In this study, a new $Δ$-evaluation method is introduced for solving a column permutation problem defined on a sparse binary matrix with the consecutive ones property. This problem models various $\mathcal{NP}$-hard problems in graph theory and industrial manufacturing contexts. The computational experiments compare the processing time of the $Δ$-evaluation method with two other methods used in well-known local search procedures. The study considers a comprehensive set of instances of well-known problems, such as Gate Matrix Layout and Minimization of Open Stacks. The proposed evaluation method is generally competitive and particularly useful for large and dense instances. It can be easily integrated into local search and metaheuristic algorithms to improve solutions without significantly increasing processing time.