In this paper, the algorithm developed in \([RK]\) for \(2\)-color Ramsey numbers is generalized to multi-colored Ramsey numbers. All the cyclic graphs yielding the lower bounds \(R(3,3,4) \geq 30\), \(R(3,3,5) \geq 45\), and \(R(3,4,4) \geq 55\) were obtained. The two last bounds are apparently new.