Abstract. When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley's method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

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This paper introduces an optimization model of a multi-terminal, multi-modal maritime container port, such as the ones in the European northern range. The decisions concern the scheduling of ships, trains and trucks on terminals, while limiting inter-terminal transport of containers and minimizing weighted turnaround time. Heuristics based on the decomposition of the resulting mixed-integer program are proposed and tested on realistic generated instances with up to four terminals. The efficiency of the restrict-and-fix heuristic allows to investigate the impact of a global management on port's performance: an average improvement of 5% was observed.

Inventory routing problems combine the optimization of product deliveries (or pickups) with inventory control at customer sites. The application that motivates this paper concerns the planning of single-product pickups over time; each site accumulates stock at a deterministic rate; the stock is emptied on each visit. At the tactical planning stage considered here, the objective is to minimize a surrogate measure of routing cost while achieving some form of regional clustering by partitioning the sites between the vehicles. The fleet size is given but can potentially be reduced. Planning consists of assigning customers to vehicles in each time period, but the routing, i.e., the actual sequence in which vehicles visit customers, is considered an "operational" decision. The planning is due to be repeated over the time horizon with constrained periodicity. We develop a truncated branch-and-price-and-cut algorithm combined with rounding and local search heuristics that yield both primal solutions and dual bounds. On a large-scale industrial test problem (with close to 6,000 customer visits to schedule), we obtain a solution within 6.25% deviation from the optimal to our model. A rough comparison between an operational routing resulting from our tactical solution and the industrial practice shows a 10% decrease in the number of vehicles as well as in the travel distance. The key to the success of the approach is the use of a state-space relaxation technique in formulating the master program to avoid the symmetry in time.