Bond Energy, Rectilinear Distance and a Worst-case Bound for the Group Technology Problem

The machine-part relation in the group technology problem can be represented by a 0-1 matrix A where the rows represent the machines and the columns stand for the parts. The grouping of machines and parts into families is then equivalent to clustering the rows and the columns of A so that the resulting matrix may review some useful patterns of the original data. One frequently used objective function is the total ‘bond energy’ between the rows and the columns, which is a quadratic assignment problem formulation. We will show that this formulation is equivalent to solving two rectilinear travelling-salesman problems. On the basis of this observation, we propose a new approach to solve the group technology problem and establish a new worst-case bound for this problem.