OR Spectrum

, Volume 38, Issue 1, pp 119–136 | Cite as

Transportation interval situations and related games

  • O. Palancı
  • S. Z. Alparslan Gök
  • M. O. Olgun
  • G.-W. Weber
Regular Article


Basically, uncertainty is present in almost every real-world situation, it is influencing and questioning our decisions. In this paper, we analyze transportation interval games corresponding to transportation interval situations. In those situations, it may affect the optimal amount of goods and consequently whether and how much of a product is transported from a producer to a retailer. Firstly, we introduce the interval Shapley value of a game arising from a transportation situation under uncertainty. Secondly, a one-point solution concept by using a one-stage producere depending on the proportional, the constrained equal awards and the constrained equal losses rule is given. We prove that transportation interval games are interval balanced (\(\mathcal {I}\)-balanced). Further, the nonemptiness of the interval core for the transportation interval games and some results on the relationship between the interval core and the dual interval optimal solutions of the underlying transportation situations are also provided. Moreover, we characterize the interval core using the square operator and addressing two scenarios such as pessimistic and optimistic.


Cooperative interval games Transportation games  Uncertainty Interval Shapley value Proportional rule 


  1. Alparslan Gök SZ, Branzei R, Tijs S (2008) Cores and stable sets for interval-valued games. In: CentER Discussion Paper, vol 17. Tilburg: Operations researchGoogle Scholar
  2. Alparslan Gök SZ, Branzei R, Tijs S (2009a) Convex interval games. J Appl Math Decis Sci (Article ID 342089)Google Scholar
  3. Alparslan Gök SZ, Miquel S, Tijs S (2009b) Cooperation under interval uncertainty. Math Methods Oper Res 69:99–109Google Scholar
  4. Alparslan Gök SZ, Branzei O, Branzei R, Tijs S (2011) Set-valued solution concepts using interval-type payoffs for interval games. J Math Econ 47:621–626CrossRefGoogle Scholar
  5. Aparicio J, Sánchez-Soriano J, Llorca N, Sancho J, Valero S (2010) Cooperative logistic games. In: Qiming Huang (ed) Game Theory. ISBN:978-953-307-132-9Google Scholar
  6. Aziz H, Cahan C, Gretton C, Kilby P, Mattei N, Walsh T (2014) A study of proxies for Shapley allocation of transport costs. J Artif Intell Res (Under review)Google Scholar
  7. Borm P, Hamers H, Hendrickx R (2001) Operations research games: a survey. TOP 9:139–216CrossRefGoogle Scholar
  8. Branzei R, Branzei O, Alparslan Gök SZ, Tijs S (2010a) Cooperative interval games: a survey. Cent Eur J Oper Res 18(3):397–411Google Scholar
  9. Branzei R, Tijs S, Alparslan Gök SZ (2010b) How to handle interval solutions for cooperative interval games. Int J Uncertain Fuzziness Knowl Based Syst 18(2):123–132Google Scholar
  10. Dantzig GB (1963) Linear programming and extensions, 10th edn. Princeton University Press, PrincetonGoogle Scholar
  11. Frisk M, Jörnsten K, Göthe-Lundgren M, Rönnqvist M (2010) Cost allocation in collaborative forest transportation. Eur J Oper Res 205:448–458CrossRefGoogle Scholar
  12. Hettich R, Kortanek KO (1993) Semi-infinite programming: theory, methods and applications. SIAM Rev 35:380–429CrossRefGoogle Scholar
  13. Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20:224–230CrossRefGoogle Scholar
  14. Llorca N, Molina E, Pulido M, Sánchez-Soriano J (2004) On the Owen set of transportation situations. Theory Decis 56:215–228CrossRefGoogle Scholar
  15. Owen G (1975) On the core of linear production games. Math Program 9:358–370CrossRefGoogle Scholar
  16. Özener OÖ, Ergun Ö (2008) Allocating costs in a collaborative transportation procurement network. Transp Sci 42(2):146–165CrossRefGoogle Scholar
  17. Pulido M, Sanchez-Soriano J, Llorca N (2002) Game theory techniques for university management: an extended bankruptcy model. Ann Oper Res 109:129–142CrossRefGoogle Scholar
  18. Pulido M, Borm P, Hendrickx R, Llorca N, Sanchez-Soriano J (2008) Compromise solutions for bankruptcy situations with references. Ann Oper Res 158:133–141CrossRefGoogle Scholar
  19. Sánchez-Soriano J (2003) The pairwise-egalitarian solution. Eur J Oper Res 150:220–231CrossRefGoogle Scholar
  20. Sánchez-Soriano J (2006) The pairwise solutions and the core of transportation situations. Eur J Oper Res 175:101–110CrossRefGoogle Scholar
  21. Sánchez-Soriano J, Llorca N, Meca A, Molina E, Pulido M (2002) An integrated transport system for Alacant’s students. UNIVERCITY. Ann Oper Res 109:41–60CrossRefGoogle Scholar
  22. Sánchez-Soriano J, López MA, García-Jurado I (2001) On the core of transportation games. Math Soc Sci 41:215–225CrossRefGoogle Scholar
  23. Schrijver A (1986) Theory of linear and integer programming. Wiley, ChichesterGoogle Scholar
  24. Shapley LS (1953) A value for \(n\)-person games. Ann Math Stud 28:307–317Google Scholar
  25. Shapley LS, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1:111–130CrossRefGoogle Scholar
  26. Soons D (2011) The determination and division of benefits among partners of a horizontal cooperation in transportation. Master’s thesis, TU/e School of Industrial Engineering, EindhovenGoogle Scholar
  27. Theys C, Dullaert W, Notteboom T (2008) Analyzing cooperative networks in intermodal transportation: a game-theoretic approach. In: Nectar Logistics and Freight Cluster Meeting, Delft, The NetherlandsGoogle Scholar
  28. Tijs S (2003) Introduction to game theory. Hindustan Book Agency, IndiaGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • O. Palancı
    • 1
  • S. Z. Alparslan Gök
    • 1
  • M. O. Olgun
    • 2
  • G.-W. Weber
    • 3
  1. 1.Department of Mathematics, Faculty of Arts and SciencesSüleyman Demirel UniversityIspartaTurkey
  2. 2.Faculty of Industrial EngineeringSüleyman Demirel UniversityIspartaTurkey
  3. 3.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

Personalised recommendations