A recently derived Multicriteria Simplex Method [Yu, P. L., M. Zeleny. 1975. The set of all nondominated solutions in linear cases and a multicriteria simplex method. J. Math. Anal. Appl. 49 (2, February) 430–468.] is used to study some basic properties in the decomposition of parametric space. A new type of parametric space, which arises naturally in its formulation, is used. Two computational methods are discussed. The first one is an indirect algebraic method which locates the corresponding set of all nondominated extreme points. The second one is a direct geometric decomposition method which is similar to that discussed by Gal and Nedoma [Gal, T., J. Nedoma. 1972 Multiparametric linear programming. Management Sci.. 18 406–421.]. The difficulties of such a geometric method are discussed through an example.