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Cooperation of customers in traveling salesman problems with profits

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Abstract

In the profitable tour problem, the carrier decides whether to visit a particular customer with respect to the prize the customer offers for being visited and traveling cost associated with the visit, all in the context of other customers. Our focus is on the prizes customers need to offer to ensure being visited by the carrier. This can be formulated as a cooperative game where customers may form coalitions and make decisions on the prize values cooperatively. We define the profitable tour game describing this situation and analyze the cost associated with each coalition of customers and prizes that help to achieve it. We derive properties of the optimal prizes to be offered when the grand coalition is formed. These properties describe relationship between the prizes and the underlying traveling salesman game to provide connection with extensive literature on core allocations in traveling salesman games. The most important result states that the set of optimal prizes coincides with the core of the underlying traveling salesman game if this core is nonempty.

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Notes

  1. A question could be raised of which ones of the customers should increase the prize. This might open a long discussion since arguably every individual customer wants to minimize its own prize. However, in the cooperative game-theoretical framework we adopt, the increment could be already reflected in the cost function value \(\text{ Cost }^{TSP}(\{1,2,3\})\). Thus, the increment gets allocated in a manner that is fair according to the chosen allocation method (the nucleolus in this example).

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Authors and Affiliations

  1. Department of Business and Management Science, NHH Norwegian School of Economics, 5045, Bergen, Norway

    Ondrej Osicka, Mario Guajardo & Kurt Jörnsten

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  1. Ondrej Osicka
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  2. Mario Guajardo
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  3. Kurt Jörnsten
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Correspondence to Ondrej Osicka.

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Osicka, O., Guajardo, M. & Jörnsten, K. Cooperation of customers in traveling salesman problems with profits. Optim Lett 14, 1219–1233 (2020). https://doi.org/10.1007/s11590-019-01429-6

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