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Fuzzy relation system with max-min composition could be applied to describe the three-tier multimedia streaming architecture. In such an architecture, the regional servers supply their local multimedia streaming services to the clients. Random fault has not been considered in the relevant existing researches. However, the random fault of a regional server is possible to occur, or even unavoidable, due to some unpredictable reasons. In this article, we consider a random fault occurring at either one of the regional servers. With such a consideration, the corresponding architecture is characterized by the so-called variable-absent fuzzy relation inequalities system with max-min composition. Some properties of our proposed variable-absent fuzzy relation inequalities system are investigated. Based on such properties, we further propose a 2-D path approach for obtaining the complete solution of the variable-absent system. In addition, we also study a corresponding weighted minimax optimization problem motivated by the practical management objective. An effective algorithm is developed for searching the optimal solution, with an illustrative example.

In recent years, the addition-min fuzzy relation inequalities have been adopted to describe the flow constraint in a P2P network system. Each solution of the inequalities represents a feasible flow control scheme. Motivated by some different managerial objectives, several optimization problems subject to the addition-min inequalities have been recently studied. For example, considering the total efficiency for decreasing the network congestion, the linear objective function was adopted. While considering the fairness among the terminals, the min–max objective function was employed. In this article, combining these two objectives, we establish a corresponding bilevel fuzzy relation programming subject to the addition-min inequalities. For solving the bilevel programming, we first investigate some properties of the first-level programming, using the concept of fixed index set. Based on the properties of the first-level programming, our studied bilevel programming could be equivalently converted into a single-level programming and then solved by the existing linear programming approach. Our resolution approach is carried out by the fixed-index-set based algorithm and illustrated by some numerical examples.