OR Spectrum

, Volume 38, Issue 1, pp 137–173 | Cite as

Pricing strategies for the site-dependent vehicle routing problem

  • Silvia SchwarzeEmail author
Regular Article


The vehicle pricing game (VPG) which addresses the vehicles’ viewpoints within a vehicle routing problem (VRP) is introduced. Each vehicle acts as a player who demands a price per kilometer. That is, vehicles represent decentralized actors in transport systems, e.g., carriers under subcontracts. Based on these prices, a VRP is solved and profits are generated. Which price should a vehicle choose to maximize its own profit, considering the competition among vehicles? To answer this question, site dependencies leading to inhomogeneous vehicles are included. More detailed, skill-levels, e.g., relating to the size of a vehicle, are used to indicate a vehicle’s ability to carry out particular services. Moreover, penalty options are added. The VPG serves as an element of a vertical collaboration in a transport scenario and thus provides decision support for cooperative models. Theoretical results for the VPG are provided for a particular case of a two-player ring network game, for which the full set of equilibria is described and their uniqueness is discussed. It is shown that the uniqueness of the higher-skilled vehicle’s payoff is guaranteed even for multiple equilibria. The competition ratio is defined; it restricts a vehicle’s price to keep its competitiveness. Moreover, the acceptance ratio gives a lower bound on prices such that a loss of market share is still accepted. Experimental results are provided for general networks including the analysis of penalty options. It is demonstrated that strict site dependencies by tendency lead to monopolistic structures. In addition, particular penalty types show a positive effect regarding load imbalances caused by universally skilled vehicles.


Vehicle routing problem Transport Equilibria  Game theory 


  1. Archer A, Tardos E (2001) Truthful mechanisms for one-parameter agents. FOCS 2001:482–491Google Scholar
  2. Arsie A, Savla K, Frazzoli E (2009) Efficient routing algorithms for multiple vehicles with no explicit communications. IEEE Trans Autom Control 54(10):2302–2317CrossRefGoogle Scholar
  3. Bachem A, Hochstättler W, Malich M (1996) The simulated trading heuristic for solving vehicle routing problems. Disc Appl Math 65:47–72CrossRefGoogle Scholar
  4. Bell MGH (2004) Games, heuristics, and risk averseness in vehicle routing problems. J Urban Plan Dev 130(1):37–41CrossRefGoogle Scholar
  5. Cappanera P, Scutellà MG (2013) Home care optimization: impact of pattern generation policies on scheduling and routing decisions. Electron Notes Discret Math 41:53–60CrossRefGoogle Scholar
  6. Cappanera P, Gouveia L, Scutellà MG (2011) The skill vehicle routing problem. In: Pahl J, Reiners T, Voß S (eds) INOC 2011, vol 6701, Springer, Heidelberg, LNCS, pp 354–364Google Scholar
  7. Cappanera P, Gouveia L, Scutellà MG (2012) Models and valid inequalities to asymmetric skill-based routing problems. EURO J Transp Logist 2:29–55CrossRefGoogle Scholar
  8. Chao IM, Golden B, Wasil E (1999) A computational study of a new heuristic for the site-dependent vehicle routing problem. INFOR 37(3):319–336Google Scholar
  9. Cordeau JF, Laporte G (2001) A tabu search algorithm for the site dependent vehicle routing problem with time windows. INFOR 39(3):292–298Google Scholar
  10. Engevall S, Göthe-Lundgren M, Värbrand P (2004) The heterogeneous vehicle-routing game. Transp Sci 38(1):71–85CrossRefGoogle Scholar
  11. Figliozzi MA, Mahmassani HS (2007) Pricing in dynamic vehicle routing problems. Transp Sci 41(3):302–318CrossRefGoogle Scholar
  12. Göthe-Lundgren M, Jörnsten K, Värbrand P (1996) On the nucleolus of the basic vehicle routing game. Math Program 72:83–100CrossRefGoogle Scholar
  13. Hollander Y, Prashker JN (2006) The applicability of non-cooperative game theory in transport analysis. Transportation 33:481–496Google Scholar
  14. Mozafari M, Karimi B (2011) Pricing for freight carriers in a competitive environment: a game theory approach. Int J Ind Eng Comput 2:467–478Google Scholar
  15. Nadarajah S, Bookbinder J (2013) Less-than-truckload carrier collaboration problem: modeling framework and solution approach. J Heuristics 19:917–942CrossRefGoogle Scholar
  16. Nag B, Golden BL, Assad A (1988) Vehicle routing with site dependencies. In: Golden BL, Assad AA (eds) Vehicle routing: methods and studies. Elsevier Science Publishers B.V., North-Holland, pp 149–159Google Scholar
  17. Nash JF (1951) Non-cooperative games. Ann Math 54(2):286–295CrossRefGoogle Scholar
  18. Rekersbrink H, Makuschewitz T, Scholz-Reiter B (2009) A distributed routing concept for vehicle routing problems. Logist Res 1:45–52CrossRefGoogle Scholar
  19. Schwarze S, Voß S (2013) Improved load balancing and resource utilization for the skill vehicle routing problem. Optim Lett 7(8):1805–1823CrossRefGoogle Scholar
  20. Solomon MM (2013) VRPTW benchmark problems. Webpage: Accessed on 29 Aug 2013
  21. Wang X, Kopfer H (2014) Collaborative transportation planning of less-than-truckload freight. OR Spectr 36:357–380CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of Information SystemsUniversity of HamburgHamburgGermany

Personalised recommendations