A note on supply chain network equilibrium models

This paper is concerned with the supply chain network equilibrium models proposed by Nagurney et al. [Nagurney, A., Dong, J., Zhang, D., 2002. A supply chain network equilibrium model. Transportation Research 38E, 281-303] and Dong et al. [Dong, J., Zhang, D., Nagurney, A., 2004. A supply chain network equilibrium model with random demands. European Journal of Operational Research 156, 194-212]. It demonstrates that these models possess the unconstrained continuously differentiable minimization formulations, whose any stationary point is the solution of the corresponding model. Accordingly, not only is the Quasi-Newton algorithm capable of finding a solution of the model, but also it can overcome the difficulty experienced by the modified projection method in choosing an appropriate predetermined step size. In addition, 11 benchmark examples are employed to show the advantage of the unconstrained minimization formulation.