A survey of algorithms for the single machine total weighted tardiness scheduling problem

AbstractThis paper surveys algorithms for the problem of scheduling jobs on a single machine to minimize total weighted tardiness. Special attention is given to two dynamic programming and four branch and bound algorithms. The dynamic programming algorithms both use the same recursion defined on sets of jobs, but they generate the sets in lexicographic order and cardinality order respectively. Two of the branch and bound algorithms use the quickly computed but possibly rather weak lower bounds obtained from linear and exponential functions of completion times problems. These algorithms rely heavily on dominance rules to restrict the search. The other two branch and bound algorithms use lower bounds obtained from the Lagrangean relaxation of machine capacity constraints and from dynamic programming state-space relaxation. They invest a substantial amount of computation time at each node of the search tree in an attempt to generate tight lower bounds and thereby generate only small search trees. A computational comparison of all these algorithms on problems with up to 50 jobs is given

The Edge Set Cost of the Vehicle Routing Problem with Time Windows

We consider an important generalization of the vehicle routing problem with time windows in which a fixed cost must be paid for accessing a set of edges. This fixed cost could reflect payment for toll roads, investment in new facilities, the need for certifications, and other costly investments. The certifications and investments impose a cost for the company while they also give unlimited usage of a set of roads to all vehicles belonging to the company. This violates the traditional assumption that the path between two destinations is well defined and independent of other choices. Different versions for defining the edge sets are discussed and formulated. Both the multigraph case and the direct path case are described, and mixed-integer-programming formulations of the problem are presented for both cases. A solution method based on branch-price-and-cut is applied to the direct path case. The computational results show that instances with up to 40 customers can be solved in a reasonable time, and that the branch-cut-and-price algorithm generally outperforms CPLEX. </jats:p

Dynamic Programming Methodologies in Very Large Scale Neighborhood Search Applied to the Traveling Salesman Problem

We provide two different neighborhood construction techniques for creating exponentially large neighborhoods that are searchable in polynomial time using dynamic programming. We illustrate both of these approaches on very large scale neighborhood search techniques for the traveling salesman problem. Our approaches are intended both to unify previously known results as well as to offer schemas for generating additional exponential neighborhoods that are searchable in polynomial time. The first approach is to define the neighborhood recursively. In this approach, the dynamic programming recursion is a natural consequence of the recursion that defines the neighborhood. In particular, we show how to create the pyramidal tour neighborhood, the twisted sequences neighborhood, and dynasearch neighborhoods using this approach. In the second approach, we consider the standard dynamic program to solve the TSP. We then obtain exponentially large neighborhoods by selecting a polynomially bounded number of states, and restricting the dynamic program to those states only. We show how the Balas and Simonetti neighborhood and the insertion dynasearch neighborhood can be viewed in this manner. We also show that one of the dynasearch neighborhoods can be derived directly from the 2-exchange neighborhood using this approach

Arbeitsbericht Nr. 2006-05, November 2006

Ilmenauer Beiträge zur Wirtschaftsinformatik Nr. 2006-05 / Technische Universität Ilmenau, Fakultät für Wirtschaftswissenschaften, Institut für Wirtschaftsinformatik, ISSN 1861-9223 ISBN 3-938940-06-

Dynamic programming for the orienteering problem with time windows

We present an exact optimization algorithm for the Orienteering Problem with Time Windows (OPTW). The algorithm is based on bi-directional and bounded dynamic programming with decremental state space relaxation. We compare different strategies proposed in the literature to guide decremental state space relaxation: our experiments on instances derived from the literature show that there is no dominance between these strategies. We also propose a new heuristic technique to initialize the critical vertex set and we provide experimental evidence of its effectiveness

VRP with Time Windows

Abstract This paper presents a survey of the research on the Vehicle Routing Problem with Time Windows (VRPTW), an extension of the Capacitated Vehicle Routing Problem. In the VRPTW, the service at each customer must start within an associated time window and the vehicle must remain at the customer location during service. Soft time windows can be violated at a cost while hard time windows do not allow for a vehicle to arrive at a customer after the latest time to begin service. We first present a multi-commodity network flow formulation with time and capacity constraints for the VRPTW. Approximation methods proposed in the literature to derive upper bounds are then reviewed. Then we explain how lower bounds can be obtained using optimal approaches, namely, Lagrangean relaxation and column generation. Next, we provide branching and cutting strategies that can be embedded within these optimal approaches to produce integer solutions. Special cases and extensions to the VRPTW follow as well as our conclusions. Résumé Cet article synthèse porte sur les récents développements concernant le problème du routage de véhicules sous des contraintes de fenêtres de temps. Dans ce problème, le serviceà un client doit débuterà l&apos;intérieur d&apos;un intervalle de temps. Celui-ci peutêtre, soit relaché au prix d&apos;une certaine pénalité, soit rigide, auquel cas, il n&apos;est pas permis de dépasser la limite supérieure. Nous présentons un modèle de réseau multi-flots avec des contraintes de temps et de capacité. Les méthodes heuristiques permettant de calculer des bornes supérieures sont d&apos;abord présentées. Suivent les modèles d&apos;optimisation basés sur la relaxation lagrangienne et la génération de colonnes pourévaluer des bornes inférieures. Enfin, on présente les stratégies de coupes et de branchements liéesà ces méthodes afin de déterminer des solutions entières. L&apos;article se termine par l&apos;étude de cas particuliers et d&apos;extensions ainsi que nos conclusions

Optimization of gendarmerie posts' main supply roads via regional ring transportation system

Cataloged from PDF version of article.The aim of this study is to find the optimum distances traveled by the vehicles distributing goods to the military units, belonging to a command center, in one day and the appropriate fleet types and sizes for each Command Center. We examine the differences in the total distances traveled by the vehicles when the types and number of the vehicles are changed in the fleet. A mixed integer programming model is proposed, and for the implementation of the model, optimization modeling software GAMS is used. The proposed model is obtained from the current system elements and shows the differences between the current distribution system and the solution of the proposed model. How the total distances are affected when distribution is made from one center is investigated, and ideal main supply routes, ideal sequences of supply plans, and fleet sizes are proposed.Karamanoğlu, Yunus EmreM.S

Dynamic Programming: Salesman to Surgeon

Dynamic Programming is an optimization technique used in computer science and mathematics. Introduced in the 1950s, it has been applied to many classic combinatorial optimization problems, such as the Shortest Path Problem, the Knapsack Problem, and the Traveling Salesman Problem, with varying degrees of practical success. In this thesis, we present two applications of dynamic programming to optimization problems. The first application is as a method to compute the Branch-Cut-and-Price (BCP) family of lower bounds for the Traveling Salesman Problem (TSP), and several vehicle routing problems that generalize it. We then prove that the BCP family provides a set of lower bounds that is at least as strong as the Approximate Linear Program (ALP) family of lower bounds for the TSP. The second application is a novel dynamic programming model used to determine the placement of cuts for a particular form of skull surgery called Cranial Vault Remodeling