1. Proposition E.1 shows that with uniformly related hubs, � j = 0 for all j ? [J] constitutes an optimal solution to (16) (this is equivalent to the flow balance constraint in (16) being redundant) and the controls d i (x, i, j) and d i (x, j, i) in the Lagrangian policy derived from (16) are identical across hubs. We can show that when hubs are uniformly related, at optimality, the flow between each hub-spoke pair is balanced;Assumption E.1 is a form of symmetry in which, for each spoke, the revenue functions and ratio of requests are identical across hubs