The BusPlus project aims at improving the off-peak hours public transit service in Canberra, Australia.To address the difficulty of covering a large geographic area, BusPlus proposes a hub and shuttle model consisting of a combination of a few high-frequency bus routes between key hubs and a large number of shuttles that bring passengers from their origin to the closest hub and take them from their last bus stop to their destination. This paper focuses on the design of bus network and proposes an efficient solving method to this multimodal network design problem based on the Benders decomposition method. Starting from a MIP formulation of the problem, the paper presents a Benders decomposition approach using dedicated solution techniques for solving independent sub-problems, Pareto optimal cuts, cut bundling, and core point update.Computational results on real-world data from Canberra's public transit system justify the design choices and show that the approach outperforms the MIP formulation by two orders of magnitude. Moreover, the results show that the hub and shuttle model may decrease transit time by a factor of 2, while staying within the costs of the existing transit system.

We describe a heuristic solution method for the Vehicle RoutingProblem with Time Windows. The method uses four improvement operators in a steepest descent search strategy. The Guided Local Search meta-heuristic is used to avoid local minima. The resulting algorithm is tested on Solomon's capacitated vehicle routing problems with time windows. We compare our results to the best heuristic approaches reported for the VRPTW. The new method performs significantly better than previous methods on classes where vehicle routes tend to be longer. The method is slightly worse on classes with shorter routes. We report 12 new best solutions for Solomon's problems. INTRODUCTIONThe Vehicle Routing Problem (VRP) is a practical problem that has been studied widely in the OR literature [7]. The problem is usually expressed as follows: given a set of customers requiring a visit, and a fleet of vehicles based at a depot that can perform the visits, construct a set of routes for the vehicles which minimizes the costs of operation. The objective function is usually expressed as costs related to the number of vehicles used and to distance travelled.Vehicle routing problems generally come at least with capacity constraints on the vehicles (indicating that they can only carry a certain quantity of goods). However, other constraints are common. In this paper we examine the Vehicle Routing Problem with Time Windows (VRPTW). In this problem, as well as

Despite being at the heart of chemical thought, the curly arrow notation of reaction mechanisms has been treated with suspicion—the connection with rigorous molecular quantum mechanics being unclear. The connection requires a view of the wavefunction that goes beyond molecular orbitals and rests on the most fundamental property of electrons. The antisymmetry of electronic wavefunctions requires that an N-electron wavefunction repeat itself in 3N dimensions, thus exhibiting tiles. Inspection of wavefunction tiles permits insight into structure and mechanism. Here, we demonstrate that analysis of the wavefunction tile along a reaction coordinate reveals the electron movements depicted by the curly arrow notation for several reactions. The Diels–Alder reaction is revealed to involve the separation and counter propagation of electron spins. This unprecedented method of extracting the movements of electrons during a chemical reaction is a breakthrough in connecting traditional depictions of chemical mechanism with state-of-the-art quantum chemical calculations.

The electronic structure of benzene is a battleground for competing viewpoints of electronic structure, with valence bond theory localising electrons within superimposed resonance structures, and molecular orbital theory describing delocalised electrons. But, the interpretation of electronic structure in terms of orbitals ignores that the wavefunction is anti-symmetric upon interchange of like-spins. Furthermore, molecular orbitals do not provide an intuitive description of electron correlation. Here we show that the 126-dimensional electronic wavefunction of benzene can be partitioned into tiles related by permutation of like-spins. Employing correlated wavefunctions, these tiles are projected onto the three dimensions of each electron to reveal the superposition of Kekulé structures. But, opposing spins favour the occupancy of alternate Kekulé structures. This result succinctly describes the principal effect of electron correlation in benzene and underlines that electrons will not be spatially paired when it is energetically advantageous to avoid one another.

Bi-objective search is a problem of finding a set of optimal solutions in a two-dimensional domain. This study proposes several enhancements to the state-of-the-art bi-objective search with A* and develops its bi-directional variant. Our experimental results on benchmark instances show that our enhanced algorithm is on average five times faster than the state of the art bi-objective search algorithms.