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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The shortest path problem is one of the most popular network optimization problems and it is of great importance in areas such as transportation, network design or telecommunications. This model deals with determining a minimum weighted path between a pair of nodes of a given network. The deterministic version of the problem can be solved easily, in polynomial time, but sometimes uncertainty or vagueness is encountered. In this work we consider the rough interval shortest path problem, where each arc's wight is represented by a lower approximation interval and an upper approximation interval, which surely contain the real weight value and that may possibly contain the real weight value, respectively. A labeling algorithm is developed to find efficient solutions of the problem.

While finding a path between two nodes is the basis for several applications, the need for alternative paths also may have various motivations. For instance, this can be of interest for ensuring reliability in a telecommunications network, for reducing the consequences of possible accidents in the transportation of hazardous materials, or to decrease the risk of robberies in money distribution. Each of these applications has particular characteristics, but they all have the common purpose of searching for a set of paths which are as dissimilar as possible with respect to the nodes/arcs that compose them.In this work we present linear integer programming formulations for finding K dissimilar paths, with the main goal of preventing the overlap of arcs in the paths for a given integer K. The different formulations are tested for randomly generated general networks and for grid networks. The obtained results are compared in terms of the solutions' dissimilarity and of the run time. Two of the new formulations are able to find 10 paths with better average and minimum dissimilarity values than an iterative approach in the literature, in less than 20 seconds, for random networks with 500 nodes and 5000 arcs.