This paper deals with the problem of scheduling a set of unit‐time jobs on a set of uniform machines. The jobs are subject to conflict constraints modeled by a graph G called the conflict graph, in which adjacent jobs cannot be processed on a same machine. The objective considered herein is the minimization of maximum job completion time in the schedule, which is famous to be NP‐hard in the strong sense. The first part of this paper is an extensive study of the computational complexity of the problem restricted to several graph classes, namely: split graphs, interval graphs, forests, trees, paths and cycles. Afterward, we focus on the resolution of the problem with arbitrary conflict graphs. For this latter, a combination of a mixed integer linear programming (MILP) formulation, lower and upper bounds is proposed. A wild range of computational experiments proved the efficiency of this technique to tremendously reduce runtime and produce more optimal solutions (around 80% in average). Furthermore, a deep analysis of the resolution process based on both the density of the conflict graph as well as machine speeds (including identical machines) is thoroughly reported.
Data-driven models have recently proved to be a very powerful tool to extract relevant information from different kinds of datasets. However, datasets are often subject to multiple anomalies, including the loss of important parts of entries. In the context of intelligent transportation, we examine in this paper the impact of data loss on the behavior of one of the frequently used approaches to address this kind of problems in the literature, namely, the k-nearest neighbors model. The method designed herein is set to perform multi-step traffic flow forecasts in urban roads. In our study, we deploy non-prepossessed real data recorded by seven inductive loop detectors and delivered by the Traffic Management Center (VMZ) of Bremen (Germany). Firstly, we measure the performance of the model on a complete dataset of 11 weeks. The same dataset is then used to artificially create 50 incomplete datasets with different gap sizes and completeness levels. Afterwards, in order to reconstruct these datasets, we propose three computationally-low techniques, which proved through empirical testing to be efficient in reproducing missing entries. Thereafter, the performance of the E-KNN model is assessed under the original dataset, incomplete and filled-in datasets. Although the accuracy of E-KNN under incomplete and reconstructed datasets depends on gap lengths and completeness levels, under original dataset, the model proves to deliver six-step forecasts with an accuracy of 83% on average over 3 weeks of the test set, which also translates to a less than one car per minute error.
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