During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50-80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.

During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50-80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.

Petrobras produces nearly 90 percent of Brazil's oil at about 80 offshore oil platforms. It transports approximately 1,900 employees daily between these platforms and four mainland bases, using more than 40 helicopters that vary in capacity, operating costs, and performance characteristics. Each day, flight planners must select the helicopter routes and schedules that satisfy passenger demands. We developed a system that requires less than one hour to generate optimized flight plans that meet operational guidelines, improve travel safety, and minimize operating costs. By using this system, Petrobras reduced its number of offshore landings by 18 percent, total flight time by 8 percent, and flight costs by 14 percent, resulting in annual savings of more than $20 million. Our optimization model is a large-scale mixed integer program that generalizes prior helicopter routing models. We designed a column-generation algorithm that exploits the problem structure to overcome its computational difficulties. As part of the solution method, we use a network flow model to optimally assign passengers to selected routes.

Studies on the kinetics of carbothermic chlorination of zirconium dioxide in gaseous chlorine were carried out with petroleum coke fines in powder form. The amounts of ZrO 2 chlorinated were found to be directly proportional to the time of chlorination in the temperature range studied (973 to 1273 K). The activation energy values for chlorination of ZrO 2 , in mixture with petroleum coke, was found to be 18.3 kJ/mole. The influence of particle size of petroleum coke on the chlorination of ZrO 2 (Ϫ38 ϩ 25 m) was studied, and it was found that the rate of chlorination increased up to the size range of Ϫ75 to ϩ53 m, and the size finer than this produced negligible increase. The amount of petroleum coke in the mixture above 17.41 pct in excess of the stoichiometry resulted in very little increase in the rate. The effect of the partial pressure of chlorine ( pCl 2 ) on the rate of chlorination was found to obey the following relationship, derived from the Langmuir adsorption isotherm: v ϭ k и K и pCl 2