We propose linear programming (LP) models for attainable region (AR) analysis by considering a rate vector field in concentration space with an arbitrarily large number of points. One model provides a method to construct candidate ARs using a fully connected network of continuously stirred tank reactors (CSTRs) of arbitrary volume. More importantly, these methods are extended to derive linear programming conditions that are stronger necessary conditions than have proposed previously by Glasser and Hildebrandt. We state the LP condition as: No combination of nonzero volume CSTRs, operating at discretized points in the complement of the candidate AR, can extend the region. We demonstrate these proposed linear programming techniques on several two-dimensional reaction mechanisms and then apply the LP methods to verify extensions for a previously published three-dimensional candidate AR.