Machine learning with artificial neural network (ANN)-based methods is a powerful tool for the prediction and exploitation of the subtle relationships between the composition and properties of materials. This work utilizes an ANN to predict the composition of high-entropy alloys (HEAs) based on non-equimolar AlCoCrFeMnNi in order to achieve the highest hardness in the system. A simulated annealing algorithm is integrated with the ANN to optimize the composition. A bootstrap approach is adopted to quantify the uncertainty of the prediction. Without any guidance, the design of new compositions of AlCoCrFeMnNi-based HEAs would be difficult by empirical methods. This work successfully demonstrates that, by applying the machine learning method, new compositions of AlCoCrFeMnNi-based HEAs can be obtained, exhibiting hardness values higher than the best literature value for the same alloy system. The correlations between the predicted composition, hardness, and microstructure are also discussed.