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Towards Asymptotically Optimal One-to-One PDP Algorithms for Capacity 2+ Vehicles

  • Lars Nørvang Andersen
  • Martin OlsenEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11184)

Abstract

We consider the one-to-one Pickup and Delivery Problem (PDP) in Euclidean Space with arbitrary dimension d, where n transportation requests are picked i.i.d. with a separate origin-destination pair for each object to be moved. First, we consider the problem from the customer perspective, where the objective is to compute a plan for transporting the objects such that the Euclidean distance traveled by the vehicles when carrying objects is minimized. We develop a polynomial time asymptotically optimal algorithm for vehicles with capacity \(o(\root 2d \of {n})\) for this case including the realistic setting where the capacity of the vehicles is a fixed constant and \(d=2\). This result also holds imposing LIFO constraints for loading and unloading objects. Secondly, we extend our algorithm to the classical single-vehicle PDP, where the objective is to minimize the total distance traveled by the vehicle and we present results indicating that the extended algorithm is asymptotically optimal for a fixed vehicle capacity, if the origins and destinations are picked i.i.d. using the same distribution.

References

  1. 1.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998).  https://doi.org/10.1145/290179.290180MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beardwood, J., Halton, J.H., Hammersley, J.M.: The shortest path through many points. Math. Proc. Camb. Philosoph. Soc. 55(4), 299–327 (1959).  https://doi.org/10.1017/S0305004100034095MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berbeglia, G., Cordeau, J.F., Gribkovskaia, I., Laporte, G.: Static pickup and delivery problems: a classification scheme and survey. TOP: Off. J. Span. Soc. Stat. Oper. Res. 15(1), 1–31 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)Google Scholar
  5. 5.
    Das, A., Mathieu, C.: A quasipolynomial time approximation scheme for Euclidean capacitated vehicle routing. Algorithmica 73(1), 115–142 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Few, L.: The shortest path and the shortest road through n points. Mathematika 2(2), 141–144 (1955).  https://doi.org/10.1112/S0025579300000784MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guan, D.: Routing a vehicle of capacity greater than one. Disc. Appl. Math. 81(1), 41–57 (1998).  https://doi.org/10.1016/S0166-218X(97)00074-7MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Math. Oper. Res. 10(4), 527–542 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Khachay, M., Dubinin, R.: PTAS for the euclidean capacitated vehicle routing problem in \(R^d\). In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 193–205. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_16CrossRefGoogle Scholar
  10. 10.
    Parragh, S.N., Doerner, K.F., Hartl, R.F.: A survey on pickup and delivery problems (part I). J. Betriebswirtschaft 58(1), 21–51 (2008)CrossRefGoogle Scholar
  11. 11.
    Parragh, S.N., Doerner, K.F., Hartl, R.F.: A survey on pickup and delivery problems (part II). J. Betriebswirtschaft 58(2), 81–117 (2008)CrossRefGoogle Scholar
  12. 12.
    Psaraftis, H.: Analysis of an o(n) heuristic for the single vehicle many-to-many Euclidean dial-a-ride problem. Transp. Res. Part B: Methodol. 17, 133–145 (1981)CrossRefGoogle Scholar
  13. 13.
    Savelsbergh, M.W.P., Sol, M.: The general pickup and delivery problem. Transp. Sci. 29, 17–29 (1995)CrossRefGoogle Scholar
  14. 14.
    Stein, D.M.: An asymptotic, probabilistic analysis of a routing problem. Math. Oper. Res. 3(2), 89–101 (1978).  https://doi.org/10.1287/moor.3.2.89MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Treleaven, K., Pavone, M., Frazzoli, E.: Asymptotically optimal algorithms for one-to-one pickup and delivery problems with applications to transportation systems. IEEE Trans. Autom. Control 58(9), 2261–2276 (2013).  https://doi.org/10.1109/TAC.2013.2259993MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhusDenmark
  2. 2.Department of Business Development and TechnologyAarhus UniversityAarhusDenmark

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