A probabilistic approach to the stochastic fluid cash management balance problem

We consider a stochastic cash balance problem in which cash receipts and cash disbursements change linearly and form a fluid process. The firm has to decide how much cash to hold in order to meet its obligations. The cash balance is monitored continuously and a four thresholds (0, S, M, m) policy is used to control the cash (0 < S< M< m). That is, (a) If, at any time, the cash level drops to an order-level 0, a cash transfer of size S is requested such that the cash level is immediately raised to level S. (b) Whenever the cash level lies within the interval (0, m), no action is taken. (c) Every time the cash balance reaches the salvage-level m, it is restored to a level M and the excess amount (m- M) is invested in other earning assets that provide a suitable return. We further study the extended (s, S, M) policy, in which s be an order level and each request for a cash transfer takes some random time (called the lead-time) until it is approved. Using fluid-flow results and multi-dimensional martingales, we construct a closed-form expression for the discounted total cost of running the cash balance. Numerical study provides several guidelines for the optimal control. For example, we show that even when the holding cost is high and receipts occur more frequently, it would be wise for the firm to hold on some extra cash.