On the use of biased-randomized algorithms for solving non-smooth optimization problems

Soft constraints are quite common in real-life applications. For example, in freight transportation, the fleet size can be enlarged by outsourcing part of the distribution service and some deliveries to customers can be postponed as well; in inventory management, it is possible to consider stock-outs generated by unexpected demands; and in manufacturing processes and project management, it is frequent that some deadlines cannot be met due to delays in critical steps of the supply chain. However, capacity-, size-, and time-related limitations are included in many optimization problems as hard constraints, while it would be usually more realistic to consider them as soft ones, i.e., they can be violated to some extent by incurring a penalty cost. Most of the times, this penalty cost will be nonlinear and even noncontinuous, which might transform the objective function into a non-smooth one. Despite its many practical applications, non-smooth optimization problems are quite challenging, especially when the underlying optimization problem is NP-hard in nature. In this paper, we propose the use of biased-randomized algorithms as an effective methodology to cope with NP-hard and non-smooth optimization problems in many practical applications. Biased-randomized algorithms extend constructive heuristics by introducing a nonuniform randomization pattern into them. Hence, they can be used to explore promising areas of the solution space without the limitations of gradient-based approaches, which assume the existence of smooth objective functions. Moreover, biased-randomized algorithms can be easily parallelized, thus employing short computing times while exploring a large number of promising regions. This paper discusses these concepts in detail, reviews existing work in different application areas, and highlights current trends and open research lines

The two-echelon capacitated vehicle routing problem: models and math-based heuristics

Multiechelon distribution systems are quite common in supply-chain and logistics. They are used by public administrations in their transportation and traffic planning strategies, as well as by companies, to model own distribution systems. In the literature, most of the studies address issues relating to the movement of flows throughout the system from their origins to their final destinations. Another recent trend is to focus on the management of the vehicle fleets required to provide transportation among different echelons. The aim of this paper is twofold. First, it introduces the family of two-echelon vehicle routing problems (VRPs), a term that broadly covers such settings, where the delivery from one or more depots to customers is managed by routing and consolidating freight through intermediate depots. Second, it considers in detail the basic version of two-echelon VRPs, the two-echelon capacitated VRP, which is an extension of the classical VRP in which the delivery is compulsorily delivered through intermediate depots, named satellites. A mathematical model for two-echelon capacitated VRP, some valid inequalities, and two math-heuristics based on the model are presented. Computational results of up to 50 customers and four satellites show the effectiveness of the methods developed

A Survey of Network Design Problems

Network design problems arise in many different application areas such as air freight, highway traffic, and communication systems. The intention of this survey is to present a coherent unified view of a number of papers in the network design literature. We discuss suggested solution procedures, computational experience, relations between various network models, and potential application areas. Promising topics of research for improving, solving, and extending the models reviewed in this survey are also indicated.Supported in part by the U.S. Department of Transportation under contract DOT-TSC-1058, Transportation Advanced Research Program (TARP)

Two-Echelon Vehicle and UAV Routing for Post-Disaster Humanitarian Operations with Uncertain Demand

Humanitarian logistics service providers have two major responsibilities immediately after a disaster: locating trapped people and routing aid to them. These difficult operations are further hindered by failures in the transportation and telecommunications networks, which are often rendered unusable by the disaster at hand. In this work, we propose two-echelon vehicle routing frameworks for performing these operations using aerial uncrewed autonomous vehicles (UAVs or drones) to address the issues associated with these failures. In our proposed frameworks, we assume that ground vehicles cannot reach the trapped population directly, but they can only transport drones from a depot to some intermediate locations. The drones launched from these locations serve to both identify demands for medical and other aids (e.g., epi-pens, medical supplies, dry food, water) and make deliveries to satisfy them. Specifically, we present two decision frameworks, in which the resulting optimization problem is formulated as a two-echelon vehicle routing problem. The first framework addresses the problem in two stages: providing telecommunications capabilities in the first stage and satisfying the resulting demands in the second. To that end, two types of drones are considered. Hotspot drones have the capability of providing cell phone and internet reception, and hence are used to capture demands. Delivery drones are subsequently employed to satisfy the observed demand. The second framework, on the other hand, addresses the problem as a stochastic emergency aid delivery problem, which uses a two-stage robust optimization model to handle demand uncertainty. To solve the resulting models, we propose efficient and novel solution approaches

Lagrangian-based methods for single and multi-layer multicommodity capacitated network design

