Optimal Solution Methods for the Minimum-Backtracking Row Layout Problem

The problem of finding a reordering of the rows (and, simultaneously, the columns) of square matrices has important applications in the areas of combinatorial data analysis, economics and engineering. One such application is the Minimum-Backtracking Row Layout Problem (MBRLP), which involves the sequencing of workstations in an automated assembly line so as to minimize the total backtracking distance. Dynamic programming and integer programming have previously been proposed as optimal solution methods for the MBRLP; however, the latter method was limited to the case of equal distances between positions in the sequence. We demonstrate that the previously suggested integer programming approach for equidistant MBRLPs can produce infeasible solutions, and develop and test a plausible integer programming model for its replacement. We also present an optimal branch-and-bound procedure for MBRLP that requires far less memory storage than dynamic programming. Our experimental tests revealed that both dynamic programming and the branch-and-bound algorithm efficiently provided optimal solutions to MBRLPs with 25 or fewer workstations, and that neither method was superior in all cases. However, the branchand-bound algorithm was appreciably faster than dynamic programming for some data conditions, and can be used for problems where computer RAM limitations preclude dynamic programming implementation.