A Model of 'Nonadditive' Routing Problem where the Costs Depend on the Set of Pending Tasks

We consider a generalization of the bottleneck (minimax) routing problem. The problem is to successively visit a number of megalopolises, complicated by precedence of constraints imposed on the order of megalopolises visited and the fact that the cost functions (of movement between megalopolises and of interior tasks) may explicitly depend on the list of tasks that are not completed at the present time. The process of movement is considered to be a sequence of steps, which include the exterior movement to the respective megalopolis and the following completion of (essentially interior) jobs connected with the megalopolis. The quality of the whole process is represented by the maximum cost of steps it consists of; the problem is to minimize the mentioned criterion (which yields a minimax problem, usually referred to as a 'bottleneck problem'). Optimal solutions, in the form of a route-track pair (a track, or trajectory, conforms to a specific instance of a tour over the megalopolises, which are numbered in accordance with the route; the latter is defined by the transposition of indices), are constructed through a 'nonstandard' variant of the dynamic programming method, which allows to avoid the process of constructing of all the values of the Bellman function whenever precedence constraints are present.

Keywords

dynamic programming; route; precedence constraints; sequential ordering problem.

References

1. Garey M.R., Johnson D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco, W.H. Freeman and Company, 1979. 338 p.
2. Melamed I.I., Sergeev S.I., Sigal I.Kh. The Travelling Salesman Problem. Issues in Theory. Automation and Remote Control, 1989, vol. 50, no. 9, pp. 1147-1173.
3. Melamed I.I., Sergeev S.I., Sigal I.Kh. The Travelling Salesman Problem. Exact Methods. Automation and Remote Control, 1989, vol. 50, no. 10, pp. 1303-1324.
4. Melamed I.I., Sergeev S.I., Sigal I.Kh. The Travelling Salesman Problem. Approximate Algorithms. Automation and Remote Control, 1989, vol. 50, no. 11, pp. 1459-1479.
5. Bellman R. Dynamic Programming Treatment of the Travelling Salesman Problem. Journal of the ACM (JACM), 1962, vol. 9, no. 1, pp. 61-63. DOI: 10.1145/321105.321111
6. Held M., Karp R. A Dynamic Programming Approach to Sequencing Problems. Journal of the Society for Industrial & Applied Mathematics, 1962, vol. 10, no. 1, pp. 196-210. DOI: 10.1137/0110015
7. Little J.D., Murty K.G., Sweeney D.W., Karel C. An Algorithm for the Travelling Salesman Problem. Operations Research, 1963, vol. 11, no. 6, pp. 972-989. DOI: 10.1287/opre.11.6.972
8. The Travelling Salesman Problem and Its Variations (Combinatorial Optimization). Dordrecht, Kluwer Academic Publishers, 2002. 830 p.
9. Sergeev S.I. Algorithms for the Minimax Problem of the Travelling Salesman. I. An Approach Based on Dynamic Programming. Automation and Remote Control, 1995, vol. 56, no. 7, pp. 1027-1032.
10. Sesekin A.N., Chentsov A.A., Chentsov A.G. [Routing with an Abstract Function of Travel Cost Aggregation]. Trudy Inst. Mat. i Mekh. UrO RAN [Proceedings of the IMM UB RAS], 2010, vol. 16, no. 3, pp 240-264. (in Russian)
11. Kuratowski K., Mostowski A. Set Theory. Amsterdam, North-Holland, 1967.
12. Dieudonn'e J. Foundations of Modern Analysis. N.Y., Academic Press, 1960. 361 p.
13. Chentsov A.G. Ekstremal'nye zadachi marshrutizatsii i raspredeleniya zadaniy: voprosy teorii [A Theoretical Treatment of Extremal Problems in Routing and Scheduling]. Moscow, Izhevsk, Regular and Chaotic Dinamics, 2008. 240 p.
14. Chentsov A.A., Chentsov A.G. Extremal Bottleneck Routing Problem with Constraints in the Form of Precedence Conditions. Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2008, vol. 263, no. 2, pp. 23-36. DOI: 10.1134/S0081543808060047
15. Cheblokov I.B., Chentsov A.G. [About one Route Problem with Interior Tasks]. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye nauki [Journal of Udmurt University. Mathematics, Mechanics and Computer Science], 2012, no. 1, pp. 96-119. (in Russian)
16. Chentsov A.G. [To Question of Routing of Works Complexes]. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye nauki [Journal of Udmurt University. Mathematics, Mechanics and Computer Science], 2013, no. 1, pp. 59-82. (in Russian)