Skip to main content
SpringerLink
Log in

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

  • Original paper
  • Published:
Operational Research Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Allahdadi M, Nehi HM, Ashayerinasab HA, Javanmard M (2016) Improving the modified interval linear programming method by new techniques. Inf Sci 339:224–236

    Article  Google Scholar 

  • Asano T, Asano Y (2000) Recent developments in maximum flow algorithms. J Oper Res Soc Jpn 43:2–31

    Google Scholar 

  • Ashayerinasab HA, Nehi HM, Allahdadi M (2018) Solving the interval linear programming problem: a new algorithm for a general case. Expert Syst Appl 93:39–49

    Article  Google Scholar 

  • Aumann RJ, Maschler M (1964) The bargaining set for cooperative games. Adv Game Theory 52:443–476

    Google Scholar 

  • Bachrach Y, Rosenschein J (2009) Power in threshold network flow games. Auton Agent Multi-Agent Syst 18:106–132

    Article  Google Scholar 

  • 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 167:270–279

    Article  Google Scholar 

  • Banez-Chicharro F, Olmos L, Ramos A, Latorre JM (2017) Estimating the benefits of transmission expansion projects: an Aumann–Shapley approach. Energy 118:1044–1054

    Article  Google Scholar 

  • 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 53–63

  • Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99:351–376

    Article  Google Scholar 

  • Bernoulli D (1954) Exposition of a new theory on the measurement of risk. Econometrica 22:23–36

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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 19:523–532

    Article  Google Scholar 

  • Chakra MA, Traulsen A (2012) Evolutionary dynamics of strategic behavior in a collective-risk dilemma. PLoS Comput Biol 8:e1002652

    Article  Google Scholar 

  • Chakra MA, Traulsen A (2014) Under high stakes and uncertainty the rich should lend the poor a helping hand. J Theor Biol 341:123–130

    Article  Google Scholar 

  • Chanas S, Kolodziejczyk W (1986) Integer flows in network with fuzzy capacity constraints. Networks 16:17–31

    Article  Google Scholar 

  • Chanas S, Kołodziejczyk W (1982) Maximum flow in a network with fuzzy arc capacities. Fuzzy Sets Syst 8:165–173

    Article  Google Scholar 

  • Chanas S, Kołodziejczyk W (1984) Real-valued flows in a network with fuzzy arc capacities. Fuzzy Sets Syst 13:139–151

    Article  Google Scholar 

  • Charnes A, Cooper WW (1959) Chance-constrained programming. Manag Sci 6:73–79

    Article  Google Scholar 

  • Chen X, Szolnoki A, Perc M (2012a) Averting group failures in collective-risk social dilemmas. EPL (Europhys Lett) 99:68003

    Article  Google Scholar 

  • Chen X, Szolnoki A, Perc M (2012b) Risk-driven migration and the collective-risk social dilemma. Phys Rev E 86:036101

    Article  Google Scholar 

  • Conitzer V, Sandholm T (2006) Complexity of constructing solutions in the core based on synergies among coalitions. Artif Intell 170(6–7):607–619

    Article  Google Scholar 

  • Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1:197–206

    Article  Google Scholar 

  • Deng X, Papadimitriou CH (1994) On the complexity of cooperative solution concepts. Math Oper Res 19:257–266

    Article  Google Scholar 

  • Diamond P (2001) A fuzzy max-flow min-cut theorem. Fuzzy Sets Syst 119:139–148

    Article  Google Scholar 

  • Evans J (1976) Maximum flow in probabilistic graphs-the discrete case. Networks 6:161–183

    Article  Google Scholar 

  • Fishman GS (1987) The distribution of maximum flow with applications to multistate reliability systems. Oper Res 35:607–618

    Article  Google Scholar 

  • Ford LR, Fulkerson DR (1956) Maximal flow through a network. Can J Math 8:399–404

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Goldberg AV, Tarjan RE (1988) A new approach to the maximum-flow problem. J ACM (JACM) 35:921–940

    Article  Google Scholar 

  • Hafezalkotob A, Makui A (2015) Cooperative maximum-flow problem under uncertainty in logistic networks. Appl Math Comput 250:593–604

    Google Scholar 

  • Hammons TJ (2001) Electricity restructuring in Latin America systems with significant hydro generation. Rev Energ Ren, Power Eng, pp 39–48

    Google Scholar 

  • 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 40:462–468

    Article  Google Scholar 

  • Hauser OP, Rand DG, Peysakhovich A, Nowak MA (2014) Cooperating with the future. Nature 511:220–223

