On the Stability of Coalitions in Supply Chain Networks via Generalized Complementarity Conditions

  • Laura ScrimaliEmail author


In this paper, we consider a supply chain network model that consists of three layers of decision-makers, namely, suppliers, manufacturers and retailers, with prices and shipments that evolve in time. We focus on the vertical integration of the levels of the supply chain and consider the retailer as the dominant player of the coalition. We give a novel reformulation of the evolutionary variational inequality related to the equilibrium conditions underlying the model. This approach, based on complementarity conditions, allows us to analyze the Lagrange multipliers associated with production capacities to understand better market’s trend. We study the behavior of marginal profits and provide some results for the efficiency and the stability of the coalitions. We apply our theoretical achievements to a duopolistic model.


SI: Evolutionary variational inequality Equilibrium conditions Lagrange multipliers 



The research was partially supported by the research project “Modelli Matematici nell’Insegnamento-Apprendimento della Matematica” DMI, University of Catania. This support is gratefully acknowledged.


  1. Amoozad Mahdiraji H, Govindan K, Zavadskas EK, Razavi Hajiagha SH (2014) Coalition or decentralization: a game-theoretic analysis of a three-echelon supply chain network. J Business Econ Manag 15(3):460–485CrossRefGoogle Scholar
  2. Barbagallo A, Daniele P, Giuffrè S, Maugeri A (2014) Variational approach for a general financial equilibrium problem: the deficit formula, the balance law and the liability formula a path to the economy recovery. Eur J Oper Res 237 (1):231–244CrossRefGoogle Scholar
  3. Daniele P, Maugeri A, Nagurney A (2017) Cybersecurity investments with nonlinear budget constraints: analysis of the marginal expected utilities. In: Daras N, Rassias T (eds) Operations research, engineering, and cyber security. Springer optimization and its applications, vol 113. Springer, ChamGoogle Scholar
  4. Daniele P, Giuffrè S, Lorino M (2016) Functional inequalities, regularity and computation of the deficit and surplus variables in the financial equilibrium problem. J Global Optimization 65(3):575–596CrossRefGoogle Scholar
  5. Daniele P, Giuffrè S, Idone G, Maugeri A (2007) Infinite dimensional duality and applications. Math Ann 339:221–239CrossRefGoogle Scholar
  6. Ding D, Chen J (2008) Coordinating a three level supply chain with flexible return policies. Omega 36(5):865–876CrossRefGoogle Scholar
  7. Giuffrè S, Maugeri A, Puglisi D (2015) Lagrange multipliers in elastic-plastic torsion problem for nonlinear monotone operators. J Diff Equation 259(3):817–837CrossRefGoogle Scholar
  8. He Y, Zhao X (2012) Coordination in multi-echelon supply chain under supply and demand uncertainty. Int J Production Economics 139(1):106–115CrossRefGoogle Scholar
  9. Jahn J (1996) Introduction to the theory of nonlinear optimization. Springer, BerlinCrossRefGoogle Scholar
  10. Lin CC, Hsieh CC (2012) A cooperative coalitional game in duopolistic Supply-Chain competition. Netw Spat Econ 12(1):129–146CrossRefGoogle Scholar
  11. Maugeri A (1987) Convex programming,variational inequalities and applications to the traffic equilibrium problem. Appl Math Optim 16:169–185CrossRefGoogle Scholar
  12. Maugeri A, Raciti F (2010) Remarks on infinite dimensional duality. J Global Optim 46(4):581–588CrossRefGoogle Scholar
  13. Maugeri A, Raciti F (2009) On existence theorems for monotone and nonmonotone variational inequalities. J Convex Anal 16(3-4):899–911Google Scholar
  14. Mirabella C, Scrimali L (2018) Cooperation in pollution control problems via evolutionary variational inequalities. J Global Optim 70:455–476CrossRefGoogle Scholar
  15. Oggioni G, Smeers Y, Allevi Em Schaible S (2012) A generalized nash equilibrium model of market coupling in the european power system. Netw Spat Econ 12(4):503–560CrossRefGoogle Scholar
  16. Scrimali L (2012) Infinite-dimensional duality theory applied to the study of investment strategies in Kyoto Protocol. J Optim Theory Appl 154:258–277CrossRefGoogle Scholar
  17. Scrimali L (2018) Coalitional games in evolutionary supply chain networks. In: Daniele P, Scrimali L (eds) Optimization and decision science: new trends in emerging complex real life problems, SpringerGoogle Scholar
  18. Shahabi M, Akbarinasaji S, Unnikrishnan A, James R (2013) Integrated inventory control and facility location decisions in a Multi-Echelon supply chain network with hubs. Netw Spat Econ 13(4):497–514CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversity of CataniaCataniaItaly

Personalised recommendations