To the best of our knowledge, there have been very few work on computational algorithms for the core or its variants in MC-nets games. One exception is the work by [Hirayama, et.al., 2014], where a constraint generation algorithm has been proposed to compute a payoff vector belonging to the least core. In this paper, we generalize this algorithm into the one for finding a payoff vector belonging to ϵ-core with pre-specified bound guarantee. The underlying idea behind this algorithm is basically the same as the previous one, but one key contribution is to give a clearer view on the pricing problem leading to the development of our new general algorithm. We showed that this new algorithm was correct and never be trapped in an infinite loop. Furthermore, we empirically demonstrated that this algorithm really presented a trade-off between solution quality and computational costs on some benchmark instances.