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On games arising from multi-depot Chinese postman problems

Author

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  • Trine Platz

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  • Herbert Hamers

Abstract

A multi-depot Chinese postman problem (MDCP) arises from a network (e.g. cityplan) in which several depots are located wherefrom edges (e.g. streets) have to be served. Since costs are involved with each visit to an edge, the objective is to find a minimum cost tour in the network that visits all edges of the network. Such a minimum cost tour consists of a collection of subtours such that the subtours originate from different depots, and each subtour starts and ends at the same depot. This typical OR problem turns into a multi decision maker problem if agents are assigned to the streets. In this new setting the cost of a minimum cost tour that visits all edges have to be paid by the agents. However, now each group of agents (coalition) has the opportunity to find its own minimum cost tour, i.e. a minimum cost tour that only visits the edges owned by the group of agents. Therefore, the main objective is to find allocations of the cost of a minimum tour that visits all agents in such a way that no coalition has higher costs than the costs incurred by its own minimum tour. We will use cooperative game theory to investigate whether these so-called core allocations exist. Therefore, we consider a cooperative Chinese postman (CP) game that is induced by an MDCP by associating every edge of the network with a different agent. In this paper, we characterize classes of networks that ensure the existence of core allocations, the so-called CP balanced graphs, and the existence of specific core allocations, the so-called CP submodular graphs. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Trine Platz & Herbert Hamers, 2015. "On games arising from multi-depot Chinese postman problems," Annals of Operations Research, Springer, vol. 235(1), pages 675-692, December.
  • Handle: RePEc:spr:annopr:v:235:y:2015:i:1:p:675-692:10.1007/s10479-015-1977-3
    DOI: 10.1007/s10479-015-1977-3
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    References listed on IDEAS

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    1. Hamers, Herbert, 1997. "On the concavity of delivery games," European Journal of Operational Research, Elsevier, vol. 99(2), pages 445-458, June.
    2. Hamers, Herbert & Borm, Peter & van de Leensel, Robert & Tijs, Stef, 1999. "Cost allocation in the Chinese postman problem," European Journal of Operational Research, Elsevier, vol. 118(1), pages 153-163, October.
    3. Granot, D. & Hamers, H.J.M. & Tijs, S.H., 1999. "On some balanced, totally balanced and submodular delivery games," Other publications TiSEM e0496604-0162-4a27-992c-a, Tilburg University, School of Economics and Management.
    4. repec:spr:compst:v:58:y:2003:i:1:p:131-139 is not listed on IDEAS
    5. Hamers, H.J.M. & Miquel, S. & Norde, H.W., 2011. "Monotonic Stable Solutions for Minimum Coloring Games," Discussion Paper 2011-016, Tilburg University, Center for Economic Research.
    6. Yoshio Okamoto, 2003. "Submodularity of some classes of the combinatorial optimization games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(1), pages 131-139, September.
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    Cited by:

    1. Behzad Hezarkhani & Marco Slikker & Tom Woensel, 2016. "A competitive solution for cooperative truckload delivery," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(1), pages 51-80, January.
    2. Platz, Trine Tornøe, 2017. "On the submodularity of multi-depot traveling salesman games," Discussion Papers of Business and Economics 8/2017, University of Southern Denmark, Department of Business and Economics.

    More about this item

    Keywords

    Chinese postman problem; Cooperative game; Balancedness; Submodularity;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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