The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, we address the primal-dual column generation technique, which relies on well-centred non-optimal solutions of the restricted master problems that are obtained by a primal-dual interior point method. Although good computational results are reported for this technique, it was only applied in a particular class of problems. Moreover, no theoretical analysis to guarantee its convergence is available. Here, we further investigate the primaldual column generation technique and present extensive computational experiments in the context of integer programming, where column generation schemes are widely employed. The results show that the primal-dual technique usually leads to substantial reductions in the number of iterations as well as less running time when compared to the classical and also analytic centre approaches.

Extreme events such as disasters cause partial or total disruption of basic services such as water, energy, communication and transportation. In particular, roads can be damaged or blocked by debris, thereby obstructing access to certain aected areas. Thus, restoration of the damaged roads is necessary to evacuate victims and distribute emergency commodities to relief centers or aected areas. The Crew Scheduling and Routing Problem (CSRP) addresses decisions in postdisaster situations with the aim of minimizing the time that aected areas remain inaccessible. The integration of crew scheduling and routing decisions makes this problem too complicated to be eectively solved for practical instances using mixed integer programming (MIP) formulations recently proposed in the literature. Therefore, we propose a branch-and-Benders-cut (BBC) algorithm that decomposes the integrated problem into a master problem (MP) with scheduling decisions and subproblems with routing decisions. Computational tests based on instances from the literature show that the proposed exact method improves the results of MIP formulations and other exact and metaheuristic methods proposed in literature. The BBC algorithm provides feasible solutions and optimality gaps for instances that thus far have not been possible to solve by exact methods in the literature.

Branch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the Vehicle Routing Problem with Time Windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm.

No abstract