    Article  Google Scholar 

  • 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 46:1452–1468

    Article  Google Scholar 

  • Hilbe C, Chakra MA, Altrock PM, Traulsen A (2013) The evolution of strategic timing in collective-risk dilemmas. PloS ONE 8:e66490

    Article  Google Scholar 

  • 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 765:3253–3257

    Google Scholar 

  • 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 6:2704–2713

    Article  Google Scholar 

  • Jacquet J, Hagel K, Hauert C, Marotzke J, Röhl T, Milinski M (2013) Intra-and intergenerational discounting in the climate game Nature. Clim Change 3:1025–1028

    Google Scholar 

  • James J, Leonard J (2008) Fuzzy max-flow problem. Stud Fuzz Soft Comput 222:195–198

    Google Scholar 

  • Ji X, Yang L, Shao Z (2006) Chance constrained maximum flow problem with fuzzy arc capacities. Lect Notes Comput Sci 4114:11–19

    Article  Google Scholar 

  • Ju-Long D (1982) Control problems of grey systems. Syst Control Lett 1:288–294

    Article  Google Scholar 

  • Kalai E, Zemel E (1982a) Generalized network problems yielding totally balanced games. Oper Res 30:998–1008

    Article  Google Scholar 

  • Kalai E, Zemel E (1982b) Totally balanced games and games of flow. Math Oper Res 7:476–478

    Article  Google Scholar 

  • Koch T, Hiller B, Pfetsch, M E, Schewe L (2015) Evaluating gas network capacities. SIAM-MOS series on Optimization. SIAM 21:xvii + 364

    Google Scholar 

  • Kopustinskas V, Praks P (2015) Probabilistic gas transmission network simulator and application to the EU gas transmission system. J Polish Safe Reliab Assoc 6:71–78

    Google Scholar 

  • Leech D (2003) Computing power indices for large voting games. Manage Sci 49:831–837

    Article  Google Scholar 

  • Li DF, Ye YF (2018) Interval-valued least square prenucleolus of interval-valued cooperative games and a simplified method. Oper Res Int J 18:205–220

    Article  Google Scholar 

  • Lozano S, Moreno P, Adenso-Díaz B, Algaba E (2013) Cooperative game theory approach to allocating benefits of horizontal cooperation. Eur J Oper Res 229:444–452

    Article  Google Scholar 

  • Mallozzi L, Scalzo V, Tijs S (2011) Fuzzy interval cooperative games. Fuzzy Sets Syst 165:98–105

    Article  Google Scholar 

  • Mann I, Shapley LS (1960) Values of large games IV: evaluating the electoral college by montecarlo techniques. RAND Corporation

  • 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 105:2291–2294

    Article  Google Scholar 

  • Milinski M, Röhl T, Marotzke J (2011) Cooperative interaction of rich and poor can be catalyzed by intermediate climate targets. Clim Change 109:807–814

    Article  Google Scholar 

  • Minoux M (2009) On robust maximum flow with polyhedral uncertainty sets. Optim Lett 3:367–376

    Article  Google Scholar 

  • Minoux M (2010) Robust network optimization under polyhedral demand uncertainty is NP-hard. Discret Appl Math 158:597–603

    Article  Google Scholar 

  • Nagamochi H, Zeng D, Kabutoya N, Ibaraki T (1997) Complexity of the minimum base game on matroids. Math Oper Res 22:146–164

    Article  Google Scholar 

  • Nawathe S, Rao B (1980) Maximum flow in probabilistic communication networks. Int J Circuit Theory Appl 8:167–177

    Article  Google Scholar 

  • Nowak MA (2006) Five rules for the evolution of cooperation. Science 314:1560–1563

    Article  Google Scholar 

  • Ordóñez F, Zhao J (2007) Robust capacity expansion of network flows. Networks 50:136–145

    Article  Google Scholar 

  • Owen G (1972) Multilinear extensions of games. Manage Sci 18:64–79

    Article  Google Scholar 

  • 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 144:254–264

    Article  Google Scholar 

  • Quiggin J (2012) Generalized expected utility theory: the rank-dependent model. Springer, Berlin

    Google Scholar 

  • Raihani N, Aitken D (2011) Uncertainty, rationality and cooperation in the context of climate change. Clim Change 108:47–55

    Article  Google Scholar 

  • Ramesh G, Ganesan K (2011) Interval linear programming with generalized interval arithmetic. Int J Sci Eng Res 2:1–8

