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

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming

Models for the optimization of promotion campaigns: exact and heuristic algorithms.

This paper presents an optimization model for the selection of sets of clients that will receive an offer for one or more products during a promotion campaign. The complexity of the problem makes it very difficult to produce optimal solutions using standard optimization methods. We propose an alternative set covering formulation and develop a branch-and-price algorithm to solve it. We also describe five heuristics to approximate an optimal solution. Two of these heuristics are algorithms based on restricted versions of the basic formulation, the third is a successive exact k-item knapsack procedure. A heuristic inspired by the Next-Product-To-Buy model and a depth-first branch-and-price heuristic are also presented. Finally, we perform extensive computational experiments for the two formulations as well as for the five heuristics.Promotion campaign; Minimum quantity commitment; Integer programming; Branch-and-price algorithm; Non-approximability; Heuristics; Business-to-business; Business-to-consumer;

Real-Time Scheduling Approaches for Vehicle-Based Internal Transport Systems

In this paper, we study the problem of scheduling and dispatching vehicles in vehicle-based internal transport systems within warehouses and production facilities. We develop and use two rolling horizon policies to solve real-time vehicle scheduling problems. To solve static instances of scheduling problems, we propose two new heuristics: combined and column-generation heuristics. We solve a real-time scheduling problem by applying a heuristic to dynamically solve a series of static instances under a rolling horizon policy. A rolling horizon can be seen either as a fixed-time interval in which advance information about loads’ arrivals is available, or as a fixed number of loads which are known to become available in the near future. We also propose a new look-ahead dynamic assignment algorithm, a different dynamic vehicle-scheduling approach. We evaluate these dynamic scheduling strategies by comparing their performance with that of two of the best online vehicle dispatching rules mentioned in the literature. Experimental results show that the new look-ahead dynamic assignment algorithm and dynamic scheduling approaches consistently outperform vehicle dispatching rules

Resilient network dimensioning for optical grid/clouds using relocation

In this paper we address the problem of dimensioning infrastructure, comprising both network and server resources, for large-scale decentralized distributed systems such as grids or clouds. We will provide an overview of our work in this area, and in particular focus on how to design the resulting grid/cloud to be resilient against network link and/or server site failures. To this end, we will exploit relocation: under failure conditions, a request may be sent to an alternate destination than the one under failure-free conditions. We will provide a comprehensive overview of related work in this area, and focus in some detail on our own most recent work. The latter comprises a case study where traffic has a known origin, but we assume a degree of freedom as to where its end up being processed, which is typically the case for e. g., grid applications of the bag-of-tasks (BoT) type or for providing cloud services. In particular, we will provide in this paper a new integer linear programming (ILP) formulation to solve the resilient grid/cloud dimensioning problem using failure-dependent backup routes. Our algorithm will simultaneously decide on server and network capacity. We find that in the anycast routing problem we address, the benefit of using failure-dependent (FD) rerouting is limited compared to failure-independent (FID) backup routing. We confirm our earlier findings in terms of network capacity savings achieved by relocation compared to not exploiting relocation (order of 6-10% in the current case studies)

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones