<p>The random-term-absent (RTA) fuzzy relation inequalities with addition-min composition have been introduced for modeling the Peer-to-Peer network system with a random line fault recently. Motivated by such an application background, the min-max programming subject to the RTA addition-min system is established to reduce the network congestion. In fact, any optimal solution of the min-max programming problem represents an optimal flow control scheme in the Peer-to-Peer network system. The major contribution of this work is developing the inverse-function-based resolution method for this min-max programming problem. We first construct a single-variable function with addition-min composition and then study its segmented function and its inverse function. Some properties of the inverse function was further investigated. Based on these properties, the min-max programming problem is converted into a single-variable optimization problem and solved. We design some detailed resolution procedures for the min-max programming. Moreover, a numerical example is provided for illustrating our proposed resolution procedures.</p>