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Evaluation of Quality of Service (QoS) is an important task in managing computer networks. In this study, an innovative QoS evaluation system was proposed. The system combines discrimination features of supervised and unsupervised neural networks to analyse and assess QoS for transmission of Voice over Internet Protocol (VoIP) in a simulated computer network. The transmitted application' QoS parameters were initially analysed by the unsupervised learning Kohonen neural network. The analysed QoS parameters were then used as inputs to a supervised learning Multi-Layer Perceptron (MLP) neural network in order to quantify the overall QoS. The QoS assessment results from the proposed method correlated closely with the previously developed QoS assessment methods that were based on fuzzy logic, regression model, and Euclidean distance measure. However, the neural network's learning ability resulted in the system's parameters to be adaptively determined, reducing the complexity of the system design and facilitating ease of further improvements in the system's capability.