Explicit flow-risk allocation for cooperative maximum flow problems under interval uncertainty

  • Adil BaykasoğluEmail author
  • Burcu Kubur Özbel
Original paper


Many decision-making problems in transportation networks can be defined as maximum flow problems. During the last five decades, several efficient solution approaches have been proposed for the deterministic maximum flow problems. On the other hand, arc capacities of networks cannot be precisely defined in many real life settings. These networks are called uncertain. In this case, it becomes challenging to maintain a stable flow on the network. This paper presents a new approach based on the framework of interval analysis for the solution of maximum flow problems. We address a multiple-owners network problem by presenting a risk explicit interval linear programming model for the desired value of the system aspiration level. Afterwards, we employ a well-known collaborative game theoretic approach (the Shapley value) in a multiple-owners network under interval uncertainty in order to solve the maximum flow problem. A detailed numerical example is provided to present the suitability of the proposed approach in devising a stable network flow. The obtained numerical results and the trade-offs between decision risk and network flow information would be very valuable for supporting decision makers in resolving maximum flow problems when facing uncertainty.


Maximum flow problem Multiple-owners graph Cooperative games Risk explicit interval linear programming Risk allocation The Shapley value 



Authors are grateful to the anonymous reviewers and editor for their helpful suggestions that concretely have contributed to ameliorate the paper from its original version.


  1. Allahdadi M, Nehi HM, Ashayerinasab HA, Javanmard M (2016) Improving the modified interval linear programming method by new techniques. Inf Sci 339:224–236CrossRefGoogle Scholar
  2. Asano T, Asano Y (2000) Recent developments in maximum flow algorithms. J Oper Res Soc Jpn 43:2–31Google Scholar
  3. Ashayerinasab HA, Nehi HM, Allahdadi M (2018) Solving the interval linear programming problem: a new algorithm for a general case. Expert Syst Appl 93:39–49CrossRefGoogle Scholar
  4. Aumann RJ, Maschler M (1964) The bargaining set for cooperative games. Adv Game Theory 52:443–476Google Scholar
  5. Bachrach Y, Rosenschein J (2009) Power in threshold network flow games. Auton Agent Multi-Agent Syst 18:106–132CrossRefGoogle Scholar
  6. Bai L, Li F, Cui H, Jiang T, Sun H, Zhu J (2016) Interval optimization based operating strategy for gas-electricity integrated energy systems considering demand response and wind uncertainty. Appl Energy 167:270–279CrossRefGoogle Scholar
  7. Banez-Chicharro F, Olmos L, Ramos A, Latorre JM (2017) Estimating the benefits of transmission expansion projects: an Aumann–Shapley approach. Energy 118:1044–1054CrossRefGoogle Scholar
  8. Baykasoğlu A, Özbel BK (2016) Cooperative interval game theory and the grey Shapley value approach for solving the maximum flow problem XIV. International Logistics and Supply Chain Congress Izmir Turkey ISBN:978-605-338-186-0, pp 53–63Google Scholar
  9. Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99:351–376CrossRefGoogle Scholar
  10. Bernoulli D (1954) Exposition of a new theory on the measurement of risk. Econometrica 22:23–36CrossRefGoogle Scholar
  11. Branzei R, Branzei O, Gök SZA, Tijs S (2010) Cooperative interval games: a survey. Cent Eur J Oper Res 18:397–411CrossRefGoogle Scholar
  12. Branzei R, Gök SA, Branzei O (2011) Cooperative games under interval uncertainty: on the convexity of the interval undominated cores. Cent Eur J Oper Res 19:523–532CrossRefGoogle Scholar
  13. Chakra MA, Traulsen A (2012) Evolutionary dynamics of strategic behavior in a collective-risk dilemma. PLoS Comput Biol 8:e1002652CrossRefGoogle Scholar
