Lagrangian Based Heuristics for the Linear Ordering Problem

Assume that the elements of a finite set N are to be ordered and that some associated precedence costs, {cij : i, j ∈ N, i = j}, are given. For any ordering of the elements of N and any pair of elements i, j ∈ N , whenever i is placed prior to j, one incurs a cost cij . Otherwise, a cost cji is incurred. The Linear Ordering Problem (LOP) is to find an ordering of the elements of N where the total sum of the costs incurred is as large as possible.