Bounds and Heuristics for Optimal Bayesian Inventory Control with Unobserved Lost Sales
Bounds and Heuristics for Optimal Bayesian Inventory Control with Unobserved Lost Sales
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DOI：
10.1287/opre.1090.0726
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发表时间：
2010-03
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通讯作者：
Li Chen
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作者：
Li Chen
在大多数零售环境中，当库存耗尽时，未满足的需求就会丢失，而不会被观察到。销售数据实际上受到库存水平的制约。在需求估计和库存控制决策中考虑这种截尾数据效应会使问题难以解决。在本文中，我们着重于开发这个问题的边界和启发式。具体地说，我们考虑了具有未知参数的需求分布和未观察到的销售损失的非易腐产品的有限视界库存控制问题。采用贝叶斯更新法对参数进行序贯估计。我们首先推导出一组适用于所有先验分布和需求分布的解的上界。对于一个相当一般的单调似然比分布族，我们推导出沿任意样本路径的松弛但易于计算的下界和上界。然后我们提出两种启发法。第一个启发式是由解界结果导出的。计算这种启发式解决方案只需要在观察到的销售损失情况下对目标函数进行评估。第二个启发式是基于一阶条件的近似。我们将观察到的更简单的销售损失和易腐库存模型的一阶导数结合起来，以获得近似。对于后一种情况，我们得到了一个简化计算的递归公式。最后，我们进行了广泛的数值研究，以评估和比较边界和启发式。数值结果表明，这两种启发式算法都有很好的效果。他们的表现远远超过了短视的政策。
In most retail environments, when inventory runs out, the unmet demand is lost and not observed. The sales data are effectively censored by the inventory level. Factoring this censored data effect into demand estimation and inventory control decision makes the problem difficult to solve. In this paper, we focus on developing bounds and heuristics for this problem. Specifically, we consider a finite-horizon inventory control problem for a nonperishable product with unobserved lost sales and a demand distribution having an unknown parameter. The parameter is estimated sequentially by the Bayesian updating method. We first derive a set of solution upper bounds that work for all prior and demand distributions. For a fairly general monotone likelihood-ratio distribution family, we derive relaxed but easily computable lower and upper bounds along an arbitrary sample path. We then propose two heuristics. The first heuristic is derived from the solution bound results. Computing this heuristic solution only requires the evaluation of the objective function in the observed lost-sales case. The second heuristic is based on the approximation of the first-order condition. We combine the first-order derivatives of the simpler observed lost-sales and perishable-inventory models to obtain the approximation. For the latter case, we obtain a recursive formula that simplifies the computation. Finally, we conduct an extensive numerical study to evaluate and compare the bounds and heuristics. The numerical results indicate that both heuristics perform very well. They outperform the myopic policies by a wide margin.