  14. Chakra MA, Traulsen A (2014) Under high stakes and uncertainty the rich should lend the poor a helping hand. J Theor Biol 341:123–130CrossRefGoogle Scholar
  15. Chanas S, Kolodziejczyk W (1986) Integer flows in network with fuzzy capacity constraints. Networks 16:17–31CrossRefGoogle Scholar
  16. Chanas S, Kołodziejczyk W (1982) Maximum flow in a network with fuzzy arc capacities. Fuzzy Sets Syst 8:165–173CrossRefGoogle Scholar
  17. Chanas S, Kołodziejczyk W (1984) Real-valued flows in a network with fuzzy arc capacities. Fuzzy Sets Syst 13:139–151CrossRefGoogle Scholar
  18. Charnes A, Cooper WW (1959) Chance-constrained programming. Manag Sci 6:73–79CrossRefGoogle Scholar
  19. Chen X, Szolnoki A, Perc M (2012a) Averting group failures in collective-risk social dilemmas. EPL (Europhys Lett) 99:68003CrossRefGoogle Scholar
  20. Chen X, Szolnoki A, Perc M (2012b) Risk-driven migration and the collective-risk social dilemma. Phys Rev E 86:036101CrossRefGoogle Scholar
  21. Conitzer V, Sandholm T (2006) Complexity of constructing solutions in the core based on synergies among coalitions. Artif Intell 170(6–7):607–619CrossRefGoogle Scholar
  22. Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1:197–206CrossRefGoogle Scholar
  23. Deng X, Papadimitriou CH (1994) On the complexity of cooperative solution concepts. Math Oper Res 19:257–266CrossRefGoogle Scholar
  24. Diamond P (2001) A fuzzy max-flow min-cut theorem. Fuzzy Sets Syst 119:139–148CrossRefGoogle Scholar
  25. Evans J (1976) Maximum flow in probabilistic graphs-the discrete case. Networks 6:161–183CrossRefGoogle Scholar
  26. Fishman GS (1987) The distribution of maximum flow with applications to multistate reliability systems. Oper Res 35:607–618CrossRefGoogle Scholar
  27. Ford LR, Fulkerson DR (1956) Maximal flow through a network. Can J Math 8:399–404CrossRefGoogle Scholar
  28. Frisk M, Göthe-Lundgren M, Jörnsten K, Rönnqvist M (2010) Cost allocation in collaborative forest transportation. Eur J Oper Res 205:448–458CrossRefGoogle Scholar
  29. Goldberg AV, Tarjan RE (1988) A new approach to the maximum-flow problem. J ACM (JACM) 35:921–940CrossRefGoogle Scholar
  30. Hafezalkotob A, Makui A (2015) Cooperative maximum-flow problem under uncertainty in logistic networks. Appl Math Comput 250:593–604Google Scholar
  31. Hammons TJ (2001) Electricity restructuring in Latin America systems with significant hydro generation. Rev Energ Ren, Power Eng, pp 39–48Google Scholar
  32. Han W-B, Sun H, Xu G-J (2012) A new approach of cooperative interval games: the interval core and Shapley value revisited. Oper Res Lett 40:462–468CrossRefGoogle Scholar
  33. Hauser OP, Rand DG, Peysakhovich A, Nowak MA (2014) Cooperating with the future. Nature 511:220–223CrossRefGoogle Scholar
  34. Heidari Gharehbolagh H, Hafezalkotob A, Makui A, Raissi S (2017) A cooperative game approach to uncertain decentralized logistic systems subject to network reliability considerations. Kybernetes 46:1452–1468CrossRefGoogle Scholar
  35. Hilbe C, Chakra MA, Altrock PM, Traulsen A (2013) The evolution of strategic timing in collective-risk dilemmas. PloS ONE 8:e66490CrossRefGoogle Scholar
  36. Huang YF, Ye WL, Zhou FF (2013) Research on the profit distribution of logistics company strategic alliance based on Shapley value. Adv Mater Res Trans Tech Publ 765:3253–3257Google Scholar
  37. Huang H, Li F, Mishra Y (2015) Modeling dynamic demand response using monte carlo simulation and interval mathematics for boundary estimation. IEEE Trans Smart Grid 6:2704–2713CrossRefGoogle Scholar
  38. Jacquet J, Hagel K, Hauert C, Marotzke J, Röhl T, Milinski M (2013) Intra-and intergenerational discounting in the climate game Nature. Clim Change 3:1025–1028CrossRefGoogle Scholar
