The Pyramidal Capacitated Vehicle Routing Problem

This paper introduces the Pyramidal Capacitated Vehicle Routing Problem (PCVRP) as a restricted version of the Capacitated Vehicle Routing Problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the Pyramidal Traveling Salesman Problem (PTSP). A pyramidal route is de ned as a route on which the vehicle rst visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index. Provided that customers are indexed in nondecreasing order of distance from the depot, the shape of a pyramidal route is such that its traversal can be divided in two parts, where on the rst part of the route, customers are visited in nondecreasing distance from the depot, and on the remaining part of the route, customers are visited in nonincreasing distance from the depot. Such a route shape is indeed found in many optimal solutions to CVRP instances. An optimal solution to the PCVRP may therefore be useful in itself as a heuristic solution to the CVRP. Further, an attempt can be made to nd an even better CVRP solution by solving a TSP, possibly leading to a non-pyramidal route, for each of the routes in the PCVRP solution. This paper develops an exact branch-and-cut-and-price (BCP) algorithm for the PCVRP. At the pricing stage, elementary routes can be computed in pseudo-polynomial time in the PCVRP, unlike in the CVRP. We have therefore implemented pricing algorithms that generate only elementary routes. Computational results suggest that PCVRP solutions are highly useful for obtaining near-optimal solutions to the CVRP. Moreover, pricing of pyramidal routes may due to its eciency prove to be very useful in column generation for the CVRP.vehicle routing; pyramidal traveling salesman; branch-and-cut-and-price

Reachability cuts for the vehicle routing problem with time windows

This paper introduces a class of cuts, called reachability cuts, for the Vehicle Routing Problem with Time Windows (VRPTW). Reachability cuts are closely related to cuts derived from precedence constraints in the Asymmetric Traveling Salesman Problem with Time Windows and to k-path cuts for the VRPTW. In particular, any reachability cut dominates one or more k-path cuts. The paper presents separation procedures for reachability cuts and reports computational experiments on well-known VRPTW instances. The computational results suggest that reachability cuts can be highly useful as cutting planes for certain VRPTW instances.Routing; time windows; precedence constraints

Scheduling participants of Assessment Centres

Assessment Centres are used as a tool for psychologists and coaches to ob- serve a number of dimensions in a person's behaviour and test his/her potential within a number of chosen focus areas. This is done in an intense course, with a number of dierent exercises which expose each participant's ability level in the chosen focus areas. The participants are observed by assessors with the purpose of gathering material for reaching a conclusion on each participant's personal pro le. We consider the particular case that arises at the company Human Equity (www.humanequity.dk), where Assessment Centres usually last two days and involve 3-6 psychologists or trained coaches as assessors. An entire course is composed of a number of rounds, with each round having its individual duration. In each round, the participants are divided into a number of groups with prespeci ed pairing of group sizes and assessors. The scheduling problem amounts to determining the allocation of participants to groups in each round. We have developed a model and solution approach for this particular scheduling problem, which may be viewed as a rather extensive generalization of the Social Golfer Problem.No keywords;

A Column Generation Approach to the Capacitated Vehicle Routing Problem with Stochastic Demands

In this article we introduce a new exact solution approach to the Capacitated Vehicle Routing Problem with Stochastic Demands (CVRPSD). In particular, we consider the case where all customer demands are distributed independently and where each customer’s demand follows a Poisson distribution. The CVRPSD can be formulated as a Set Partitioning Problem. We show that, under the above assumptions on demands, the associated column generation subproblem can be solved using a dynamic programming scheme which is similar to that used in the case of deterministic demands. To evaluate the potential of our approach we have embedded this column generation scheme in a branch-and-price algorithm. Computational experiments on a large set of test instances show promising resultsRouting; Stochastic programming; Logistics; Branch and Bound; Dynamic programming

Column generation approaches to ship scheduling with flexible cargo sizes

We present a Dantzig-Wolfe procedure for the ship scheduling problem with flexible cargo sizes. This problem is similar to the well-known pickup and delivery problem with time windows, but the cargo sizes are defined by an interval instead of a fixed value. We show that the introduction of flexible cargo sizes to the column generation framework is not straightforward, and we handle the flexible cargo sizes heuristically when solving the subproblems. This leads to convergence issues in the branch-and-price search tree, and the optimal solution cannot be guaranteed. Hence we have introduced a method that generates an upper bound on the optimal objective. We have compared our method with an a priori column generation approach, and our computational experiments on real world cases show that the Dantzig-Wolfe approach is faster than the a priori generation of columns, and we are able to deal with larger or more loosely constrained instances. By using the techniques introduced in this paper, a more extensive set of real world cases can be solved either to optimality or within a small deviation from optimalityTransportation; integer programming; dynamic programming

A Branch-and-Cut Algorithm for the Capacitated Open Vehicle Routing Problem

In open vehicle routing problems, the vehicles are not required to return to the depot after completing service. In this paper, we present the first exact optimization algorithm for the open version of the well-known capacitated vehicle routing problem (CVRP). The algorithm is based on branch-and-cut. We show that, even though the open CVRP initially looks like a minor variation of the standard CVRP, the integer programming formulation and cutting planes need to be modified in subtle ways. Computational results are given for several standard test instances, which enables us for the first time to assess the quality of existing heuristic methods, and to compare the relative difficulty of open and closed versions of the same problem.Vehicle routing; branch-and-cut; separation

Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;

Genetic algorithms for computational materials discovery accelerated by machine learning

Abstract Materials discovery is increasingly being impelled by machine learning methods that rely on pre-existing datasets. Where datasets are lacking, unbiased data generation can be achieved with genetic algorithms. Here a machine learning model is trained on-the-fly as a computationally inexpensive energy predictor before analyzing how to augment convergence in genetic algorithm-based approaches by using the model as a surrogate. This leads to a machine learning accelerated genetic algorithm combining robust qualities of the genetic algorithm with rapid machine learning. The approach is used to search for stable, compositionally variant, geometrically similar nanoparticle alloys to illustrate its capability for accelerated materials discovery, e.g., nanoalloy catalysts. The machine learning accelerated approach, in this case, yields a 50-fold reduction in the number of required energy calculations compared to a traditional “brute force” genetic algorithm. This makes searching through the space of all homotops and compositions of a binary alloy particle in a given structure feasible, using density functional theory calculations

The atomic simulation environment — a python library for working with atoms

The Atomic Simulation Environment (ASE) is a software package written in the Python programming language with the aim of setting up, steering, and analyzing atomistic simula- tions. In ASE, tasks are fully scripted in Python. The powerful syntax of Python combined with the NumPy array library make it possible to perform very complex simulation tasks. For example, a sequence of calculations may be performed with the use of a simple "for-loop" construction. Calculations of energy, forces, stresses and other quantities are performed through interfaces to many external electronic structure codes or force fields using a uniform interface. On top of this calculator interface, ASE provides modules for performing many standard simulation tasks such as structure optimization, molecular dynamics, handling of constraints and performing nudged elastic band calculations