# Analysis of the stochastic cash balance problem using a level crossing technique

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## Abstract

The simple cash management problem includes the following considerations: the opportunity cost of holding too much cash versus the penalty cost of not having enough cash to meet current needs; the cost incurred (or profit generated) when making changes to cash levels by increasing or decreasing them when necessary; the uncertainty in timing and magnitude of cash receipts and cash disbursements; and the type of control policy that should be used to minimize the required level of cash balances and related costs. In this paper, we study a version of this problem in which cash receipts and cash disbursements occur according to two independent compound Poisson processes. The cash balance is monitored continuously and an order-point, order-up-to-level, and keep-level \( \left( {s, S, M} \right) \) policy is used to monitor the content, where \( s \le S \le M \). That is, (a) if, at any time, the cash level is below *s*, an order is immediately placed to raise the level to *S*; (b) if the cash level is between *s* and *M*, no action is taken; (c) if the cash level is greater than *M*, the amount in excess of *M* is placed into an earning asset. We seek to minimize the expected total costs per unit time of running the cash balance. We use a level-crossing approach to develop a solution procedure for finding the optimal policy parameters and costs. Several numerical examples are given to illustrate the tradeoffs.

## Keywords

Cash balance problem Stationary analysis Level-crossing method## Notes

### Acknowledgements

The author thanks the referee for constructive comments and suggestions that have led to many improvements in the paper’s content.

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