Fuzzy relation inequalities about the data transmission mechanism in BitTorrent-like Peer-to-Peer file sharing systems

Min-Max Programming Problem Subject to Addition-Min Fuzzy Relation Inequalities

“…(See [1], [2]) Let x = (x 1 , x 2 , · · · , x n ) be a solution of (1), then x is a lower solution of (1) if and only if…”

“…Li and S.-J. Yang [1], [2] introduced addition-min fuzzy relation equalities and its related optimization problem with application background in the data transmission mechanism in BitTorrent-like Peer-to-Peer file sharing systems. As shown in the problem statement in Section 3, for avoiding the network congestion and improve the stability of data transmission, minimizing the biggest quality level, i.e.…”

“…According to Theorem 4, it is clear that one of the optimal solutions of (4) can be selected from the lower solutions of system (1). However, the objective function, i.e.…”

“…A solutionx ∈ X(A, b) is said to be a lower (or minimal) solution of system (1) if and only if x ≤x implies x =x for any x ∈ X(A, b). A solutioṅ x ∈ X(A, b) is said to be the minimum solution of system (1) if and only if x ≥ẋ for all x ∈ X(A, b).…”

“…Yang [1] introduced system of fuzzy relation inequalities with addition-min composition operator (addition-min fuzzy relation inequalities) for the first time. It was shown in [1] that the data transmission in BitTorrent-like Peer-to-Peer (P2P) file sharing systems might be reduced into a system of addition-min fuzzy relation inequalities, which was described as:          a 11 ∧ x 1 + a 12 ∧ x 2 + · · · + a 1n ∧ x n ≥ b 1 , a 21 ∧ x 1 + a 22 ∧ x 2 + · · · + a 2n ∧ x n ≥ b 2 , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · a m1 ∧ x 1 + a m2 ∧ x 2 + · · · + a mn ∧ x n ≥ b m .…”