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Heuristic evaluation of the characteristic function in the Cooperative Inventory Routing Game


This paper is concerned with the problem of the characteristic function evaluation for the game theory model of horizontal cooperation of carriers on large transportation networks. We consider a mathematical formalization of the Cooperative Inventory Routing Game (CIRG) in the form of characteristic function. Due to non-exact evaluation of the minimal cost for the coalitions, we have the subadditivity property of the characteristic function to be not fulfilled in general case. To calculate the values of characteristic function that satisfy the subadditivity condition, the Direct Coalition Induction Algorithm (DCIA) is used. To solve the multi-depot Inventory Routing Problem for each coalition, the Dynamic Adaptation procedure for Large Neighborhood Search (ALNS) is applied. Calculation of the characteristic function vales, DCIA performance, and advantages of Dynamic Adaptation for ALNS are demonstrated with computational experiments. It is shown that an experimental level of time consistency of the solutions generated by DALNS for given IRP benchmark instances appears to be higher in average in comparison with the solutions generated by ALNS.

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  1. Archetti, C., Bianchessi, N., Irnich, S., Speranza, M.G.: Formulations for an inventory routing problem. Int. Trans. Oper. Res. 21(3), 353–374 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  2. Archetti, C., Bertazzi, L., Hertz, A., Speranza, M.G.: A hybrid heuristic for an inventory routing problem. INFORMS J. Comput. 24(1), 101–116 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  3. Archetti, C., Bertazzi, L., Laporte, G., Speranza, M.G.: A branch-and-cut algorithm for a vendor-managed inventory-routing problem. Transp. Sci. 41(3), 382–391 (2007)

    Article  Google Scholar 

  4. Audy, J.F., D’Amours, S., Rousseau, L.M.: Cost allocation in the establishment of a collaborative transportation agreement–an application in the furniture industry. J. Oper. Res. Soc. 62(6), 960–970 (2011)

    Article  Google Scholar 

  5. Bellman, R.: Dynamic programming. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  6. Bertazzi, L., Paletta, G., Speranza, M.G.: Deterministic order-up-to level policies in an inventory routing problem. Transp. Sci. 36(1), 119–132 (2002)

    Article  MATH  Google Scholar 

  7. Coelho, L.C., Cordeau, J.F., Laporte, G.: The inventory-routing problem with transshipment. Comput. Oper. Res. 39(11), 2537–2548 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  8. Coelho, L.C., Cordeau, J.F., Laporte, G.: Thirty years of inventory routing. Transp. Sci. 48(1), 1–19 (2013)

    Article  Google Scholar 

  9. Cordeau, J.F., Laganà, D., Musmanno, R., Vocaturo, F.: A decomposition-based heuristic for the multiple-product inventory-routing problem. Comput. Oper. Res. 55, 153–166 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. Desaulniers, G., Rakke, J.G., Coelho, L.C.: A branch-price-and-cut algorithm for the inventory-routing problem. Transp. Sci. 50(3), 1060–1076 (2015)

    Article  Google Scholar 

  11. Guajardo, M., Rönnqvist, M.: A review on cost allocation methods in collaborative transportation. Int. Trans. Oper. Res. 23(3), 371–392 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  12. Juan, A.A., Faulin, J., Pérez-Bernabeu, E., Jozefowiez, N.: Horizontal cooperation in vehicle routing problems with backhauling and environmental criteria. Proc. Soc. Behav. Sci. 111, 1133–1141 (2014)

    Article  Google Scholar 

  13. Kimms, A., Kozeletskyi, I.: Core-based cost allocation in the cooperative traveling salesman problem. Eur. J. Oper. Res. 248(3), 910–916 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  14. Krajewska, M.A., Kopfer, H., Laporte, G., Ropke, S., Zaccour, G.: Horizontal cooperation among freight carriers: request allocation and profit sharing. J. Oper. Res. Soc. 59(11), 1483–1491 (2008)

    Article  MATH  Google Scholar 

  15. Park, H., Park, D., Jeong, I.J.: An effects analysis of logistics collaboration in last-mile networks for CEP delivery services. Transp. Policy 50, 115–125 (2016)

    Article  Google Scholar 

  16. Shirokikh, V.A., Zakharov, V.V.: Dynamic adaptive large neighborhood search for inventory routing problem. In: Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 231–241. Springer International Publishing (2015)

  17. Shi, Y., Boudouh, T., Grunder, O.: A hybrid genetic algorithm for a home jhealth care routing problem with time window and fuzzy demand. Expert Syst. Appl. 72, 160–176 (2017)

    Article  Google Scholar 

  18. Soysal, M., Bloemhof-Ruwaard, J.M., Haijema, R., van der Vorst, J.G.: Modeling a green inventory routing problem for perishable products with horizontal collaboration. Comput. Oper. Res. (2016). doi:10.1016/j.cor.2016.02.003

  19. Verdonck, L., Caris, A.N., Ramaekers, K., Janssens, G.K.: Collaborative logistics from the perspective of road transportation companies. Transp. Rev. 33(6), 700–719 (2013)

  20. Zakharov, V.V., Shchegryaev, A.N.: Stable cooperation in dynamic vehicle routing problems. Autom. Remote Control 76(5), 935–943 (2015)

    MathSciNet  Article  Google Scholar 

  21. Zibaei, S., Hafezalkotob, A., Ghashami, S.S.: Cooperative vehicle routing problem: an opportunity for cost saving. J. Indus. Eng. Int. 12(3), 271–286 (2016)

    Article  Google Scholar 

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Correspondence to Viacheslav A. Shirokikh.

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Zakharov, V.V., Shirokikh, V.A. Heuristic evaluation of the characteristic function in the Cooperative Inventory Routing Game. J Veh Routing Algorithms 1, 19–32 (2018).

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  • Inventory routing problem (IRP)
  • Cooperative inventory routing game (CIRG)and Characteristic function
  • Heuristic algorithm
  • ALNS
  • Time consistency
  • Dynamic adaptive large neighborhood search (DALNS)