We introduce a method to generate a new class of lifetime models based on the bounded distributions such that the defined models are exclusively a special case of the new class. A new subfamily, Generalized Alpha Power (GAP) is discussed and some stochastic orders in this subfamily are investigated to identify the proposed method effect. The performance of the maximum likelihood estimators based on the simulation is studied and in the end, the importance and flexibility of the new family for the models are illustrated by a real data set. Our results indicate that using the proposed method substantially improves the fitness of any G-family model and can be extended to any real data set. Finally, the GAPTW regression model is applied to the kidney infection data.