The extreme vertices design is developed as a procedure for conducting experimen& with mixtures when several factors have constraints placed on them. The constraints so imposed reduce the size of the factor space which would result had the factor levels been restricted to only 0 to 100 per cent. The selection of the vertices and the various centroids of the resulting hyper-polyhedron as the design is a method of determining a unique set of treatment combinations. This selection is motivated by the desire to explore the extremes as well as the center of the factor space.A non-linear programming procedure is used in determining the optimum treatment combination.