques, Novosibirsk, 264-271. - doi:10.1007/BFb0007243

Osborne, M.J. 2004. An introduction to game theory. New York: Oxford University Press.

Perez, T. 2005. Ship motion control. London: Springer.

Polak, E., Lee. S., Bustany, I., Madhan, A. 2016. Method of approximations and adaptive approximations for a class of matrix games. Journal of Optimization Theory and Applications, Vol. 170, No. 3: 876-899. - doi:10.1007/s10957-016-0953-7

Straffin, P.D. 2001. Game theory and strategy. Warszawa: Scholar.

Szlapczynski, R. & Szlapczynska, J. 2016. An analysis of domain-based ship collision risk parameters, Ocean Eng., Vol. 126: 47-56, - doi:10.1016/j.oceaneng.2016.08.030

Tomera, M. 2012. Nonlinear Observers Design for Multivariable Ship Motion Control, Polish Maritime Research, Vol. 19, Special Issue 1: 50-56 - doi:10.2478/v10012-012-0023-5

Xu, Q. & Wang, N. 2014. A survey on ship collision risk evaluation. Promet – Traffic & Transportation, Vol. 26, No. 6: 475-486. - doi:10.7307/ptt.v26i6.1386

Wells, D. 2013. Games and mathematics. Cambridge: Cambridge University Press.

Citation note:

Lisowski J.: Multi-criteria Optimization of Multi-step Matrix Game in Collision Avoidance of Ships. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, Vol. 13, No. 1, doi:10.12716/1001.13.01.12, pp. 125-131, 2019

BibTeX EndNote

Authors in other databases:

Józef Lisowski: ORCID iD icon

orcid.org/0000-0001-9281-376X Scopus icon