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This article describes a method for solving the crew rostering problem in air transportation. This problem consists of constructing personalized schedules that assign pairings, days off, and other activities to airline crew members. A generalized set partitioning model and a method using column generation have been used. This method has been adapted in a number of ways to take advantage of the nature of the problem and to accelerate solution. Numerical tests on problems from Air France have demonstrated that this method is capable of solving very large scale problems with thousands of constraints and hundreds of subproblems. The tests have also shown that these adaptations are capable of reducing solution time by a factor of about a thousand. Finally, results from this method are compared with those obtained with the method currently used at Air France.

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This article describes the problem in which the edges of a network represent customers, and a quantity of material is delivered to them so that each one achieves a desired inventory level while finding the lowest-cost route of delivery. Routing and inventory decisions are made at the same time. An example of an application of this problem is dust suppression in open-pit mines. A fleet of trucks spray water along the roads of a mine. Humidity increases the effectiveness of dust-particle retention. Because the level of humidity decreases, replenishment is done periodically. Other examples of applications include dust suppression in forest roads and plants watering in street medians and sidewalks. We develop a mathematical model that combines two objectives: An inventory objective that minimizes the penalty for the lack of humidity and a routing objective that minimizes watering and traversing costs. Due to the complexity of the mathematical model, we developed an adaptive large neighborhood search algorithm that combines several destroy and repair operators dynamically.

This paper describes the Preferential Bidding Problem solved in the airline industry to construct personalized monthly schedules for pilots and officers. This problem consists in assigning to crew members pairings, days off, annual leaves, training periods, etc., while considering a set of weighted bids that reflect individual preferences. This assignment must be done under strict seniority restrictions: the construction of a maximum-score schedule for a particular crew member must never be done at the expense of a more senior employee. This research and development project has resulted in the Preferential Bidding System that has been used at Air Canada since May 1995. The solution process is summarized as follows. For each employee, from the most senior to the most junior, a so-called residual problem is solved: given an employee and a set of unassigned pairings, the solution to an integer linear program determines the employee's maximum-score schedule while taking into account all the remaining employees. The residual problem is solved by column generation embedded in a branch-and-bound tree. Integer solutions are obtained by using very efficient cutting planes, without which it would have been impossible to solve some of these residual problems.

A solution approach based on the column-generation technique is presented for solving a time-indexed formulation of the total weighted tardiness problem. An acceleration strategy based on a decomposition of the time horizon into subperiods, where each subperiod is associated with a subproblem of the column-generation approach, is used to solve the linear relaxation. Branching strategies and dominance rules are also applied to find the optimal integer solution. Using this new approach, it is possible to solve to optimality 117 out of 125 open problems of the OR-Library.

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