Methodological Foundations

  • Anna Nagurney
  • Dong Li
Part of the Springer Series in Supply Chain Management book series (SSSCM, volume 2)


In this chapter, we set the groundwork for the understanding and application of the methodological tools that are utilized for the supply chain network models with quality competition in this book. We first overview the basics of variational inequality theory and the connections with optimization. We provide conditions for existence and uniqueness of solutions, along with the definitions of the essential properties. We relate the variational inequality problem to game theory since game theory models are developed throughout this book in order to formulate competition among supply chain network decision-makers. In addition, we recall the fundamentals of projected dynamical systems theory and the relationships with variational inequality theory in order to enable the description of dynamic interactions among decision-makers in supply chains. For completeness, we also provide results on stability analysis. We discuss some fundamentals of multicriteria decision-making since supply chain decision-makers may be faced with multiple criteria, even conflicting ones, that they wish to optimize. Finally, we present algorithms that are used for solving the supply chain network models with quality competition.


Supply Chain Nash Equilibrium Variational Inequality Euler Method Variational Inequality Problem 
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  1. Cohon, J. L. (1978). Multiobjective programming and planning. New York: Academic.Google Scholar
  2. Dhanda, K. K., Nagurney, A., & Ramanujam, P. (1999). Environmental networks: A framework for economic decision-making and policy analysis. Cheltenham: Edward Elgar Publishing.Google Scholar
  3. Dupuis, P., & Nagurney, A. (1993). Dynamical systems and variational inequalities. Annals of Operations Research, 44, 9–42.CrossRefGoogle Scholar
  4. Gabay, D., & Moulin, H. 1980. On the uniqueness and stability of Nash equilibria in noncooperative games. In A. Bensoussan, P. Kleindorfer, & C. S. Tapiero (Eds.), Applied stochastic control in econometrics and management science (pp. 271–294). Amsterdam: North-Holland.Google Scholar
  5. Gal, T., Stewart, T. J., & Hanne, T. (1999). Multicriteria decision making: Advances in MCDM models, algorithms, theory and applications. Boston: Kluwer Academic.CrossRefGoogle Scholar
  6. Hartman, P., & Stampacchia, G. (1966). On some nonlinear elliptic differential functional equations. Acta Mathematica, 115, 271–310.CrossRefGoogle Scholar
  7. Jones, D. F., Mirrazavi, S. K., & Tamiz, M. (2002). Multi-objective meta-heuristics: An overview of the current state-of-the-art. European Journal of Operational Research, 137(1), 1–9.CrossRefGoogle Scholar
  8. Karamardian, S. (1969). The nonlinear complementarity problem with applications, Part 1. Journal of Optimization Theory and Applications, 4, 87–98.CrossRefGoogle Scholar
  9. Keeney, R. L., & Raffa, H. (1976). Decisions with multiple objectives. New York: Wiley.Google Scholar
  10. Korpelevich, G. M. (1977). The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49.Google Scholar
  11. Marler, R. T., & Arora, J. S. (2004). Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization, 26(6), 369–395.CrossRefGoogle Scholar
  12. Mullon, C. (2014). Network economics of marine ecosystems and their exploitation. Boca Raton, FL: CRC.Google Scholar
  13. Nagurney, A. (1999). Network economics: A variational inequality approach (2nd and Rev. ed.). Boston: Kluwer Academic.CrossRefGoogle Scholar
  14. Nagurney, A. (2006). Supply chain network economics: Dynamics of prices, flows, and profits. Cheltenham: Edward Elgar Publishing.Google Scholar
  15. Nagurney, A., & Dong, J. (2002). Supernetworks: Decision-making for the information age. Cheltenham: Edward Elgar Publishing.Google Scholar
  16. Nagurney, A., & Siokos, S. (1997). Financial networks: Statics and dynamics. Berlin: Springer.CrossRefGoogle Scholar
  17. Nagurney, A., & Zhang, D. (1996). Projected dynamical systems and variational inequalities with applications. Boston: Kluwer Academic.CrossRefGoogle Scholar
  18. Nagurney, A., Yu, M., Masoumi, A. H., & Nagurney, L. S. (2013). Networks against time: Supply chain analytics for perishable products. New York: Springer.CrossRefGoogle Scholar
  19. Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49.Google Scholar
  20. Nash, J. F. (1951). Noncooperative games. Annals of Mathematics, 54, 286–298.CrossRefGoogle Scholar
  21. Pareto, V. (1971). Manual of political economy. New York: Augustus M. Kelley Publishers.Google Scholar
  22. Patriksson, M. (2015). The traffic assignment problem: Models and methods. New York: Courier Dover Publications.Google Scholar
  23. Ran, B., & Boyce, D. E. (1996). Modeling dynamic transportation networks. Berlin: Springer.CrossRefGoogle Scholar
  24. Rosen, J. B. (1965). Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3), 520–533.CrossRefGoogle Scholar
  25. Triantaphyllou, E. (2000). Multi-criteria decision making methods: A comparative study. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  26. Zadeh, L. A. (1963). Optimality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control, 8(1), 59–60.CrossRefGoogle Scholar
  27. Zhang, D., & Nagurney, A. (1995). On the stability of projected dynamical systems. Journal of Optimization Theory and Its Applications, 85, 97–124.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Anna Nagurney
    • 1
  • Dong Li
    • 2
  1. 1.Isenberg School of ManagementUniversity of MassachusettsAmherstUSA
  2. 2.Department of Management and Marketing College of BusinessArkansas State UniversityState UniversityUSA

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