Network reliability is extensively used to measure the degree of stability of the quality of infrastructure services. The performance of an infrastructure network and its components degrades over time in real situations. Multi-state reliability modeling that allows a finite number of different states for the performance of the network and its components is more appropriate for the reliability assessment, and provides a more realistic view of the network performance than the traditional binary reliability modeling. Due to the computational complexity of the enumerative methods in evaluating the multi-state reliability, the problem can be reduced to searching lower boundary points, and using them to evaluate reliability. Lower boundary points can be used to compute the exact reliability value and reliability bounds. We present an algorithm to search for lower boundary points. The proposed algorithm has considerable improvement in terms of computational efficiency by significantly reducing the number of iterations to obtain lower reliability bounds.