Annals of Operations Research

, Volume 264, Issue 1–2, pp 89–121 | Cite as

A variational inequality formulation for designing a multi-echelon, multi-product supply chain network in a competitive environment

  • Isa Feyzian-Tary
  • Jafar Razmi
  • Mohamad Sadegh Sangari
Original Research


In a competitive environment, supply chains are competing with each other to gain the market share and competition is a critical factor influencing the supply chain network structure. The current paper presents a variational inequality formulation and provides the results for a competitive supply chain network design model. The new-entrant supply chain competes against an existing one in a non-cooperative behavior. The networks include raw material suppliers, manufacturers, retailers, and the same demand markets. The manufacturers produce multiple products with deterministic, price-dependent demand. The goal is to maximize the future revenue of both chains. The problem is modeled by mathematical programming and the governing Nash equilibrium conditions are derived. Then, a finite-dimensional variational inequality formulation is presented to solve the equilibrium problem. Qualitative properties of the equilibrium pattern are provided to establish existence and uniqueness results under reasonable conditions. The modified projection algorithm is used to solve the variational inequality problem. A numerical example is presented in order to show the efficiency of the proposed model and to investigate the behavior of the model under different conditions.


Supply chain network design (SCND) Variational inequality Nash equilibrium Multi-product Dynamic competition Modified projection method 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Isa Feyzian-Tary
    • 1
  • Jafar Razmi
    • 1
  • Mohamad Sadegh Sangari
    • 2
  1. 1.School of Industrial Engineering, College of EngineeringUniversity of TehranTehranIran
  2. 2.Faculty of Fouman, College of EngineeringUniversity of TehranFoumanIran

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