Presently the only available method of exploring the 15-dimensional entropy region formed by the entropies of four random variables is the one of Zhang and Yeung from 1998. It is argued that their method is equivalent to solving linear multiobjective optimization problems. Benson's outer approximation algorithm is a fundamental tool for solving these optimization problems. An improved version of Benson's algorithm is described which requires solving one scalar linear program in each iteration rather than two or three as in previous versions. During the algorithm design special care was taken for numerical stability. The implemented algorithm was used to check previous statements about the entropy region, and to gain new information on that region. The experimental results demonstrate the viability of the method for determining the extremal set of medium size, numerically ill-posed optimization problems. With growing problem size two limitations of Benson's algorithm have been observed: the inefficiency of the scalar LP solver on one hand and the unexpectedly large number of intermediate vertices on the other.