Consider a depot and a set of client nodes divided into non-disjoint subsets. Each subset is associated with a route. The aim of the Travelling Salesman Problem with positional consistency constraints is to generate a minimum cost set of routes, one for each subset, such as all the routes start and end at the depot, and each route visits all the nodes of the corresponding subset. Positional consistency means that client nodes should be visited in the same relative position in all the routes they appear in. The problem was inspired by an application in healthcare services. Several compact formulations were developed to solve the problem with the current ILP packages. Due to the positional constraints, the study focused on so-called time-dependent formulations with specially developed consistency constraints. To overcome the size of the previously proposed formulations, an aggregated formulation that does not use route-indexed variables was also proposed and shown to be valid. Some enhancements were proposed for disaggregated and aggregated models, some including exponentially sized sets of constraints for which separation routines were implemented. Theoretical relations of the corresponding LP relaxations were also established. An empirical comparison was carried out using instances with characteristics from the healthcare application and more general instances adapted from the literature. Computational results show that, in some cases, the aggregated formulation is the most efficient one. An iterated local search algorithm was also developed to address the instances that the exact methods could not solve to optimality. The heuristic showed a robust performance for the instances with known optimal solutions and was able to improve, in some cases, considerably, the upper bounds provided by the ILP packages combined with the proposed models, for most of the remaining instances.

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.

The Minimum Weighted Tree Reconstruction (MWTR) problem consists of finding a minimum length weighted tree connecting a set of terminal nodes in such a way that the length of the path between each pair of terminal nodes is greater than or equal to a given distance between the considered pair of terminal nodes. This problem has applications in several areas, namely, the inference of phylogenetic trees, the modeling of traffic networks and the analysis of internet infrastructures. In this paper, we investigate the MWTR problem and we present two compact mixed-integer linear programming models to solve the problem. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that the best of the two models is able to solve instances of the problem having up to 15 terminal nodes.

As client demands grow, optical network operators are required to introduce lightpaths of higher line rates in order to groom more demand into their network capacity. For a given fiber network and a given set of client demands, the minimum cost network design is the task of assigning routing paths and wavelengths for a minimum cost set of lightpaths able to groom all client demands. The variant of the optical network design problem addressed in this paper considers a transparent optical network, single hop grooming, client demands of a single interface type, and lightpaths of two line rates. We discuss two slightly different mixed integer linear programming models that define the network design problem combining grooming, routing, and wavelength assignment. Then, we propose a parameters increase rule and three types of additional constraints that, when applied to the previous models, make their linear relaxation solutions closer to the integer solutions. Finally, we use the resulting models to derive a hybrid heuristic method, which combines a relax-and-fix approach with an integer linear programming-based local search approach. We present the computational results showing that the proposed heuristic method is able to find solutions with cost values very close to the optimal ones for a real nation-wide network and considering a realistic fiber link capacity of 80 wavelengths. Moreover, when compared with other approaches used in the problem variants close to the one addressed here, our heuristic is shown to compute solutions, on average, with better cost values and/or in shorter runtimes.

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.

Given a network defined by a graph, a weight associated to each node pair and a positive parameter p, the CND problem addressed here is to identify a set of at most p critical nodes minimizing the total weight of the node pairs that remain connected when all critical nodes are removed. We improve previously known compact models and present computational results, based on telecommunication backbone networks, showing that the proposed models are much more efficiently solved and enable us to obtain optimal solutions for networks up to 200 nodes and p values up to 20 critical nodes within a few minutes in the worst cases.

For a given fiber network and a given set of client demands, the transparent optical network design problem is the task of assigning routing paths and wavelengths for a set of lightpaths able to groom all client demands. We address this design problem minimizing the impact of a given set of critical nodes. The problem is tackled in two steps: first, we minimize the demand that is disrupted by the simultaneous failure of all critical nodes; second, we minimize the network design cost guaranteeing that the minimum disrupted demand is met. We present MILP models for each step, together with valid inequalities strengthening both models. For the second step, an efficient hybrid heuristic is also proposed.

We consider the Minimum Weighted Tree Reconstruction (MWTR) problem and two matheuristic methods to obtain optimal or near-optimal solutions: the Feasibility Pump heuristic and the Local Branching heuristic. These matheuristics are based on a Mixed Integer Programming (MIP) model used to find feasible solutions. We discuss the applicability and effectiveness of the matheuristics to obtain solutions to the MWTR problem. The purpose of the MWTR problem is to find a minimum weighted tree connecting a set of leaves in such a way that the length of the path between each pair of leaves is greater than or equal to a given distance between the considered pair of leaves. The Feasibility Pump matheuristic starts with the Linear Programming solution, iteratively fixes the values of some variables and solves the corresponding problem until a feasible solution is achieved. The Local Branching matheuristic, in its turn, improves a feasible solution by using a local search. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that these matheuristics are quite effective in finding feasible solutions and present small gap values. Each matheuristic can be used independently; however, the best results are obtained when used together. For instances of the problem having up to 17 leaves, the feasible solution obtained by the Feasibility Pump heuristic is improved by the Local Branching heuristic. Noticeably, when comparing with existing based models processes that solve instances having up to 15 leaves, this achievement of the matheuristic increases the size of solved instances.

The facility layout problem is concerned with finding an arrangement of non-overlapping indivisible departments within a facility so as to minimize the total expected flow cost. In this paper we consider the special case of multi-row layout in which all the departments are to be placed in three or more rows, and our focus is on, for the first time, solutions for large instances. We first propose a new mixed integer linear programming formulation that uses continuous variables to represent the departments’ location in both x and y coordinates, where x represents the position of a department within a row and y represents the row assigned to the department. We prove that this formulation always achieves an optimal solution with integer values of y, but it is limited to solving instances with up to 13 departments. This limitation motivates the application of a two-stage optimization algorithm that combines two mathematical optimization models by taking the output of the first-stage model as the input of the second-stage model. This algorithm is, to the best of our knowledge, the first one in the literature reporting solutions for instances with up to 100 departments.

In a previous work we showed that hardlimit multilayer neural networks have more computational power than sigmoidal multilayer neural networks [1]. In 1962 Minsky and Papert showed the limitations of a single perceptron which can only solve linearly separable classification problems and since at that time there was no algorithm to find the weights of a multilayer hardlimit perceptron research on neural networks stagnated until the early eighties with the invention of the Backpropagation algorithm [2]. Nevertheless since the sixties there have arisen some proposals of algorithms to implement logical functions with threshold elements or hardlimit neurons that could have been adapted to classification problems with multilayer hardlimit perceptrons and this way the stagnation of research on neural networks could have been avoided. Although the problem of training a hardlimit neural network is NP-Complete, our algorithm based on mathematical programming, a mixed integer linear model (MILP), takes few seconds to train the two input XOR function and a simple logical function of three variables with two minterms. Since any linearly separable logical function can be implemented by a perceptron with integer weights, varying them between -1 and 1 we found all the 10 possible solutions for the implementation of the two input XOR function and all the 14 and 18 possible solutions for the implementation of two logical functions of three variables, respectively, with a two layer architecture, with two neurons in the first layer. We describe our MILP model and show why it consumes a lot of computational resources, even a small hardlimit neural network translates into a MILP model greater than 1G, implying the use of a more powerful computer than a common 32 bits PC. We consider the reduction of computational resources as the near future work main objective to improve our novel MILP model and we will also try a nonlinear version of our algorithm based on a MINLP model that will consume less memory.