A dynamic ordering policy for a stochastic inventory problem with cash constraints

DOI10.1016/J.OMEGA.2020.102378arXiv1912.07438MaRDI QIDQ6331198

Roberto Rossi, Unnamed Author

Publication date: 16 December 2019

Abstract: This paper investigates a stochastic inventory management problem in which a cash-constrained small retailer periodically purchases a product from suppliers and sells it to a market while facing non-stationary demands. In each period, the retailer's available cash restricts the maximum quantity that can be ordered. There exists a fixed ordering cost for the retailer when purchasing. We partially characterize the optimal ordering policy by showing it has an $$ structure: for each period, when initial inventory is above the $$s threshold, no product should be ordered no matter how much initial cash it has; when initial inventory is not large enough to be a $$ threshold, it is also better to not order when initial cash is below the threshold $C$. The values of $C$ may be state-dependent and related to each period's initial inventory. A heuristic policy $(s,C(x),S)$ is proposed: when initial inventory $x$ is less than $s$ and initial cash is greater than $C(x)$, order a quantity that brings inventory as close to $S$ as possible; otherwise, do not order. We first determine the values of the controlling parameters $s$, $C(x)$ and $S$ based on the results of stochastic dynamic programming and test their performance via an extensive computational study. The results show that the $(s,C(x),S)$ policy performs well with a maximum optimality gap of less than 1% and an average gap of approximately 0.01%. We then develop a simple and time-efficient heuristic method for computing policy $(s,C(x),S)$ by solving a mixed-integer linear programming problem and approximate newsvendor models: the average gap for this heuristic is approximately 2% on our test bed.

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