    Google Scholar 

  • Raub W, Snijders C (1997) Gains, losses, and cooperation in social dilemmas and collective action: the effects of risk preferences. J Math Sociol 22:263–302

    Article  Google Scholar 

  • Reyes PM (2005) Logistics networks: a game theory application for solving the transshipment problem. Appl Math Comput 168:1419–1431

    Google Scholar 

  • Rosenthal EC (2017) A cooperative game approach to cost allocation in a rapid-transit network. Transp Res B: Methodol 97:64–77

    Article  Google Scholar 

  • Sabater-Grande G, Georgantzis N (2002) Accounting for risk aversion in repeated prisoners’ dilemma games: an experimental test. J Econ Behav Organ 48:37–50

    Article  Google Scholar 

  • Santos FC, Pacheco JM (2011) Risk of collective failure provides an escape from the tragedy of the commons. Proc Natl Acad Sci 108:10421–10425

    Article  Google Scholar 

  • Schmeidler D (1969) The nucleolus of a characteristic function game SIAM. J Appl Math 17:1163–1170

    Google Scholar 

  • Sengupta A, Pal TK, Chakraborty D (2001) Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets Syst 119:129–138

    Article  Google Scholar 

  • Shaocheng T (1994) Interval number and fuzzy number linear programmings. Fuzzy Sets Syst 66:301–306

    Article  Google Scholar 

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

    Google Scholar 

  • Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787–792

    Article  Google Scholar 

  • Sheng Y, Gao J (2014) Chance distribution of the maximum flow of uncertain random network. J Uncert Anal Appl 2:1–15

    Article  Google Scholar 

  • Sohrabi MK, Azgomi H (2018) A survey on the combined use of optimization methods and game theory. Arch Comput Methods Eng. https://doi.org/10.1007/s11831-018-9300-5

    Article  Google Scholar 

  • Tang Z-Y, Tang B-B (2001) The maximum flow problem with fuzzy constraints. Fuzzy Syst Math 15:77–80

    Google Scholar 

  • 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 108:11825–11829

    Article  Google Scholar 

  • Tran TH, French S, Ashman R, Kent E (2018) Impact of compressor failures on gas transmission network capability. Appl Math Model 55:741–757

    Article  Google Scholar 

  • Tversky A, Kahneman D (1992) Advances in prospect theory: cumulative representation of uncertainty. J Risk Uncert 5:297–323

    Article  Google Scholar 

  • 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 38–65

  • Vasconcelos VV, Santos FC, Pacheco JM (2013) A bottom-up institutional approach to cooperative governance of risky commons Nature. Clim Change 3:797–801

    Google Scholar 

  • Vasconcelos VV, Santos FC, Pacheco JM, Levin SA (2014) Climate policies under wealth inequality. Proc Natl Acad Sci 111:2212–2216

    Article  Google Scholar 

  • Von Neumann J, Morgenstern O (2007) Theory of games and economic behavior. Princeton University Press, Princeton

    Google Scholar 

  • Wang X, Huang G (2014) Violation analysis on two-step method for interval linear programming. Inf Sci 281:85–96

    Article  Google Scholar 

  • Wang J, Fu F, Wang L (2010) Effects of heterogeneous wealth distribution on public cooperation with collective risk. Phys Rev E 82:016102

    Article  Google Scholar 

  • Zhou F, Huang GH, Chen GX, Guo HC (2009) Enhanced-interval linear programming. Eur J Oper Res 199:323–333

    Article  Google Scholar 

  • 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 432–437

  • Zou R, Liu Y, Liu L, Guo H (2009) REILP approach for uncertainty-based decision making in civil engineering. J Comput Civil Eng 24:357–364

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Faculty of Engineering, Dokuz Eylül University, Tinaztepe Campus, 35397, Buca, Izmir, Turkey

    Adil Baykasoğlu & Burcu Kubur Özbel

  2. The Graduate School of Natural and Applied Sciences, Dokuz Eylül University, Izmir, Turkey

    Burcu Kubur Özbel

Authors
  1. Adil Baykasoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Burcu Kubur Özbel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adil Baykasoğlu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baykasoğlu, A., Kubur Özbel, B. Explicit flow-risk allocation for cooperative maximum flow problems under interval uncertainty. Oper Res Int J 21, 2149–2179 (2021). https://doi.org/10.1007/s12351-019-00500-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12351-019-00500-5

Keywords

Search

Navigation