Abstract
AbstractThe purpose of the dissimilar paths problem is to find a set of 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 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 ‐constraint method is adapted to solve them using a decreasing and an increasing strategy for updating . These methods are tested for finding sets of 10 paths in random and grid instances to assess the efficiency of the ‐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.
Highlights
The present paper addresses the determination of sets of K paths between two nodes in a network, with two goals: the minimization of the total cost of the K paths; and the maximization
The term dissimilarity is often found in the literature intending to measure the diversity between two entities, and, in practical terms, it is useful in a number of situations, whenever we look for alternative solutions
We focus on arc dissimilarity
Summary
The present paper addresses the determination of sets of K paths between two nodes in a network, with two goals: the minimization of the total cost of the K paths; and the maximization. The conclusions to be drawn from the present work will be useful for the applications community for two reasons: the proposed models can be used in various contexts by performing adequate adjustments; and, they allow a better understanding on the trade-off between the two objective functions in different types of networks, avoiding the distorting effects caused by the presence of other constraints, involved in the above mentioned studies This is done by revisiting four of the models presented: one which intends to minimize the number of arcs repeated in the paths; another one which intends to minimize the number of arc repetitions; as well as variants of these two resulting from imposing an upper bound on the number of presences of each arc in the solution.
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