Dynamic programming algorithm for the vehicle routing problem with time windows and EC social legislation

In practice, apart from the problem of vehicle routing, schedulers also face the problem of nding feasible driver schedules complying with complex restrictions on drivers' driving and working hours. To address this complex interdependent problem of vehicle routing and break scheduling, we propose a dynamic programming approach for the vehicle routing problem with time windows including the EC social legislation on drivers' driving and working hours. Our algorithm includes all optional rules in these legislations, which are generally ignored in the literature. To include the legislation in the dynamic programming algorithm we propose a break scheduling method that does not increase the time-complexity of the algorithm. This is a remarkable eect that generally does not hold for local search methods, which have proved to be very successful in solving less restricted vehicle routing problems. Computational results show that our method finds solutions to benchmark instances with 18% less vehicles and 5% less travel distance than state of the art approaches. Furthermore, they show that including all optional rules of the legislation leads to an additional reduction of 4% in the number of vehicles and of 1.5%\ud regarding the travel distance. Therefore, the optional rules should be exploited in practice

Direct knock-on of desolvated ions governs strict ion selectivity in K+ channels

The seeming contradiction that K+ channels conduct K+ ions at maximal throughput rates while not permeating slightly smaller Na+ ions has perplexed scientists for decades. Although numerous models have addressed selective permeation in K+ channels, the combination of conduction efficiency and ion selectivity has not yet been linked through a unified functional model. Here, we investigate the mechanism of ion selectivity through atomistic simulations totalling more than 400 μs in length, which include over 7,000 permeation events. Together with free-energy calculations, our simulations show that both rapid permeation of K+ and ion selectivity are ultimately based on a single principle: the direct knock-on of completely desolvated ions in the channels' selectivity filter. Herein, the strong interactions between multiple 'naked' ions in the four filter binding sites give rise to a natural exclusion of any competing ions. Our results are in excellent agreement with experimental selectivity data, measured ion interaction energies and recent two-dimensional infrared spectra of filter ion configurations