  This paper investigates a Multistage Distributionally Robust Optimization
(MDRO) approach to water allocation under climate uncertainty. The MDRO is
formed by creating sets of conditional distributions (called conditional
ambiguity sets) on a finite scenario tree. The distributions in the conditional
ambiguity sets remain close to a nominal conditional distribution according a
$\phi$-divergence (e.g., Kullback-Liebler divergence, Hellinger distance, Burg
entropy, etc.). The paper discusses a decomposition algorithm to solve the
resulting MDRO and applies the modeling and solution techniques to allocate
water in a rapidly-developing area of Tucson, Arizona. Tucson, like many arid
and semi-arid regions around the world, faces considerable uncertainty in its
ability to provide water for its citizens in the future. The primary sources of
uncertainty in the Tucson region include (1) unpredictable population growth,
(2) the availability of water from the Colorado River, and (3) the effects of
climate variability on water consumption. This paper integrates forecasts for
all these sources of uncertainty into a single optimization model for robust
and sustainable water allocation. Then, it uses this model to analyze the value
of constructing additional treatment facilities to reduce future water
shortages. The results indicate that the MDRO approach can be very valuable for
water managers by providing insights to minimize their risks and help them plan
for the future.
