This paper studies an open question in the warehouse problem where a merchant trading a commodity tries to find an optimal inventory-trading policy to decide on purchase and sale quantities during a fixed time horizon in order to maximize their total pay-off, making use of fluctuations in sale and cost prices. We provide the first known polynomial-time algorithms for the case when there are fixed costs for purchases and sales, optional complementarity constraints that prohibit purchasing and selling during the same time period, and bounds on purchase and sales quantities. We do so by providing an exact characterization of the extreme points of the feasible region and using this to construct a suitable network where a min-cost flow computation provides an optimal solution. We are also able to provide polynomial extended linear formulations for the original feasible regions. Our methods build on the work by Wolsey and Yaman (Discrete Optimization 2018). We also consider the problem without fixed costs and provide a fully polynomial time approximation scheme in a setting with time-dependent bounds.

翻译：本文研究仓库问题中的一个未解问题：某商品交易商在固定时间范围内，为最大化总收益，需利用销售价格与成本价格的波动，制定最优的库存交易策略以决定购买与销售数量。针对存在购买与销售固定成本、禁止同一时期同时购买与销售的可选互补约束、以及购买与销售数量界约束的情形，本文首次提出了多项式时间算法。为此，我们精确刻画了可行域的极点特征，并基于此构造了一个适当的网络，通过最小成本流计算得到最优解。同时，我们给出了原可行域的多项式扩展线性规划公式。本文方法基于Wolsey与Yaman（《离散优化》2018年）的工作。此外，我们进一步考虑了无固定成本情形，并在时间依赖界约束的设置下提出了完全多项式时间近似方案。