Decision-making is significant in economics, education, management, industries, and many real-life situations. Every decision-making problem does not have crisp parameters and restrictions, and may have uncertainties and qualitative information. Trapezoidal Hesitant fuzzy sets are highly beneficial for dealing with situations that are more uncertain and have some qualitative aspects. First, this paper introduces four score functions on the class of trapezoidal hesitant fuzzy elements, namely mean-position, right-spread, inference, and support position score function to define a complete ranking principle on trapezoidal hesitant fuzzy elements. Secondly, we propose a complete ranking principle on the class of trapezoidal Hesitant fuzzy sets by generalizing the four proposed score functions on the set of trapezoidal hesitant fuzzy elements. Thirdly, we extend our work to suggest another complete ranking principle on the new important sub-class of the trapezoidal Hesitant fuzzy set. Fourthly, we compare our proposed complete ranking principle with a few existing important ranking principles discussed in the trapezoidal Hesitant fuzzy sets (TrHFS) class. Further, we establish two new simplex algorithms on a special class of trapezoidal Hesitant fuzzy sets to solve fully Hesitant fuzzy linear programming problems (HFLPP). These new algorithms incorporate the proposed complete ranking principle on the trapezoidal Hesitant fuzzy set. Finally, we solve a few numerical examples of linear programming problems with fully Hesitant fuzzy information to show the applicability and potentiality of the proposed linear programming method. Complete ranking principle in the class of TrHFS and the use of complete ranking principle in solving a fully HFLPP is proposed for the first time in the literature.