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.

Column generation is often used to solve problems involving set-partitioning constraints, such as vehicle-routing and crew-scheduling problems. When these constraints are in large numbers and the columns have on average more than 8-12 nonzero elements, column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy. To overcome this difficulty, we introduce a dynamic constraint aggregation method that reduces the number of set-partitioning constraints in the master problem by aggregating some of them according to an equivalence relation. To guarantee optimality, this equivalence relation is updated dynamically throughout the solution process. Tests on the linear relaxation of the simultaneous vehicle and crew-scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem, degeneracy, and solution times, especially for larger problems. In fact, for an instance involving 1,600 set-partitioning constraints, the master problem solution time is reduced by a factor of 8.

Traditionally, the airline crew scheduling problem has been decomposed into a crew pairing problem and a crew assignment problem, both of which are solved sequentially. The first consists of generating a set of least-cost crew pairings (sequences of flights starting and ending at the same crew base) that cover all flights. The second aims at finding monthly schedules (sequences of pairings) for crew members that cover all pairings previously built. Pairing and schedule construction must respect all safety and collective agreement rules. In this paper, we focus on the pilot crew scheduling problem in a bidline context where anonymous schedules must be built for pilots and high fixed costs are considered to minimize the number of scheduled pilots. We propose a model that completely integrates the crew pairing and crew assignment problems, and we develop a combined column generation/dynamic constraint aggregation method for solving them. Computational results on real-life data show that integrating crew pairing and crew assignment can yield significant savings—on average, 3.37% on the total cost and 5.54% on the number of schedules for the 7 tested instances. The integrated approach, however, requires much higher computational times than the sequential approach.