The integral column generation algorithm (ICG) was recently introduced to solve set partitioning problems involving a very large number of variables. This primal algorithm generates a sequence of integer solutions with decreasing costs, leading to an optimal or near-optimal solution. ICG combines the well-known column generation algorithm and a primal algorithm called the integral simplex using decomposition algorithm (ISUD). In this paper, we develop a generalized version of ICG, denoted I2CG, that can solve efficiently large-scale set partitioning problems with side constraints. This new algorithm can handle the side constraints in the reduced problem of ISUD, in its complementary problem, or in both components. Computational experiments on instances of the airline crew pairing problem (CPP) and the multidepot vehicle routing problem with time windows show that the latter strategy is the most efficient one and I2CG significantly outperforms basic variants of two popular column generation heuristics, namely, a restricted master heuristic and a diving heuristic. For the largest tested CPP instance with 1,761 constraints, I2CG can produce in less than one hour of computational time more than 500 integer solutions leading to an optimal or near-optimal solution. Summary of Contribution: In this paper, we develop a new integral column generation algorithm that can solve efficiently large-scale set partitioning problems with side constraints. The latter alter the quasi-integrality property needed for primal integral algorithms. The paper adds a methodological contribution remedying this issue. This remedy should, in our opinion, boost the use of primal exact methods, especially in the column generation context. The paper also has a computational contribution. Effectively, computational experiments on instances of the airline crew pairing problem and the multidepot vehicle routing problem with time windows are extensively discussed. We compare the proposed algorithm to basic variants of two popular column generation heuristics.

Abstract The primal adjacency‐based algorithm and the multidirectional dynamic programming algorithm are two exact methods that have recently been developed to efficiently solve the shortest path problem with resource constraints (SPPRCs). These methods are primal in the sense that they are able to produce sequences of feasible solutions using iterative exploration of the search space. Since the SPPRCs often appear as a subproblem (SP) in the solution of vehicle and crew scheduling problems (VCSP) using column generation (CG), we propose a new primal column generation framework that embeds these primal methods in a CG scheme. The primal column generation solves at each iteration a sequence of appropriate restricted SP and stops solving the SP when there is no need to continue. This approach introduces a large degree of flexibility, and allows performing good cost improvements in a very limited time. Computational experiments on VCSP instances show that the proposed approach is able to find optimal solutions while reducing the time spent solving the SP by factors of up to seven compared to the standard CG algorithm. This leads to significant improvements in the overall solution times, with an average reduction factor of 3.5.

Since its appearance in 1947, the primal simplex algorithm has been one of the most popular algorithms for solving linear programs. It is often very efficient when there is very little degeneracy, but it often struggles in the presence of high degeneracy, executing many pivots without improving the objective function value. In this paper, we propose an improved primal simplex algorithm that deals with this issue. This algorithm is based on new theoretical results that shed light on how to reduce the negative impact of degeneracy. In particular, we show that, from a nonoptimal basic solution with p positive-valued variables, there exists a sequence of at most m - p + 1 simplex pivots that guarantee the improvement of the objective value, where m is the number of constraints in the linear program. These pivots can be identified by solving an auxiliary linear program. Finally, we briefly summarize computational results that show the effectiveness of the proposed algorithm on degenerate linear programs.