Private blockchains tend to apply deterministic con-sensus mechanisms as a more efficient alternative to the proof-based consensus. Deterministic mechanisms tolerate two types of failures, byzantine and crash-fault. Byzantine fault-tolerant consensus assumes restrictive assumptions of time and number of failures to guarantee validity, while the termination depends on message broadcasting among nodes. Crash-Fault tolerant consensus is less rigorous to ensure termination and higher throughput while sacrificing agreement. This paper proposes a lightweight consensus mechanism based on vicinity voting with confirmed message broadcasting. Formation rules in the neighborhoods of the peer-to-peer network relax the trade-off between agreement and termination. Experimental results show that the proposal guarantees agreement and termination in case of more permissive formation rules. Besides, the cost of achieving consensus is reduced by up to 46% in more rigorous formation rules with limited impact on termination and agreement.