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

The roots and routes of environmental and sustainability education policy research

Environmental Education Research has developed a Virtual Special Issue (VSI)(http://explore.tandfonline.com/content/ed/ceer-vsi) focusing on studies of environmental and sustainability education (ESE) policy. The VSI draws on key examples of research on this topic published in the Journal from the past two decades, for three reasons. First, to provide readers with a series of snapshots into the genealogy of ESE policy research in this field. Second, to encourage renewed attention to previously published work. And third, to offer commentary on the evolution of research trends, approaches and findings

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

Ballet de la Nuit: Staging the Absolute Monarchy of Louis XIV

This thesis researches the ballet de cour spectacles of the seventeenth century French court, with a careful examination of the political and social importance of such spectacles in anticipation of reconstructing Louis XIV’s Ballet de la Nuit (Ballet of the Night), originally performed February 23, 1653 at the Louvre’s Salle de Petit Bourbon. This ballet de cour was a visual spectacle that combined music, dance, poetry, and allegory with elaborate staging and costumes. Such spectacles were first produced in France during the sixteenth century and involved the royal court not only as spectators but also as performers. The Ballet de la Nuit was influential in numerous aspects. This ballet marked for the first time in the history of the ballet de cour, a work presented as a complete and united whole. This meant the libretto was a unified piece, expertly divided into four veilles (watches of the night) and a concluding grand ballet. The entire ballet moved toward a single goal: the rising of the Sun (King). This influential court ballet changed the direction of ballet from a court etiquette requirement to a professional performing art. The reconstruction, The Sun King Dances, performed at San José State University on February 22nd and 23rd, 2018, 365 years after its original performance, showcases the developmental changes from the seventeenth century to the twenty-first, all while allowing the original production to come alive once more

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