A stochastic card balance management problem with continuous and batch-type bilateral transactions

We study a stochastic continuous-review card balance management problem with two transaction patterns, namely, continuous and batch-type bilateral transactions, both in a Markovian environment. Motivated by the Autoload program used in public transit systems, the card is managed using a two-parameter band policy. Our cost structure includes activation and loading costs, and a fine for a negative balance. By applying hitting time theory and martingales, we derive the cost functionals and obtain, numerically, the optimal thresholds minimizing the expected discounted total cost. Surprisingly, a numerical study shows that the optimal policy is inherently linked with the outflow patterns, and is more sensitive to changes in withdrawal rates than to changes in batch sizes. We further show that timing is a significant factor in determining the policy: a high discount factor leads to frequent activations with smaller amounts.