  Product ranking is the core problem for revenue-maximizing online retailers.
To design proper product ranking algorithms, various consumer choice models are
proposed to characterize the consumers' behaviors when they are provided with a
list of products. However, existing works assume that each consumer purchases
at most one product or will keep viewing the product list after purchasing a
product, which does not agree with the common practice in real scenarios. In
this paper, we assume that each consumer can purchase multiple products at
will. To model consumers' willingness to view and purchase, we set a random
attention span and purchase budget, which determines the maximal amount of
products that he/she views and purchases, respectively. Under this setting, we
first design an optimal ranking policy when the online retailer can precisely
model consumers' behaviors. Based on the policy, we further develop the
Multiple-Purchase-with-Budget UCB (MPB-UCB) algorithms with $\~O(\sqrt{T})$
regret that estimate consumers' behaviors and maximize revenue simultaneously
in online settings. Experiments on both synthetic and semi-synthetic datasets
prove the effectiveness of the proposed algorithms.
