In this paper we present a new formulation for the Time-Dependent Travelling Salesman Problem (TDTSP). We start by reviewing well known natural formulations with some emphasis on the formulation by Picard and Queyranne (1978). The main feature of this formulation is that it uses, as a subproblem, an exact description of the n-circuit problem. Then, we present a new formulation that uses more variables and is based on using, for each node, a stronger subproblem, namely a ncircuit subproblem with the additional constraint that the corresponding node is not repeated in the circuit. Although the new model has more variables and constraints than the original PQ model, the results given from our computational experiments show that the linear programming relaxation of the new model gives, for many of the instances tested, gaps that are close to zero. Thus, the new model is worth investigating for solving TDTSP instances. We have also provided a complete characterization of the feasible set of the corresponding linear programming relaxation in the space of the variables of the PQ model. This characterization permits us to suggest alternative methods of using the proposed formulations.
We consider two types of hop-indexed models for the unit-demand asymmetric Capacitated Vehicle Routing Problem (CVRP): (a) capacitated models guaranteeing that the number of commodities (paths) traversing any given arc does not exceed a specified capacity; and (b) hop-constrained models guaranteeing that any route length (number of nodes) does not exceed a given value. The latter might, in turn, be divided into two classes: (b1) those restricting the length of the path from the depot to any node k, and (b2) those restricting the length of the circuit passing through any node k. Our results indicate that formulations based upon circuit lengths (b2) lead to models with a linear programming relaxation that is tighter than the linear programming relaxation of models based upon path lengths (b1), and that combining features from capacitated models with those of circuit lengths can lead to formulations for the CVRP with a tight linear programming bound. Computational results on a small number of problem instances with up to 41 nodes and 440 edges show that the combined model with capacities and circuit lengths produce average gaps of less than one percent. We also briefly examine the asymmetric travelling salesman problem (ATSP), showing the potential use of the ideas developed for the vehicle routing problem to derive models for the ATSP with a linear programming relaxation bound that is tighter than the linear programming relaxation bound of the standard Dantzig, Fulkerson and Johnson [G. Dantzig, D. Fulkerson, D. Johnson, Solution of large-scale travelling salesman problem, Operations Research 2 (1954) 393-410] formulation. : Vehicle routing problem; Hop-indexed network flow models; Travelling salesman problem "Preamble. George Dantzig's fingerprints permeate essentially the entire field of mathematical programming. In particular, he was the first person to recognize the extraordinary modelling power of integer programming [12], the first, with co-authors, to suggest the value of adding valid inequalities to improve the modelling of integer programming models [13], and the first, with a co-author, to study solution approaches to vehicle routing
Agriculture has undergone some very important changes over the last few decades. The emergence and evolution of precision agriculture has allowed to move from the uniform site management to the site-specific management, with both economic and environmental advantages. However, to be implemented effectively, site-specific management requires within-field spatial variability to be well-known and characterized. In this paper, an algorithm that delineates within-field management zones in a maize plantation is introduced. The algorithm, based on triclustering, mines clusters from temporal remote sensing data. Data from maize crops in Alentejo, Portugal, have been used to assess the suitability of applying triclustering to discover patterns over time, that may eventually help farmers to improve their harvests.
The purpose of the K dissimilar paths problem is to find a set of K paths, between the same pair of nodes, which share few arcs. The problem has been addressed from an application point of view, and integer programming formulations have also been introduced recently. In the present work, it is assumed that each arc is assigned with a cost, and the goal is then to find K dissimilar paths while simultaneously minimizing the total cost. Some of the previous formulations: one minimizing the number of repeated arcs, another one minimizing the number of arc repetitions, as well as modifications that bound the number of paths in which the arcs appear, are extended with a cost function. Properties of the resulting biobjective problems are studied and the ε‐constraint method is adapted to solve them using a decreasing and an increasing strategy for updating ε. These methods are tested for finding sets of 10 paths in random and grid instances to assess the efficiency of the ε‐constraint methods and the performance of the formulations to calculate shortest and dissimilar paths. Results show that minimizing the number of arc repetitions produces efficient solutions with higher dissimilarities faster than minimizing the number of repeated arcs. The cost range of the solutions is similar in both approaches. Additionally, bounding the number of paths in which each arc appears improves the quality of the solutions as to the dissimilarity while worsening its cost.
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