In this paper, we analyze a cooperative communication network with multi energy-harvesting and decode-and-forward relays in which the best relay is selected based on criteria such as Maximizing First-Hop Signal to Noise Ratios (SNRs) (MFHS protocol), Maximizing Second-Hop SNRs (MSHS protocol), and Maximizing End-to-End SNRs (MEES protocol). In these protocols, the relays apply power-splitting receivers to harvest energy from radio frequency signals emitted from a source. Thus, each received SNR in the second hop is a function of a direct relay-destination gain and an indirect source-relay gain. The system performance of the proposed protocols is evaluated via exact outage probability analyses and Monte Carlo simulations. For further comparisons, an energy-harvesting decode-and-forward scheme with randomly relay selection (RRS protocol) and an energy-harvesting amplify-and-forward scheme (BAF protocol) are investigated and discussed. The simulation results show that 1) the MEES protocol outperforms the MFHS and MSHS protocols, and the MFHS protocol is more efficient than the MSHS protocol in the low SNR regions; 2) the proposed protocols achieve the best performance at the specific optimal power splitting ratios for which the MEES protocol has a balanced ratio for energy harvesting and decoding capacity; and 3) the theoretical analyses agree well with the simulation results.