Continuous knapsack sets with divisible capacities

We study two continuous knapsack sets (Formula presented.) and (Formula presented.) with (Formula presented.) integer, one unbounded continuous and (Formula presented.) bounded continuous variables in either (Formula presented.) or (Formula presented.) form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and (Formula presented.) polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the (Formula presented.) case. We also provide an extended formulation for the (Formula presented.) case. It follows that, given a specific objective function, optimization over both (Formula presented.) and (Formula presented.) can be carried out by solving (Formula presented.) polynomial size linear programs. A further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality of the convex hull of such sets. © 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society

Polyhedral techniques in combinatorial optimization II: applications and computations

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems

Reformulation and Decomposition Approaches for Traffic Routing in Optical Networks

International audienceWe consider a multi-layer network design model arising from a real-life telecommunication application where traffic routingdecisions imply the installation of expensive nodal equipment. Customer requests come in the form of bandwidthreservations for a given origin destination pair. Bandwidth demands are expressed as multiples of nominal granularities. Each request must be single-path routed. Grooming several requests on the same wavelength and multiplexing wavelengths in the same optical stream allow a more efficient use of network capacity. However, each addition or withdrawal of a request from a wavelength requires optical to electrical conversion and the use of cross-connect equipment with expensive ports of high densities. The objective is to minimize the number of required ports of the cross-connect equipment. We deal with backbone optical networks, therefore with networks with a moderate number of nodes (14 to 20) but thousands of requests. Further difficulties arise from the symmetries in wavelength assignment and traffic loading. Traditional multi-commodity network flowapproaches are not suited for this problem. Instead, four alternative models relying on Dantzig-Wolfe and/or Benders' decomposition areintroduced and compared. The formulations are strengthened using symmetry breaking restrictions, variable domain reduction, zero-onediscretization of integer variables, and cutting planes. The resulting dual bounds are compared to the values of primal solutions obtained through hierarchical optimization and rounding procedures. For realistic size instances, our best approaches provide solutions with optimality gap of approximately 5% on average in around two hours of computing time

Reformulation and Decomposition Approaches for Traffic Routing in Optical Networks

International audienceWe consider a multi-layer network design model arising from a real-life telecommunication application where traffic routingdecisions imply the installation of expensive nodal equipment. Customer requests come in the form of bandwidthreservations for a given origin destination pair. Bandwidth demands are expressed as multiples of nominal granularities. Each request must be single-path routed. Grooming several requests on the same wavelength and multiplexing wavelengths in the same optical stream allow a more efficient use of network capacity. However, each addition or withdrawal of a request from a wavelength requires optical to electrical conversion and the use of cross-connect equipment with expensive ports of high densities. The objective is to minimize the number of required ports of the cross-connect equipment. We deal with backbone optical networks, therefore with networks with a moderate number of nodes (14 to 20) but thousands of requests. Further difficulties arise from the symmetries in wavelength assignment and traffic loading. Traditional multi-commodity network flowapproaches are not suited for this problem. Instead, four alternative models relying on Dantzig-Wolfe and/or Benders' decomposition areintroduced and compared. The formulations are strengthened using symmetry breaking restrictions, variable domain reduction, zero-onediscretization of integer variables, and cutting planes. The resulting dual bounds are compared to the values of primal solutions obtained through hierarchical optimization and rounding procedures. For realistic size instances, our best approaches provide solutions with optimality gap of approximately 5% on average in around two hours of computing time

Operational Research and Machine Learning Applied to Transport Systems

The New Economy, environmental sustainability and global competitiveness drive inno- vations in supply chain management and transport systems. The New Economy increases the amount and types of products that can be delivered directly to homes, challenging the organisation of last-mile delivery companies. To keep up with the challenges, deliv- ery companies are continuously seeking new innovations to allow them to pack goods faster and more efficiently. Thus, the packing problem has become a crucial factor and solving this problem effectively is essential for the success of good deliveries and logistics. On land, rail transportation is known to be the most eco-friendly transport system in terms of emissions, energy consumption, land use, noise levels, and quantities of people and goods that can be moved. It is difficult to apply innovations to the rail industry due to a number of reasons: the risk aversion nature, the high level of regulations, the very high cost of infrastructure upgrades, and the natural monopoly of resources in many countries. In the UK, however, in 2018 the Department for Transport published the Joint Rail Data Action Plan, opening some rail industry datasets for researching purposes. In line with the above developments, this thesis focuses on the research of machine learning and operational research techniques in two main areas: improving packing operations for logistics and improving various operations for passenger rail. In total, the research in this thesis will make six contributions as detailed below. The first contribution is a new mathematical model and a new heuristic to solve the Multiple Heterogeneous Knapsack Problem, giving priority to smaller bins and consid- ering some important container loading constraints. This problem is interesting because many companies prefer to deal with smaller bins as they are less expensive. Moreover, giving priority to filling small bins (rather than large bins) is very important in some industries, e.g. fast-moving consumer goods. The second contribution is a novel strategy to hybridize operational research with ma- chine learning to estimate if a particular packing solution is feasible in a constant O(1) computational time. Given that traditional feasibility checking for packing solutions is an NP-Hard problem, it is expected that this strategy will significantly save time and computational effort. The third contribution is an extended mathematical model and an algorithm to apply the packing problem to improving the seat reservation system in passenger rail. The problem is formulated as the Group Seat Reservation Knapsack Problem with Price on Seat. It is an extension of the Offline Group Seat Reservation Knapsack Problem. This extension introduces a profit evaluation dependent on not only the space occupied, but also on the individual profit brought by each reserved seat. The fourth contribution is a data-driven method to infer the feasible train routing strate- gies from open data in the United Kingdom rail network. Briefly, most of the UK network is divided into sections called berths, and the transition point from one berth to another is called a berth step. There are sensors at berth steps that can detect the movement when a train passes by. The result of the method is a directed graph, the berth graph, where each node represents a berth and each arc represents a berth-step. The arcs rep- resent the feasible routing strategies, i.e. where a train can move from one berth. A connected path between two berths represents a connected section of the network. The fifth contribution is a novel method to estimate the amount of time that a train is going to spend on a berth. This chapter compares two different approaches, AutoRe- gressive Moving Average with Recurrent Neural Networks, and analyse the pros and cons of each choice with statistical analyses. The method is tested on a real-world case study, one berth that represent a busy junction in the Merseyside region. The sixth contribution is an adaptive method to forecast the running time of a train journey using the Gated Recurrent Units method. The method exploits the TD’s berth information and the berth graph. The case-study adopted in the experimental tests is the train network in the Merseyside region