On weighting two criteria with a parameter in combinatorial optimization problems

Two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1. For special problem types, e.g., when one of the criteria is a bottleneck value, efficient algorithms are known that solve for a given value of the weighting parameter. We transform the underlying enumeration method into a parametric algorithm solving simultaneously for all values of the weighting parameter. Efficient implementations are presented for combinatorial problems with criteria as balanced optimization, min-sum with min-max, and min-sum with balanced optimization, considering the spanning tree, the linear assignment and the single machine scheduling problem. Further the new algorithmic scheme can easily incorporate a trade-off of the criteria by means of penalty functions, again without consequences for the algorithm and its complexity order. (C) 2012 Elsevier B.V. All rights reserved.