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Tools of mathematical modeling of arbitrary object packing problems. (English) Zbl 1201.90167
Summary: The article reviews the concept of and further develops phi-functions (Φ-functions) as an efficient tool for mathematical modeling of two-dimensional geometric optimization problems, such as cutting and packing problems and covering problems. The properties of the phi-function technique and its relationship with Minkowski sums and the nofit polygon are discussed. We also describe the advantages of phi-functions over these approaches. A clear definition of the set of objects for which phi-functions may be derived is given and some exceptions are illustrated. A step by step procedure for deriving phi-functions illustrated with examples is provided including the case of continuous rotation.
90C27Combinatorial optimization
[1]Bennell, J. A., & Song, X. (2008). A comprehensive and robust procedure for obtaining the nofit polygon using Minkowski sums. Computers and Operational Research, 35(1), 267–281. · Zbl 1136.65023 · doi:10.1016/j.cor.2006.02.026
[2]Burke, E. K., Hellier, R. S. R., Kendall, G., & Whitwell, G. (2007). Complete and robust no-fit polygon generation for the irregular stock cutting problem. European Journal of Operational Research, 179(1), 27–49. · Zbl 1175.90325 · doi:10.1016/j.ejor.2006.03.011
[3]Culberson, J. C., & Reckhow, R. A. (1988). Covering polygons is hard. In Proceedings of 29th IEEE conference on foundations of computer science (pp. 601–611).
[4]Cunninghame-Green, R. (1989). Geometry. Shoemaking and the milk tray problem. New Scientist, 1677, 50–53.
[5]Daniels, K., & Inkulu, R. (2001). Translational polygon covering using intersection graphs. In Proceedings of the thirteenth Canadian conference on computational geometry (pp. 61–64).
[6]Daniels, K., Mathur, A., & Grinde, R. (2003). A combinatorial maximum cover approach to 2D translational geometric covering. In Proceeedings of 15th Canadian conference on computational geometry (pp. 2–5). Halifax, Nova Scotia, Canada, August 11–13, 2003.
[7]Fomenko, A., Fuchs, D., & Gutenmacher, V. (1986). Homotopic topology. Akademiai Kiado: Budapest.
[8]Ghosh, P. K. (1991). An algebra of polygons through the notation of negative shapes. CVGIP: Imag Understand, 17, 357–378.
[9]Scheithauer, G., Stoyan, Y., Gil, N., & Romanova, T. (2003). Phi-functions for circular segments. Prepr. Technische Univarsitat Dresden, MATH-NM-7-2003. Dresden.
[10]Stoyan, Y. (1983). Mathematical methods for geometric design. In Advances in CAD/CAM//. Proceeding PROLAMAT82. Leningrad, USSR, 16–18 May, 1982 (pp. 67–86). Amsterdam: North-Holland.
[11]Stoyan, Y. (2003). Phi-function of non-convex polygons with rotations. Journal of Mechanical Engineering, 6(1), 74–86.
[12]Stoyan, Y., & Gil, N. (1976). Methods and algorithms of placement of 2D geometric objects. Ukrainian SSR academy of sciences. Kiev: Naukova Dumka (In Russian).
[13]Stoyan, Y., & Pankratov, A. V. (1999). Regular packing of congruent polygons on the rectangular sheet. European Journal of Operational Research, 113, 653–675. · Zbl 0941.90065 · doi:10.1016/S0377-2217(98)00050-2
[14]Stoyan, Y., & Patsuk, V. N. (2000). A method of optimal lattice packing of congruent oriented polygons in the plane. European Journal of Operational Research, 124, 204–216. · Zbl 0990.90103 · doi:10.1016/S0377-2217(99)00115-0
[15]Stoyan, Y., & Ponomarenko, L. D. (1977). Minkowski’s sum and the hodograph of the dense allocation vector function. Reports of the Ukrainian SSR Academy of Science, A(10), 888–890 (In Russian).
[16]Stoyan, Y., Novozhilova, M., & Kartashov, A. (1996). Mathematical model and method of searching for a local extremum for the non-convex oriented polygons allocation problem. European Journal of Operational Research, 92, 193–210. · Zbl 0916.90234 · doi:10.1016/0377-2217(95)00038-0
[17]Stoyan, Y., Terno, J., Scheithauer, G., Gil, N., & Romanova, T. (2002a). Phi-functions for primary 2D-objects. Studia Informatica Universalis, 2(1), 1–32.
[18]Stoyan, Y., Terno, J., Gil, M., Romanova, T., & Scheithauer, G. (2002b). Construction of a Phi-function for two convex polytopes. Applicationes Mathematicae, 29(2), 199–218. · Zbl 1053.90009 · doi:10.4064/am29-2-6
[19]Stoyan, Y., Scheithauer, G., Gil, N., & Romanova, T. (2004). Phi-functions for complex 2D-objects. 4OR (Operations Research): Quarterly Journal of the Belgian, French and Italian Operations Research Societies, 2, 69–84.
[20]Stoyan, Y., Scheithauer, G., & Romanova, T. (2005a). Mathematical modeling of interaction of primary geometric 3D objects. Cybernetics and Systems Analysis, 41(3), 332–342. Translated from Kibernetika i Sistemnyi Analiz, 3, 19–31. · Zbl 1102.68684 · doi:10.1007/s10559-005-0067-y
[21]Stoyan, Y., Gil, N., Scheithauer, G., Pankratov, A., & Magdalina, I. (2005b). Packing of convex polytopes into a parallelepiped. Optimization, 54(2), 215–235. · Zbl 1134.90550 · doi:10.1080/02331930500050681
[22]Toth, G. F. (1997). Packing and covering. In J. Goodman & J. O’Rourke (Eds.), Handbook of discrete and computational geometry. New York: CRC Press.