A Multiplier Adjustment Approach for the Set Partitioning Problem

Abstract

We introduce an effective branch-and-bound algorithm for solving the set partitioning problem. The new algorithm employs a new multiplier-adjustment-based bounding procedure, and a complementary branching strategy which results in relatively small search trees. Computational results based on 20 moderately sized crew scheduling problems indicate that our new algorithm is on average 16.6 times faster than the popular code, SETPAR. The improvements are mainly due to the bounding procedure, which is fast, easy to use, and provides tight lower bounds. On average, the bounds are 97.6% of the optimal objective value of the linear programming relaxation after only five iterations, and 98.5% after ten iterations. Moreover, the lower bounds are observed to be monotonically nondecreasing. We also apply the technique of variable elimination, which is very effective in reducing the size of the problems. On average, 89% of the variables are eliminated from the problem at the root node.

Optimal consolidation of municipalities: An analysis of alternative designs

“…Equation (3) implies that the optimal population of each district will be "larger" if the fixed cost of serving the district is larger, and will be "larger" if the (marginal) cost a3 of serving each additional person or population unit is smaller. The optimal number of districts is obtained by computing (P/p*) and rounding up or down to an integer, rounding in both directions to see which gives the lower total cost.…”

“…Special algorithms for solving this problem are available: see Ref. [8], and more recent work on solving set partitioning problems [3], of which the current problem is a special case.…”

“…We did not use the algorithms from Refs [8] or [3], since we were here not very concerned with computing time, and wished to experiment with some types of constraints we found easier to code-up from 'scratch', rather than seeking to adapt and code-up existing algorithms. (Also, we didn't have a code for Refs [8] or [3] available.) Our algorithm starts, as in Ref.…”

“…If the number of potential candidate districts (Stage 1 above) and/or the number of candidate districts (in Stage 2) had been very much larger, then the above solution approach would not have been tractable. In that case we could have adapted the algorithms already referred to, in Refs [8] and [3], as these are known to easily solve very much larger problems. …”

A Fast Lower Bound for the Minimum Cost Perfect 2-Matching Linear Program

“…Let P be the linear programming relaxation of P obtained by relaxing (4). Our approach for bounding this relaxation is a modification of the multiplier adjustment method given in Chan and Yano (1992) for the set partitioning problem (see also Fisher and Kedia (1990)). …”