A carrier head for a polishing system includes a housing, a flexible membrane, a first plurality of pressure supply lines, a second plurality of pressure supply lines, and a valve assembly. The flexible membrane defines a multiplicity of independently pressurizable chambers. The valve assembly has a multiplicity of valves with each respective valve of the multiplicity of valves coupled to a respective pressure chamber from the multiplicity of independently pressurizable chambers. Each respective valve is configured to selectively couple the respective pressure chamber to one pressure supply line from a pair of pressure supply lines that include a pressure supply line from the first plurality of pressure supply lines and a pressure supply line from the second plurality of pressure supply lines.