We develop an algorithm to solve a multiple objective linear programming problem with bounded variables. It is based on the scalarization theorem of optimal solutions of multiobjective linear programs and the single objective adaptive method. We suggest a process for the search for the first efficient solution without having to calculate a feasible solution, and we elaborate a method to generate efficient solutions, weakly efficient solutions, and $\epsilon$-efficient solutions. Supporting theoretical results are established and the method is demonstrated on a numerical example.