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Transportation interval situations and related games

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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.

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  • Alparslan Gök SZ, Branzei R, Tijs S (2008) Cores and stable sets for interval-valued games. In: CentER Discussion Paper, vol 17. Tilburg: Operations research

  • Alparslan Gök SZ, Branzei R, Tijs S (2009a) Convex interval games. J Appl Math Decis Sci (Article ID 342089)

  • Alparslan Gök SZ, Miquel S, Tijs S (2009b) Cooperation under interval uncertainty. Math Methods Oper Res 69:99–109

  • Alparslan Gök SZ, Branzei O, Branzei R, Tijs S (2011) Set-valued solution concepts using interval-type payoffs for interval games. J Math Econ 47:621–626

    Article  Google Scholar 

  • Aparicio J, Sánchez-Soriano J, Llorca N, Sancho J, Valero S (2010) Cooperative logistic games. In: Qiming Huang (ed) Game Theory. ISBN:978-953-307-132-9

  • 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)

  • Borm P, Hamers H, Hendrickx R (2001) Operations research games: a survey. TOP 9:139–216

    Article  Google Scholar 

  • Branzei R, Branzei O, Alparslan Gök SZ, Tijs S (2010a) Cooperative interval games: a survey. Cent Eur J Oper Res 18(3):397–411

  • 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):123–132

  • Dantzig GB (1963) Linear programming and extensions, 10th edn. Princeton University Press, Princeton

    Google Scholar 

  • Frisk M, Jörnsten K, Göthe-Lundgren M, Rönnqvist M (2010) Cost allocation in collaborative forest transportation. Eur J Oper Res 205:448–458

    Article  Google Scholar 

  • Hettich R, Kortanek KO (1993) Semi-infinite programming: theory, methods and applications. SIAM Rev 35:380–429

    Article  Google Scholar 

  • Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20:224–230

    Article  Google Scholar 

  • Llorca N, Molina E, Pulido M, Sánchez-Soriano J (2004) On the Owen set of transportation situations. Theory Decis 56:215–228

    Article  Google Scholar 

  • Owen G (1975) On the core of linear production games. Math Program 9:358–370

    Article  Google Scholar 

  • Özener OÖ, Ergun Ö (2008) Allocating costs in a collaborative transportation procurement network. Transp Sci 42(2):146–165

    Article  Google Scholar 

  • Pulido M, Sanchez-Soriano J, Llorca N (2002) Game theory techniques for university management: an extended bankruptcy model. Ann Oper Res 109:129–142

    Article  Google Scholar 

  • Pulido M, Borm P, Hendrickx R, Llorca N, Sanchez-Soriano J (2008) Compromise solutions for bankruptcy situations with references. Ann Oper Res 158:133–141

    Article  Google Scholar 

  • Sánchez-Soriano J (2003) The pairwise-egalitarian solution. Eur J Oper Res 150:220–231

    Article  Google Scholar 

  • Sánchez-Soriano J (2006) The pairwise solutions and the core of transportation situations. Eur J Oper Res 175:101–110

    Article  Google Scholar 

  • Sánchez-Soriano J, Llorca N, Meca A, Molina E, Pulido M (2002) An integrated transport system for Alacant’s students. UNIVERCITY. Ann Oper Res 109:41–60

    Article  Google Scholar 

  • Sánchez-Soriano J, López MA, García-Jurado I (2001) On the core of transportation games. Math Soc Sci 41:215–225

    Article  Google Scholar 

  • Schrijver A (1986) Theory of linear and integer programming. Wiley, Chichester

    Google Scholar 

  • Shapley LS (1953) A value for \(n\)-person games. Ann Math Stud 28:307–317

    Google Scholar 

  • Shapley LS, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1:111–130

    Article  Google Scholar 

  • 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, Eindhoven

  • 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 Netherlands

  • Tijs S (2003) Introduction to game theory. Hindustan Book Agency, India

    Google Scholar 

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Correspondence to S. Z. Alparslan Gök.

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Palancı, O., Alparslan Gök, S.Z., Olgun, M.O. et al. Transportation interval situations and related games. OR Spectrum 38, 119–136 (2016).

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