# American Institute of Mathematical Sciences

## Applications of mathematics to maritime search

 1 School of Computer and Computing, Zhejiang University City College, Hangzhou 310015, China 2 Department of investigation, Zhejiang Police College, Hangzhou 310053, China 3 School of Business, Zhejiang University City College, Hangzhou 310015, China 4 School of Business and Economics, Australian National University, Acton ACT 2061, Australia

* Corresponding author: Jinming Zhang

Received  October 2017 Revised  January 2018 Published  November 2018

The issue of searching missing aircraft is valuable after the event of MH370. This paper provides a global optimal model to foster the efficiency of maritime search. Firstly, the limited scope, a circle whose center is the last known position of the aircraft, should be estimated based on the historical data recorded before the disappearance of the aircraft. And Bayes' theorem is applied to calculate the probability that the plane falling in the region can be found. Secondly, the drift of aircraft debris under the influence of wind and current is considered via Finite Volume Community Ocean Model(FVCOM) and Monte Carlo Method(MC), which make the theory more reasonable. Finally, a global optimal model about vessel and aircraft quantitative constraints is established, which fully considers factors including the area of sea region to be searched, the maximum speed, search capabilities, initial distance of the vessels by introducing 0-1 decision variables.

Citation: Jinling Wei, Jinming Zhang, Meishuang Dong, Fan Zhang, Yunmo Chen, Sha Jin, Zhike Han. Applications of mathematics to maritime search. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019064
##### References:
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##### References:
 [1] A. Adcroft and J.-M. Campin, Rescaled height coordinates for accurate representation of free-surface flows in ocean circulation models, Ocean Modelling, 7 (2004), 269-284. [2] F. Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Applied Mathematics and Nonlinear Sciences, 1 (2016), 391-404. [3] J.-M. Campin, A. Adcroft, C. Hill and J. Marshall, Conservation of properties in a free-surface model, Ocean Modelling, 6 (2004), 221-244. [4] W. Chapline, Estimating the drift of distressed small craft, New London (CT): Coast Guard Alumni Association Bulletin, US Coast Guard Academy, 22 (1960), 39-42. [5] C. Chen, G. Cowles and R. Beardsley, An unstructured grid, finite-volume coastal ocean model: Fvcom user manual, SMAST/UMASSD. [6] M. Fischetti and M. Monaci, Proximity search for 0-1 mixed-integer convex programming, Journal of Heuristics, 20 (2014), 709-731. [7] S. M. Griffies, Fundamentals of Ocean Climate Models, vol. 518, Princeton university Press, 2004. [8] U. C. Guard, Us coast guard addendum to the united states national search and rescue supplement to the international aeronautical and maritime search and rescue manual, United States Coast Guard Publication. [9] W. Guo, Z. Zhu and Y. Hou, Bayesian network based cooperative area coverage searching for uavs, in Frontiers in Computer Education, Springer, 2012, 611–618. [10] E. Hernández, M. Carreras, J. Antich, P. Ridao and A. Ortiz, A topologically guided path planner for an auv using homotopy classes, in Robotics and Automation (ICRA), 2011 IEEE International Conference on, IEEE, 2011, 2337–2343. [11] M. I. Karavelas, Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs, Computational Geometry, 44 (2011), 20-51. doi: 10.1016/j.comgeo.2010.07.002. [12] E. Klarreich, In search of bayesian inference, Communications of the ACM, 58 (2015), 21-24. [13] T. M. Kratzke, L. D. Stone and J. R. Frost, Search and rescue optimal planning system, in Information Fusion (FUSION), 2010 13th Conference on, IEEE, 2010, 1–8. [14] K. Li, L. Li, T. Tesfazghi and E. H.-M. Sha, Adaptive and cost-optimal parallel algorithm for the 0-1 knapsack problem, in Parallel, Distributed and Network-Based Processing (PDP), 2011 19th Euromicro International Conference on, IEEE, 2011, 537–544. [15] Y. Nakano, Quasi-monte carlo methods for choquet integrals, Journal of Computational and Applied Mathematics, 287 (2015), 63-66. doi: 10.1016/j.cam.2015.03.026. [16] M. H. Overmars and J. Van Leeuwen, Maintenance of configurations in the plane, Journal of computer and System Sciences, 23 (1981), 166-204. doi: 10.1016/0022-0000(81)90012-X. [17] M. M. Polycarpou, Y. Yang and K. M. Passino, A cooperative search framework for distributed agents, in Intelligent Control, 2001.(ISIC'01). Proceedings of the 2001 IEEE International Symposium on, IEEE, 2001, 1–6. [18] S. Rajasekaran, Efficient parallel hierarchical clustering algorithms, IEEE Transactions on Parallel & Distributed Systems, 497–502. [19] H. R. Richardson and L. D. Stone, Operations analysis during the underwater search for scorpions, Naval Research Logistics Quarterly, 18 (1971), 141-157. [20] M. Rosa and M. Gandarias, Multiplier method and exact solutions for a density dependent reaction-diffusion equation, Appl. Math. Nonlinear Sci, 1 (2016), 311-320. [21] C. A. A. Sanches, N. Y. Soma and H. H. Yanasse, Parallel time and space upper-bounds for the subset-sum problem, Theoretical Computer Science, 407 (2008), 342-348. doi: 10.1016/j.tcs.2008.06.051. [22] R. Smith, J. Dukowicz and R. Malone, Parallel ocean general circulation modeling, Physica D: Nonlinear Phenomena, 60 (1992), 38-61. [23] L. Štrubelj, I. Tiselj and B. Mavko, Simulations of free surface flows with implementation of surface tension and interface sharpening in the two-fluid model, International Journal of Heat and Fluid Flow, 30 (2009), 741-750. [24] B.-F. Wang, S.-C. Ku and K.-H. Shil, Cost-optimal parallel algorithms for the tree bisector and related problems, Parallel and Distributed Systems, IEEE Transactions on, 12 (2001), 888-898.
The bounding limit of aircraft in distress
Wind-induced drift component diagram
Ocean Surface Zonal Currents-Ocean Surface Meridional Currents(meter/sec)
Sketch of calculation process
Time taken by exhaustive method for optimal solution with quantitative constraint
 Problem size $n$ Number of package $S$ Number of package $t$ 30 $2^{30}$=1073741824 About 1s 50 $2^{50}$=1125899906842624 About 13 days 100 $2^{100}$ About $4\times{}10^{13}$ days
 Problem size $n$ Number of package $S$ Number of package $t$ 30 $2^{30}$=1073741824 About 1s 50 $2^{50}$=1125899906842624 About 13 days 100 $2^{100}$ About $4\times{}10^{13}$ days
Time taken by exhaustive method for optimal solution with quantitative constraint
 Problem size $n$ Number of package $S$ Computation time $t$ 30(20 ships, 10 planes) $\binom{20}{10}\cdot\binom{10}{5}=46558512$ About 0.05 s 50(40 ships, 10 planes) $\binom{40}{10}\cdot\binom{10}{5}=213610453056$ About 4 s 100(90 ships, 10 planes) $\binom{100}{10}\cdot\binom{10}{5}=1441602661439556$ About 400 hours
 Problem size $n$ Number of package $S$ Computation time $t$ 30(20 ships, 10 planes) $\binom{20}{10}\cdot\binom{10}{5}=46558512$ About 0.05 s 50(40 ships, 10 planes) $\binom{40}{10}\cdot\binom{10}{5}=213610453056$ About 4 s 100(90 ships, 10 planes) $\binom{100}{10}\cdot\binom{10}{5}=1441602661439556$ About 400 hours
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