A new approach for the stochastic cash balance problem with fixed costs.

*(English)*Zbl 1176.90011This paper considers the stochastic cash balance problem which is a cost minimization problem faced by a firm that has to decide how much cash to hold in order to meet the transaction requirements for a given planning horizon with multiple time periods. At the beginning of each time period, the firm can decide to increase or decrease the cash levels. Both the ordering and the return costs include a fixed and a variable component. At the end of the transaction period, cash levels may be positive or negative depending on whether cash is held or owed. Now a holding cost is charged when the cash level is positive while a penalty cost is charged when the cash level is negative. The objective is to find a policy for holding cash which minimizes the total expected (possibly discounted) cost over the whole planning horizon. This paper presents new insights about this stochastic cash balance problem with fixed costs for both orders and returns. The authors utilize the concept of symmetric \(K\)-convexity developed by X. Chen and D. Simchi-Levi [Oper. Res. 52, No. 6, 887–896 (2004; Zbl 1165.90308); Math. Oper. Res. 29, No. 3, 698–723 (2004; Zbl 1082.90025)] and the concept of \((K,Q)\)-convexity (which reduces to symmetric \(K\)-convexity when \(K = Q\)) introduced by Ye and Duenyas (2003) to characterize the structure of an optimal policy.

Reviewer: Frank Werner (Magdeburg)

##### MSC:

90B05 | Inventory, storage, reservoirs |

91B70 | Stochastic models in economics |

90C39 | Dynamic programming |

##### Keywords:

cash balance; stochastic problems; holding cost; penalty cost; dynamic programming; optimal policy
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\textit{X. Chen} and \textit{D. Simchi-Levi}, Probab. Eng. Inf. Sci. 23, No. 4, 545--562 (2009; Zbl 1176.90011)

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