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Abstract: Non-negative matrix factorization (NMF) has been proposed as a mathematical tool for identifying the components of a dataset. However, popular NMF algorithms tend to operate slowly and do not always identify the components which are most representative of the data. In this paper, an alternative algorithm for performing NMF is developed using the geometry of the problem. The computational costs of the algorithm are explored, and it is shown to successfully identify the components of a simulated dataset. The development of the geometric algorithm framework illustrates the ill-posedness of the NMF problem and suggests that NMF is not sufficiently constrained to be applied successfully outside of a particular class of problems. [Copyright &y& Elsevier]

The article focuses on the model traffic with a combination of queuing theory and cellular automata in order the order the optimum number of tollbooths. The number of cars that arrived in the toll plaza is analytically described in a Poisson process can be managed through the modified Nagel-Schreckenberg cellular automata scheme. The consistency of the model with a variety of theory and experiment suggests that its accurate and robust. The changes in the distribution of the tollbooth wait time and the probability of random braking does not significantly alter the recommendations.