Abstract: Lexicographic Linear Goal programming within a pre-emptive priority structure has been one of the most widely used techniques considered in solving multiple objective problems. In the past several years, the modified simplex algorithm has been shown to be widely used and very accurate in computational formulation. Orumie and Ebong recently developed a generalized linear goal programming algorithm that is efficient. A new approach for solving lexicographic linear Goal programming problem is developed, together with an illustrative example. The method is efficient in reaching solution.