Le problème de conception de réseau avec coûts fixes et capacités (MCFND) et le problème de conception de réseau multicouches (MLND) sont parmi les problèmes de conception de réseau les plus importants. Dans le problème MCFND monocouche, plusieurs produits doivent être acheminés entre des paires origine-destination différentes d’un réseau potentiel donné. Des liaisons doivent être ouvertes pour acheminer les produits, chaque liaison ayant une capacité donnée. Le problème est de trouver la conception du réseau à coût minimum de sorte que les demandes soient satisfaites et que les capacités soient respectées. Dans le problème MLND, il existe plusieurs réseaux potentiels, chacun correspondant à une couche donnée. Dans chaque couche, les demandes pour un ensemble de produits doivent être satisfaites. Pour ouvrir un lien dans une couche particulière, une chaîne de liens de support dans une autre couche doit être ouverte. Nous abordons le problème de conception de réseau multiproduits multicouches à flot unique avec coûts fixes et capacités (MSMCFND), où les produits doivent être acheminés uniquement dans l’une des couches. Les algorithmes basés sur la relaxation lagrangienne sont l’une des méthodes de résolution les plus efficaces pour résoudre les problèmes de conception de réseau. Nous présentons de nouvelles relaxations à base de noeuds, où le sous-problème résultant se décompose par noeud. Nous montrons que la décomposition lagrangienne améliore significativement les limites des relaxations traditionnelles. Les problèmes de conception du réseau ont été étudiés dans la littérature. Cependant, ces dernières années, des applications intéressantes des problèmes MLND sont apparues, qui ne sont pas couvertes dans ces études. Nous présentons un examen des problèmes de MLND et proposons une formulation générale pour le MLND. Nous proposons également une formulation générale et une méthodologie de relaxation lagrangienne efficace pour le problème MMCFND. La méthode est compétitive avec un logiciel commercial de programmation en nombres entiers, et donne généralement de meilleurs résultats.The multicommodity capacitated fixed-charge network design problem (MCFND) and the multilayer network design problem (MLND) are among the most important network design problems. In the single-layer MCFND problem, several commodities have to be routed between different origin-destination pairs of a given potential network. Appropriate capacitated links have to be opened to route the commodities. The problem is to find the minimum cost design and routing such that the demands are satisfied and the capacities are respected. In the MLND, there are several potential networks, each at a given layer. In each network, the flow requirements for a set of commodities must be satisfied. However, the selection of the links is interdependent. To open a link in a particular layer, a chain of supporting links in another layer has to be opened. We address the multilayer single flow-type multicommodity capacitated fixed-charge network design problem (MSMCFND), where commodities are routed only in one of the layers. Lagrangian-based algorithms are one of the most effective solution methods to solve network design problems. The traditional Lagrangian relaxations for the MCFND problem are the flow and knapsack relaxations, where the resulting Lagrangian subproblems decompose by commodity and by arc, respectively. We present new node-based relaxations, where the resulting subproblem decomposes by node. We show that the Lagrangian dual bound improves significantly upon the bounds of the traditional relaxations. We also propose a Lagrangian-based algorithm to obtain upper bounds. Network design problems have been the object of extensive literature reviews. However, in recent years, interesting applications of multilayer problems have appeared that are not covered in these surveys. We present a review of multilayer problems and propose a general formulation for the MLND. We also propose a general formulation and an efficient Lagrangian-based solution methodology for the MMCFND problem. The method is competitive with (and often significantly better than) a state-of-the-art mixedinteger programming solver on a large set of randomly generated instances

Arc Routing with Time-Dependent Travel Times and Paths

Vehicle routing algorithms usually reformulate the road network into a complete graph in which each arc represents the shortest path between two locations. Studies on time-dependent routing followed this model and therefore defined the speed functions on the complete graph. We argue that this model is often inadequate, in particular for arc routing problems involving services on edges of a road network. To fill this gap, we formally define the time-dependent capacitated arc routing problem (TDCARP), with travel and service speed functions given directly at the network level. Under these assumptions, the quickest path between locations can change over time, leading to a complex problem that challenges the capabilities of current solution methods. We introduce effective algorithms for preprocessing quickest paths in a closed form, efficient data structures for travel time queries during routing optimization, as well as heuristic and exact solution approaches for the TDCARP. Our heuristic uses the hybrid genetic search principle with tailored solution-decoding algorithms and lower bounds for filtering moves. Our branch-and-price algorithm exploits dedicated pricing routines, heuristic dominance rules and completion bounds to find optimal solutions for problem counting up to 75 services. Based on these algorithms, we measure the benefits of time-dependent routing optimization for different levels of travel-speed data accuracy