Preprocessing on a diversity of tasks for a lexicographic bi-criteria line balancing problem with task relatives

A lexicographic bi-criteria line balancing problem with task relatives is considered, where the primary objective is to minimize the maximum machine load, and the second objective is to minimize the maximum degree of diverseness of assigned tasks to a machine. The precedence constraint is generally specified by a partial order of given tasks. The problem is NP-hard for the general case, while an O(n² max{m, r}) time algorithm has recently been presented for a special case in which the precedence constraint is specified by a total order of given tasks, where m denotes the number of available machines, n denotes the number of tasks to be processed, and r denotes the number of task relatives. In this paper, an O(n²) time preprocessing procedure is proposed to compute the degrees of diverseness for all possible O(n²) task subsequences in a total order of given tasks. The proposed preprocessing procedure improves the known time complexity O(n² max{m, r}) to O(mn²) for the special case, and it runs actually faster than the previous preprocessing one in numerical experiments.