Block seriation problems: A unified approach. Reply to the problem of H. Garcia and J. M. Proth (Applied Stochastic Models and Data Analysis, 1, (1), 25–34 (1985))

In their article, entitled ‘Group technology in production management, the short horizon planning level’, H. Garcia and J. M. Proth have stated the following problem: starting from a (0, 1) binary matrix of size (N x M), how to divide into independent subsets the rows of this matrix simultaneously with a one-to-one corresponding partition of the columns, maximizing the presence of 1s in the intersecting blocks with a joint minimization of the presence of 0s outside of these blocks. The authors have proposed an efficient and very fast heuristic algorithm in comparison with the existing methods of a fast-growing literature on the subject. The only drawback of this algorithm is its dependence on the initial partition. In this paper, we try to improve this algorithm slightly, first in rewriting the objective function in a linear form and secondly in giving computational improvements related to this linear formulation.