  39. James J, Leonard J (2008) Fuzzy max-flow problem. Stud Fuzz Soft Comput 222:195–198Google Scholar
  40. Ji X, Yang L, Shao Z (2006) Chance constrained maximum flow problem with fuzzy arc capacities. Lect Notes Comput Sci 4114:11–19CrossRefGoogle Scholar
  41. Ju-Long D (1982) Control problems of grey systems. Syst Control Lett 1:288–294CrossRefGoogle Scholar
  42. Kalai E, Zemel E (1982a) Generalized network problems yielding totally balanced games. Oper Res 30:998–1008CrossRefGoogle Scholar
  43. Kalai E, Zemel E (1982b) Totally balanced games and games of flow. Math Oper Res 7:476–478CrossRefGoogle Scholar
  44. Koch T, Hiller B, Pfetsch, M E, Schewe L (2015) Evaluating gas network capacities. SIAM-MOS series on Optimization. SIAM 21:xvii + 364Google Scholar
  45. Kopustinskas V, Praks P (2015) Probabilistic gas transmission network simulator and application to the EU gas transmission system. J Polish Safe Reliab Assoc 6:71–78Google Scholar
  46. Leech D (2003) Computing power indices for large voting games. Manage Sci 49:831–837CrossRefGoogle Scholar
  47. Li DF, Ye YF (2018) Interval-valued least square prenucleolus of interval-valued cooperative games and a simplified method. Oper Res Int J 18:205–220CrossRefGoogle Scholar
  48. Lozano S, Moreno P, Adenso-Díaz B, Algaba E (2013) Cooperative game theory approach to allocating benefits of horizontal cooperation. Eur J Oper Res 229:444–452CrossRefGoogle Scholar
  49. Mallozzi L, Scalzo V, Tijs S (2011) Fuzzy interval cooperative games. Fuzzy Sets Syst 165:98–105CrossRefGoogle Scholar
  50. Mann I, Shapley LS (1960) Values of large games IV: evaluating the electoral college by montecarlo techniques. RAND CorporationGoogle Scholar
  51. Milinski M, Sommerfeld RD, Krambeck HJ, Reed FA, Marotzke J (2008) The collective-risk social dilemma and the prevention of simulated dangerous climate change. Proc Natl Acad Sci 105:2291–2294CrossRefGoogle Scholar
  52. Milinski M, Röhl T, Marotzke J (2011) Cooperative interaction of rich and poor can be catalyzed by intermediate climate targets. Clim Change 109:807–814CrossRefGoogle Scholar
  53. Minoux M (2009) On robust maximum flow with polyhedral uncertainty sets. Optim Lett 3:367–376CrossRefGoogle Scholar
  54. Minoux M (2010) Robust network optimization under polyhedral demand uncertainty is NP-hard. Discret Appl Math 158:597–603CrossRefGoogle Scholar
  55. Nagamochi H, Zeng D, Kabutoya N, Ibaraki T (1997) Complexity of the minimum base game on matroids. Math Oper Res 22:146–164CrossRefGoogle Scholar
  56. Nawathe S, Rao B (1980) Maximum flow in probabilistic communication networks. Int J Circuit Theory Appl 8:167–177CrossRefGoogle Scholar
  57. Nowak MA (2006) Five rules for the evolution of cooperation. Science 314:1560–1563CrossRefGoogle Scholar
  58. Ordóñez F, Zhao J (2007) Robust capacity expansion of network flows. Networks 50:136–145CrossRefGoogle Scholar
  59. Owen G (1972) Multilinear extensions of games. Manage Sci 18:64–79CrossRefGoogle Scholar
  60. Praks P, Kopustinskas V, Masera M (2015) Probabilistic modelling of security of supply in gas networks and evaluation of new infrastructure. Reliab Eng Syst Safe 144:254–264CrossRefGoogle Scholar
  61. Quiggin J (2012) Generalized expected utility theory: the rank-dependent model. Springer, BerlinGoogle Scholar
  62. Raihani N, Aitken D (2011) Uncertainty, rationality and cooperation in the context of climate change. Clim Change 108:47–55CrossRefGoogle Scholar
  63. Ramesh G, Ganesan K (2011) Interval linear programming with generalized interval arithmetic. Int J Sci Eng Res 2:1–8Google Scholar
  64. Raub W, Snijders C (1997) Gains, losses, and cooperation in social dilemmas and collective action: the effects of risk preferences. J Math Sociol 22:263–302CrossRefGoogle Scholar
  65. Reyes PM (2005) Logistics networks: a game theory application for solving the transshipment problem. Appl Math Comput 168:1419–1431Google Scholar
