The minimum utility cost problem is investigated for a single mass exchanger which is required to meet composition targets in multiple components. The problem is formulated as a nonlinear program. Properties for this nonlinear program are then established. These lead to a combinatorial global solution methodology whose complexity grows as the square of the number of components considered. The minimum utility cost is shown to not necessarily be the maximum of the single component minimum utility costs. The solution procedure is illustrated with two example applications.