In terms of the multiple attribute decision making(MADM) problems under hesitant triangular fuzzy environment, the distance measure for the hesitant triangular fuzzy sets is first introduced, and then the concept of the hesitant triangular fuzzy power average(HTFPA) is proposed. In order to effectively alleviate the influence of these ``false'' or ``biased'' arguments on the final decision-making result, the generalized hesitant triangular fuzzy power average(GHTFPA) is introduced and its related properties are also investigated. Then, a nonlinear programming model is constructed on the basis of the least absolute deviation method to reasonably attain the parameter of the GHTFPA operator. Furthermore, a novel approach, with respect to the MADM problem in which there are interactions among attributes and the attribute weights are unknown, is provided by utilizing the GHTFPA operator. Finally, a numerical example and a comparison are provided to illustrate the feasibility and rationality of the proposed approach.