Various types of support vector machines (SVMs) have been proposed for multiclass classification tasks. Recently, hypersphere and hyperellipsoid classifiers have attracted much attention since they outperform the existing hyperplane classifiers. The twin hypersphere multi-classification support vector machine (THKSVM) is an effective SVM-based method that exploits multiple hyperspheres to classify data effectively. The THKSVM outperforms the existing methods based on the SVM in terms of computational time, even though its prediction accuracy is competitive. In this paper, we modify the THKSVM to achieve better classifiers within a reasonable timeframe. Firstly, we adopt hyperellipsoids instead of hyperspheres for classification purposes. Secondly, we replace the multiple constraints in the THKSVM with a single, specific constraint. Consequently, our proposed optimization model comprises a single nonconvex quadratic constraint and a convex quadratic objective function. Although the problem remains nonconvex optimization, the uniqueness of the constraint allows us to analytically obtain its global optimum. Therefore, we anticipate that our proposed method will surpass the THKSVM in terms of computational efficiency and prediction accuracy. Finally, we conducted several numerical experiments to show the effectiveness of our proposed method.