A Multiplier - Adjustment - Based Branch - and - Bound Algorithm for Solving the Set Partitioning Problem.

Abstract

We introduce a new branch- and -bound algorithm (called BB-SPP) for solving the set partitioning problem (SPP). The algorithm consists of a new lower bounding procedure (called MAM) and a new branching strategy, which complements MAM. MAM is a multiplier adjustment method which attempts to satisfy the constraints of SPP by adjusting the values of the dual variables. At each iteration, selected dual variables are increased to improve the lower bound, while others may be decreased to maintain dual feasibility. Computational results show that MAM is superior to subgradient optimization, the most popular bounding method currently used for SPP. In addition, MAM requires no fine tuning nor an upper bound, and the lower bound obtained at each iteration is monotonically nondecreasing. On average, based on the test problems, MAM provided a lower bound within 4% of the LP optimal objective value in only one iteration, and within 1% in four iterations. We also extend the concept of variable elimination of SPP and describe how it can be implemented within BB-SPP to reduce the computational effort. Finally, we apply BB-SPP to a set of ten airline crew scheduling problems. Computational results show that BB-SPP outperforms SETPAR, the fastest code currently available for SPP. On average, the computation times were 9.5 times faster than SETPAR. This ratio increases as the size of the problem increases.