Reliability analysis is particularly relevant for industrial plants where plant failures can lead to large financial losses. Existing reliability analysis approaches mostly rely on heavyweight simulations that are computationally expensive and require extensive modeling effort. On the other hand, there is an industrial need for quickly evaluating plant reliability for developing new services and business models. In this paper, we extend and apply the reliability bound approach using linear programming to address this need. The reliability bound approach is based on a system model in the form of a graph, an event vector, and estimates for component reliabilities. Based on this model, lower and upper reliability bounds are calculated by solving a linear programming problem. The advantage of this approach is the ubiquity of solvers for linear programming. Furthermore, the approach is guaranteed to produce the narrowest bound with respect to the reliability data. We demonstrate the applicability of the approach to a subsystem of an industrial plant as a test case. Future work consists applying the method to whole plants and comparing the results with simulation-based approaches. Moreover, the approach is planned to be extended to system attributes such as buffers and multiple failure states.