  66. Rosenthal EC (2017) A cooperative game approach to cost allocation in a rapid-transit network. Transp Res B: Methodol 97:64–77CrossRefGoogle Scholar
  67. Sabater-Grande G, Georgantzis N (2002) Accounting for risk aversion in repeated prisoners’ dilemma games: an experimental test. J Econ Behav Organ 48:37–50CrossRefGoogle Scholar
  68. Santos FC, Pacheco JM (2011) Risk of collective failure provides an escape from the tragedy of the commons. Proc Natl Acad Sci 108:10421–10425CrossRefGoogle Scholar
  69. Schmeidler D (1969) The nucleolus of a characteristic function game SIAM. J Appl Math 17:1163–1170Google Scholar
  70. Sengupta A, Pal TK, Chakraborty D (2001) Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets Syst 119:129–138CrossRefGoogle Scholar
  71. Shaocheng T (1994) Interval number and fuzzy number linear programmings. Fuzzy Sets Syst 66:301–306CrossRefGoogle Scholar
  72. Shapley LS (1953) A value for n-person games. Ann Math Stud 28:307–317Google Scholar
  73. Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787–792CrossRefGoogle Scholar
  74. Sheng Y, Gao J (2014) Chance distribution of the maximum flow of uncertain random network. J Uncert Anal Appl 2:1–15CrossRefGoogle Scholar
  75. Sohrabi MK, Azgomi H (2018) A survey on the combined use of optimization methods and game theory. Arch Comput Methods Eng. Google Scholar
  76. Tang Z-Y, Tang B-B (2001) The maximum flow problem with fuzzy constraints. Fuzzy Syst Math 15:77–80Google Scholar
  77. Tavoni A, Dannenberg A, Kallis G, Löschel A (2011) Inequality, communication, and the avoidance of disastrous climate change in a public goods game. Proc Natl Acad Sci 108:11825–11829CrossRefGoogle Scholar
  78. Tran TH, French S, Ashman R, Kent E (2018) Impact of compressor failures on gas transmission network capability. Appl Math Model 55:741–757CrossRefGoogle Scholar
  79. Tversky A, Kahneman D (1992) Advances in prospect theory: cumulative representation of uncertainty. J Risk Uncert 5:297–323CrossRefGoogle Scholar
  80. Van Assen M, Snijders C (2004) Effects of risk preferences in social dilemmas: a game-theoretical analysis and evidence from two experiments. Contemp Psychol Res Soc Dilemmas 38–65Google Scholar
  81. Vasconcelos VV, Santos FC, Pacheco JM (2013) A bottom-up institutional approach to cooperative governance of risky commons Nature. Clim Change 3:797–801CrossRefGoogle Scholar
  82. Vasconcelos VV, Santos FC, Pacheco JM, Levin SA (2014) Climate policies under wealth inequality. Proc Natl Acad Sci 111:2212–2216CrossRefGoogle Scholar
  83. Von Neumann J, Morgenstern O (2007) Theory of games and economic behavior. Princeton University Press, PrincetonGoogle Scholar
  84. Wang X, Huang G (2014) Violation analysis on two-step method for interval linear programming. Inf Sci 281:85–96CrossRefGoogle Scholar
  85. Wang J, Fu F, Wang L (2010) Effects of heterogeneous wealth distribution on public cooperation with collective risk. Phys Rev E 82:016102CrossRefGoogle Scholar
  86. Zhou F, Huang GH, Chen GX, Guo HC (2009) Enhanced-interval linear programming. Eur J Oper Res 199:323–333CrossRefGoogle Scholar
  87. Zlotkin G, Rosenschein J (1994) Coalition, cryptography, and stability: mechanisms foe coalition formation in task oriented domains. In: Proceedings of the National Conference on Artificial Intelligence (AAAI-94), pp 432–437Google Scholar
  88. Zou R, Liu Y, Liu L, Guo H (2009) REILP approach for uncertainty-based decision making in civil engineering. J Comput Civil Eng 24:357–364CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Industrial Engineering, Faculty of EngineeringDokuz Eylül UniversityBucaTurkey
  2. 2.The Graduate School of Natural and Applied SciencesDokuz Eylül UniversityIzmirTurkey

Personalised recommendations