This paper attempts to optimize the flow patterns in a perishable food supply chain network for a high-speed rail catering service. The proposed variational inequality models describe the uncertain demand on trains using the Newsvendor model and impose time deadline constraints on paths considering flow-dependent lead time. The constraints are then reformulated based on the Dirac delta function so that they can be directly dualized. An Euler algorithm with an Augmented Lagrangian Dual algorithm is developed to solve the model. A case study using 246 trains in the Beijing-Shanghai high-speed corridor is applied to demonstrate the applicability of the method.@en