Issmail Elhallaoui

Multi-phase dynamic constraint aggregation for set partitioning type problems

Dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems. This speed up is mostly due to the use, at each iteration, of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model. This aggregation is updated when needed to ensure the exactness of the overall approach. In this paper, we propose a new version of this method, called the multi-phase dynamic constraint aggregation method, which essentially adds to the original method a partial pricing strategy that involves multiple phases. This strategy helps keeping the size of the aggregated problem as small as possible, yielding a faster average computation time per iteration and fewer iterations. We also establish theoretical results that provide some insights explaining the success of the proposed method. Tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2,000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4.5.

An Improved Primal Simplex Algorithm for Degenerate Linear Programs

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.

Integrated airline crew scheduling: A bi-dynamic constraint aggregation method using neighborhoods

The integrated crew scheduling (ICS) problem consists of determining, for a set of available crew members, least-cost schedules that cover all flights and respect various safety and collective agreement rules. A schedule is a sequence of pairings interspersed by rest periods that may contain days off. A pairing is a sequence of flights, connections, and rests starting and ending at the same crew base. Given its high complexity, the ICS problem has been traditionally tackled using a sequential two-stage approach, where a crew pairing problem is solved in the first stage and a crew assignment problem in the second stage. Recently, Saddoune et al. (2010b) developed a model and a column generation/dynamic constraint aggregation method for solving the ICS problem in one stage. Their computational results showed that the integrated approach can yield significant savings in total cost and number of schedules, but requires much higher computational times than the sequential approach. In this paper, we enhance this method to obtain lower computational times. In fact, we develop a bi-dynamic constraint aggregation method that exploits a neighborhood structure when generating columns (schedules) in the column generation method. On a set of seven instances derived from real-world flight schedules, this method allows to reduce the computational times by an average factor of 2.3, while improving the quality of the computed solutions.

Bi-dynamic constraint aggregation and subproblem reduction

Dynamic constraint aggregation was recently introduced by Elhallaoui et al. [Dynamic aggregation of set partitioning constraints in column generation. Operations Research 2005; 53: 632-45] for efficiently solving the linear relaxation of a class of set partitioning type problems in a column generation context. It reduces the master problem size by aggregating some of its constraints and updates this aggregation when needed. In this paper, we present an advanced version of the dynamic constraint aggregation that reduces both the master problem and the subproblem sizes. This version is called the bi-dynamic constraint aggregation method because aggregation is dynamically applied to both the master problem and the subproblem. We also discuss solution integrality. Computational results for the mass transit simultaneous vehicle and crew scheduling problem are reported.

Integrated Airline Crew Pairing and Crew Assignment by Dynamic Constraint Aggregation

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.

Integral Simplex Using Decomposition with Primal Cuts

The integral simplex using decomposition (ISUD) algorithm [22] is a dynamic constraint reduction method that aims to solve the popular set partitioning problem (SPP). It is a special case of primal algorithms, i.e. algorithms that furnish an improving sequence of feasible solutions based on the resolution, at each iteration, of an augmentation problem that either determines an improving direction, or asserts that the current solution is optimal. To show how ISUD is related to primal algorithms, we introduce a new augmentation problem, MRA. We show that MRA canonically induces a decomposition of the augmentation problem and deepens the understanding of ISUD. We characterize cuts that adapt to this decomposition and relate them to primal cuts. These cuts yield a major improvement over ISUD, making the mean optimality gap drop from 33.92% to 0.21% on some aircrew scheduling problems.

Column generation decomposition with the degenerate constraints in the subproblem

In this paper, we propose a new Dantzig-Wolfe decomposition for degenerate linear programs with the non degenerate constraints in the master problem and the degenerate ones in the subproblem. We propose three algorithms. The first one, where some set of variables of the original problem are added to the master problem, corresponds to the Improved Primal Simplex algorithm (IPS) presented recently by Elhallaoui et al. [7]. In the second one, some extreme points of the subproblem are added as columns in the master problem. The third algorithm is a mixed implementation that adds some original variables and some extreme points of a subproblem to the master problem. Experimental results on some degenerate instances show that the proposed algorithms yield computational times that are reduced by an average factor ranging from 3.32 to 13.16 compared to the primal simplex of CPLEX.

Influence of the normalization constraint on the integral simplex using decomposition

Since its introduction in 1969, the set partitioning problem has received much attention, and the structure of its feasible domain has been studied in detail. In particular, there exists a decreasing sequence of integer feasible points that leads to the optimum, such that each solution is a vertex of the polytope of the linear relaxation and adjacent to the previous one. Several algorithms are based on this observation and aim to determine that sequence; one example is the integral simplex using decomposition (ISUD) of Zaghrouti etźal. (2014). In ISUD, the next solution is often obtained by solving a linear program without using any branching strategy. We study the influence of the normalization-weight vector of this linear program on the integrality of the next solution. We extend and strengthen the decomposition theory in ISUD, prove theoretical properties of the generic and specific normalization constraints, and propose new normalization constraints that encourage integral solutions. Numerical tests on scheduling instances (with up to 500,000 variables) demonstrate the potential of our approach.

Dynamic penalization of fractional directions in the integral simplex using decomposition: Application to aircrew scheduling

To solve integer linear programs, primal algorithms follow an augmenting sequence of integer solutions leading to an optimal solution. In this work, we focus on a particular primal algorithm, the integral simplex using decomposition (ISUD). To find the next point, one solves a linear program to select an augmenting direction for the current point from a cone of feasible directions. To ensure that this linear program is bounded, a normalization constraint is added and the optimization is performed on a section of the cone. The solution of the linear program, i.e., the direction proposed by the algorithm, strongly depends on the chosen normalization weights, and so does the likelihood that the next solution is integer. We modify ISUD so that the normalization is dynamically updated whenever the direction leads to a fractional solution, to penalize that direction. We propose several update strategies, based on theoretical and experimental results. To prove the efficiency of our strategies, we show that our version of the algorithm yields better results than the former version and than classical branch-and-bound techniques on a benchmark of industrial aircrew scheduling instances. The benchmark that we propose here is, to the best of our knowledge, comparable to no other from the literature. It provides large-scale instances with up to 1,700 flights and 115,000 pairings, hence as many constraints and variables, and the instances are given in a set-partitioning form together with initial solutions that accurately mimic those of industrial applications. Our work shows the strong potential of primal algorithms for the crew scheduling problem, which is a key challenge for large airlines. Acknowledgments: This work was supported by a Collaborative research and development grant from NSERC and Kronos Inc. (RDCPJ 477127-14). Les Cahiers du GERAD G–2016–01 1