Abstract
A multistatic sonar system consists of one or more sources that are able to emit underwater sound, and receivers that listen to the reflected sound waves. Knowing the speed of sound in water, the time when the sound was sent from a source, and the arrival time of the sound at one or more receivers, it is possible to determine the location of surrounding objects. The propagation of underwater sound is a complex phenomenon that depends on various attributes of the water (density, pressure, temperature, and salinity) and the emitted sound (pulse length and volume), as well as the reflection properties of the water’s surface. These effects can be approximated by nonlinear equations. Furthermore, natural obstacles in the water, such as the coastline, need to be taken into consideration. Given an area of the ocean that should be endowed with a sonar system for surveillance, this paper formulates two natural sensor placement problems. In the first, the goal is to maximize the area covered by a fixed number of sources and receivers. In the second, the goal is to cover the entire area with a minimum-cost set of equipment. For each problem, this paper considers two different sensor models: definite range (“cookie-cutter”) and probabilistic. It thus addresses four problem variants using integer nonlinear formulations. Each variant can be reformulated as an integer linear program in one of several ways; this paper discusses these reformulations, then compares them numerically using a test bed from coastlines around the world and a state-of-the-art mixed-integer program solver (IBM ILOG CPLEX).
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References
Adams WP, Forrester RJ (2005) A simple recipe for concise mixed 0–1 linearizations. Oper Res Lett 33(1):55–61
Balas E (1964) Extension de l’algorithme additif à la programmation en nombres entiers et à la programmation non linéaire. Comptes rendus de l’Académie des Sciences. Technical report, Paris
Beraldi P, Ruszczynski A (2002) The probabilistic set-covering problem. Oper Res 50(6):956–967
Bliek C, Bonami P, Lodi A (2014) Solving mixed-integer quadratic programming problems with IBM-CPLEX: a progress report. In: Proceedings of the twenty-sixth RAMP symposium, Hosei University, Tokyo, October 16–17, 2014
Chaovalitwongse W, Pardalos PM, Prokopyev OA (2004) A new linearization technique for multi-quadratic \(0-1\) programming problems. Oper Res Lett 32:517–522
Conover WJ (1999) Practical nonparametric statistics. Wiley, New York
Cox AW (1974) Sonar and underwater sound. Lexington Books, Lanham
Craparo EM, Fügenschuh A, Hof C, Karatas M (2018) Optimizing source and receiver placement in multistatic sonar networks to monitor fixed targets. Eur J Oper Res 272:816–831 (online available, in print)
Craparo EM, Karatas M, Kuhn TU (2017) Sensor placement in active multistatic sonar networks. Nav Res Logist 64(4):287–304
DelBalzo DR, Stangl KC (2009) Design and performance of irregular sonobuoy patterns in complicated environments. In: Proceedings of the IEEE OCEANS 2009, Biloxi
Fortet R (1959) L’algèbre de Boole et ses applications en recherche opérationelle. Cah Centre d’Études de Rech Opér 4:5–36
Friedman M (1937) The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J Am Stat Assoc, 32 (200):675–701. ISSN 01621459
Glover F (1975) Improved linear integer programming formulations of nonlinear integer problems. Manag Sci 22(4):455–460
Glover F, Woolsey E (1974) Converting the 0–1 polynomial programming problem to a 0–1 linear program. Oper Res 22(1):180–182
Gong X, Zhang J, Cochran D, Xing K (2013) Barrier coverage in bistatic radar sensor networks: cassini oval sensing and optimal placement. In: Proceedings of the 14th ACM international symposium on mobile ad hoc networking and computing, pp 49–58
Hansen P, Meyer C (2009) Improved compact linearizations for the unconstrained quadratic 0–1 minimization problem. Discrete Appl Math 157:1267–1290
Karatas M, Craparo EM (2015) Evaluating the direct blast effect in multistatic sonar networks using Monte Carlo simulation. In: Yilmaz L et al (ed) Proceedings of the 2015 winter simulation conference. IEEE Press, Piscataway, NJ
Karatas M, Craparo EM, Washburn A (2014) A cost effectiveness analysis of randomly placed multistatic sonobuoy fields. In: Bruzzone C. et al (ed) The international workshop on applied modeling and simulation
Karatas M, Gunal MM, Craparo EM (2016) Performance evaluation of mobile multistatic search operations via simulation. In SpringSim-ANSS, Society for Modeling and Simulation Internation (SCS)
Ngatchou PN, Fox WLJ, El-Sharkawi MA (2006) Multiobjective multistatic sonar sensor placement. In Proceedings of the IEEE congress on evolutionary computations. Vancouver, Canada
Oral M, Kettani O (1992) A linearization procedure for quadratic and cubic mixed-integer problems. Oper Res 40(1):109–116
Ozols S, Fewell MP (2011) On the design of multistatic sonobuoy fields for area search. Technical report, Maritime Operations Division, Defence Science and Technology Organisation (DSTO), Australia
Padberg M (1989) The Boolean quadric polytope: some characteristics, facets and relatives. Math Program 45:139–172
Pardalos PM, Chaovalitwongse W, Iasemidis LD, Sackellares JC, Shiau D-S, Carney PR, Prokopyev OA, Yatsenko VA (2004) Seizure warning algorithm based on optimization and nonlinear dynamics. Math Program Ser B 101:365–385
Prim RC (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36:1389–1401
Przemieniecki JS (2000) Mathematical methods in defense analyses. American Institute of Aeronautics and Astronautics (AIAA), Reston
Ryan WBF, Carbotte SM, Coplan JO, O’Hara S, Melkonian A, Arko R, Weissel RA, Ferrini V, Goodwillie A, Nitsche F, Bonczkowski J, Zemsky R (2009) Global multi-resolution topography synthesis. Geochem Geophys Geosyst 10(3):Q03014. https://doi.org/10.1029/2008GC00233
Strode C, Mourre B, Rixen M (2012) Decision support using the multistatic tactical planning aid (MSTPA). Ocean Dyn 62:161–175
Urick RJ (1983) Principles of underwater sound, 3rd edn. McGraw-Hill, New York
Watters LJ (1967) Letter to the editor—reduction of integer polynomial programming problems to zero-one linear programming problems. Oper Res 6(15):1171–1174
Zangwill WI (1965) Media selection by decision programming. J Advert Res 5(3):30–36
Acknowledgements
Dr. Craparo is funded by the Office of Naval Research. The authors thank the two anonymous referees for their various helpful comments.
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Fügenschuh, A.R., Craparo, E.M., Karatas, M. et al. Solving multistatic sonar location problems with mixed-integer programming. Optim Eng 21, 273–303 (2020). https://doi.org/10.1007/s11081-019-09445-2
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DOI: https://doi.org/10.1007/s11081-019-09445-2