Automated vision inspection has become a vital part of computer related industries. Most of the existing inspection systems mainly utilize black and white images. In this paper, we consider an application of automated vision inspection in which cable color has to be recognized in order to detect the quality status of assembled wire harness. A back-propagation neural network is proposed to classify seven different cable colors. To represent a single point in image space, we use the ($L^*,\;a^*,\;b^*$) model which is one of commonly used color-coordinate systems in image processing. After training the neural network with ($L^*,\;a^*,\;b^*$) data obtained from color image, we tested its performance. The results show that the neural network is able to classify cable colors with high performance.