A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

Distributionally robust optimization for the berth allocation problem under uncertainty

Berth allocation problems are amongst the most important problems occurring in port terminals, and they are greatly affected by several unpredictable events. As a result, the study of these problems under uncertainty has been a target of more and more researchers. Following this research line, we consider the berth allocation problem under uncertain handling times. A distributionally robust two-stage model is presented to minimize the worst-case of the expected sum of delays with respect to a set of possible probability distributions of the handling times. The solutions of the proposed model are obtained by an exact decomposition algorithm for which several improvements are discussed. An adaptation of the proposed algorithm for the case where the assumption of relatively complete recourse fails is also presented. Extensive computational tests are reported to evaluate the effectiveness of the proposed approach and to compare the solutions obtained with those resulting from the stochastic and robust approaches.Fundação para a Ciência e a Tecnologiapublishe

Discrete time and continuous time formulations for a short sea inventory routing problem

We consider a fuel oil distribution problem where an oil company is responsible for the routing and scheduling of ships between ports such that the demand for various fuel oil products is satisfied during the planning horizon. The production/consumption rates are given and assumed to be constant. We provide two alternative mixed integer formulations: a discrete time model adapted from the case where the production/consumption rates are varying and a classical continuous time formulation. We discuss different extended formulations and valid inequalities that allow us to reduce the linear gap of the two initial formulations. A computational study comparing the various models accordingly to their size, linear gap and running time, was conducted based on real small-size instances, using a commercial software

A hybrid heuristic for a stochastic production-inventory-routing problem

We consider a stochastic single item production-inventory-routing problem with a single producer and multiple clients. At the clients, demand is allowed to be backlogged incurring a penalty cost. Demands are considered uncertain. A recourse model is presented where the production and routing decisions are taken before the scenario is known, and the quantities to deliver to the clients and the inventory levels are adjustable to the scenario. Valid inequalities are introduced and a hybrid heuristic that combines ideas from the sample average approximation method and from relax-and-fix approaches is proposed. Preliminary tests based on randomly generated instances are reported showing that the hybrid heuristic performs better than the classical sample approximation algorithm for hard instances.publishe

The minimum cost network upgrade problem with maximum robustness to multiple node failures

The design of networks which are robust to multiple failures is gaining increasing attention in areas such as telecommunications. In this paper, we consider the problem of upgrading an existent network in order to enhance its robustness to events involving multiple node failures. This problem is modeled as a bi-objective mixed linear integer formulation considering both the minimization of the cost of the added edges and the maximization of the robustness of the resulting upgraded network. As the robustness metric of the network, we consider the value of the Critical Node Detection (CND) problem variant which provides the minimum pairwise connectivity between all node pairs when a set of c critical nodes are removed from the network. We present a general iterative framework to obtain the complete Pareto frontier that alternates between the minimum cost edge selection problem and the CND problem. Two different approaches based on a cover model are introduced for the edge selection problem. Computational results conducted on different network topologies show that the proposed methodology based on the cover model is effective in computing Pareto solutions for graphs with up to 100 nodes, which includes four commonly used telecommunication networks.publishe

A computational comparison of compact MILP formulations for the zero forcing number

Consider a graph where some of its vertices are colored. A colored vertex with a single uncolored neighbor forces that neighbor to become colored. A zero forcing set is a set of colored vertices that forces all vertices to become colored. The zero forcing number is the size of a minimum forcing set. Finding the minimum forcing set of a graph is NP-hard. We give a new compact mixed integer linear programming formulation (MILP) for this problem, and analyse this formulation and establish relation to an existing compact formulation and to two variants. In order to solve large size instances we propose a sequential search algorithm which can also be used as a heuristic to derive upper bounds for the zero forcing number. A computational study using Xpress (a MILP solver) is conducted to test the performances of the discussed compact formulations and the sequential search algorithm. We report results on cubic, Watts-Strogatz and randomly generated graphs with 10, 20 and 30 vertices.publishe

Lagrangian duality for robust problems with decomposable functions: the case of a robust inventory problem

We consider a class of min-max robust problems in which the functions that need to be “robustified” can be decomposed as the sum of arbitrary functions. This class of problems includes many practical problems, such as the lot-sizing problem under demand uncertainty. By considering a Lagrangian relaxation of the uncertainty set, we derive a tractable approximation, called the dual Lagrangian approach, that we relate with both the classical dualization approximation approach and an exact approach. Moreover, we show that the dual Lagrangian approach coincides with the affine decision rule approximation approach. The dual Lagrangian approach is applied to a lot-sizing problem, in which demands are assumed to be uncertain and to belong to the uncertainty set with a budget constraint for each time period. Using the insights provided by the interpretation of the Lagrangian multipliers as penalties in the proposed approach, two heuristic strategies, a new guided iterated local search heuristic, and a subgradient optimization method are designed to solve more complex lot-sizing problems in which additional practical aspects, such as setup costs, are considered. Computational results show the efficiency of the proposed heuristics that provide a good compromise between the quality of the robust solutions and the running time required in their computation.publishe

A maritime inventory routing problem with stochastic sailing and port times

This paper describes a stochastic short sea shipping problem where a company is responsible for both the distribution of oil products between islands and the inventory management of those products at consumption storage tanks located at ports. In general, ship routing and scheduling is associated with uncertainty in weather conditions and unpredictable waiting times at ports. In this work, both sailing times and port times are considered to be stochastic parameters. A two-stage stochastic programming model with recourse is presented where the first stage consists of routing, loading and unloading decisions, and the second stage consists of scheduling and inventory decisions. The model is solved using a decomposition approach similar to an L-shaped algorithm where optimality cuts are added dynamically, and this solution process is embedded within the sample average approximation method. A computational study based on real-world instances is presented

Mixed integer formulations for a routing problem with information collection in wireless networks

We study a routing-collecting problem where a system of stations is considered. A vehicle is responsible for collecting information generated continuously in the stations and to deliver it to a base station. The objective is to determine the vehicle route and the collection operations, both physical and wireless, in order to maximize the amount of information collected during a time horizon. Three mixed integer programming models are introduced and a computational study is reported to compare the performance of a solver based on each one of the models.publishe

Heuristics for a vehicle routing problem with information collection in wireless networks

International audienceWe consider a wireless network where a given set of stations is continuously generating information. A single vehicle, located at a base station, is available to collect the information via wireless transfer. The wireless transfer vehicle routing problem (WTVRP) is to decide which stations should be visited in the vehicle route, how long shall the vehicle stay in each station, and how much information shall be transferred from the nearby stations to the vehicle during each stay. The goal is to collect the maximum amount of information during a time period after which the vehicle returns to the base station. The WTVRP is NP-hard. Although it can be solved to optimality for small size instances, one needs to rely on good heuristic schemes to obtain good solutions for large size instances. In this work, we consider a mathematical formulation based on the vehicle visits. Several heuristics strategies are proposed, most of them based on the mathematical model. These strategies include constructive and improvement heuristics. Computational experiments show that a strategy that combines a combinatorial greedy heuristic to design a initial vehicle route, improved by a fix-and-optimize heuristic to provide a local optimum, followed by an exchange heuristic, affords good solutions within reasonable amount of running time