On segmenting logistical zones for servicing continuously developed consumers. (English. Russian original) Zbl 1301.49110
Autom. Remote Control 74, No. 6, 968-977 (2013); translation from Avtom. Telemekh. 2013, No. 6, 87-100 (2013).
Summary: We study the optimal placement problem for several logistical objects. A characteristic feature of this problem is the need for sequential segmentation into servicing zones and accounting for population distributed continuously across the entire region. We reduce this problem to a variational calculus problem in a special form. To study this problem, we develop numerical algorithms that are able to determine the optimal placement of a logistical object inside a given segment. The algorithms are based on constructing wavefronts for a light wave emitted from the boundary of the chosen region. The wave moves inside the region, which lets us account for all inhabitants in this region. In constructing the solution, we have accounted for the loss of smoothness in the wavefront, have developed a software implementation for the computational algorithms, and have conducted a numerical experiment for a number of model problems.
MSC:
| 49N90 | Applications of optimal control and differential games |
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
| 90B06 | Transportation, logistics and supply chain management |
Software:
VIGOLTReferences:
| [1] | Kazakov, A.L. and Lempert, A.A., An Approach to Optimization in Transport Logistics, Autom. Remote Control, 2011, vol. 72, no. 7, pp. 1398–1404. · Zbl 1226.90018 · doi:10.1134/S0005117911070071 |
| [2] | Matviichuk, A.R. and Ushakov, V.N., On Constructing Resolving Controls in Control Problems with Phase Constraints, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2006, no. 1, pp. 5–20. |
| [3] | Ushakov, V.N. and Matviichuk, A.R., AMethod for Solving Control Problems for Elongated Objects over a Finite Time Interval, in Tr. IX Mezhd. Chetaevskoi konf. ”Analiticheskaya mekhanika, ustoichivost’ i upravlenie dvizheniem” (Proc. IX Int. Chetaev Conf. ”Analytic Mechanics, Stability, and Motion Control”), Irkutsk: Inst. Dinamiki Sist. Teor. Upravlen., 2007, vol. 3, pp. 253–261. |
| [4] | Bashurov, V.V. and Filimonenkova, T.I., Matematicheskie modeli bezopasnosti (Mathematical Models of Security), Novosibirsk: Nauka, 2009. |
| [5] | Propoi, A.I., Models of Excitable Media, Autom. Remote Control, 1995, vol. 56, no. 6, part 2, pp. 861–868. · Zbl 0922.93004 |
| [6] | Propoi, A.I., Excitable Media and the Nonlocal Search, Autom. Remote Control, 1995, vol. 56, no. 7, part 2, pp. 1042–1050. · Zbl 0987.90535 |
| [7] | Hwang, F.K., Richards, D.S., and Winter, P., The Steiner Tree Problem, Amsterdam: Elsevier, 1992. · Zbl 0774.05001 |
| [8] | Gordeev, E.N. and Tarastsov, O.G., The Steiner Problem. A Survey, Diskret. Mat., 1993, no. 2, pp. 3–28. · Zbl 0805.05018 |
| [9] | Cormen, T.H., Leiserson, C.E., Rivest, R.L., and Stein, C., Introduction to Algorithms, Boston: MIT Press, 2001. Translated under the title Algoritmy. Postroenie i analiz, Moscow: Will’yams, 2005. · Zbl 1047.68161 |
| [10] | Panyukov, A.V., The Steiner Problem in Graphs: Topological Methods of Solution, Autom. Remote Control, 2004, vol. 65, no. 3, pp. 439–448. · Zbl 1090.90165 · doi:10.1023/B:AURC.0000019376.31168.20 |
| [11] | Lotarev, D.T. and Uzdemir, A.P., Conversion of the Steiner Problem on the Euclidean Plane to the Steiner Problem on Graph, Autom. Remote Control, 2005, vol. 66, no. 10, pp. 1603–1613. · Zbl 1126.90414 · doi:10.1007/s10513-005-0194-y |
| [12] | Arnol’d, V.I., Osobennosti kaustik i volnovykh frontov (Singularities of Caustics and Wavefronts), Moscow: FAZIS, 1996. |
| [13] | Mikhalevich, V.S., Trubin, V.A., and Shor, N.Z., Optimizatsionnye zadachi proizvodstvenno-transportnogo planirovaniya. Modeli, metody, algoritmy (Optimization Problems in Industrial Transportation Planning: Models, Methods, and Algorithms), Moscow: Nauka, 1986. · Zbl 0701.90025 |
| [14] | Preparata, F.P. and Shamos, M.G., Computational Geometry. An Introduction, New York: Springer, 1985. Translated under the title Vychislitel’naya geometriya: Vvedenie, Moscow: Mir, 1989. · Zbl 0744.68131 |
| [15] | Du, D. and Hu, X., Steiner Tree Problems in Computer Communication Networks, Singapore: World Scientific, 2008. · Zbl 1156.90014 |
| [16] | Bukharov, D.S. and Kazakov, A.L., The Software Suite ”VIGOLT” for Solving Optimization Problems Arising in Transportation Logistics, Vychisl. Met. Programmir., 2012, part 2, pp. 65–74, http://nummeth.srcc.msu.ru/ . |
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