SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 267-295, January 2021.
This paper studies a continuous-review backlogged inventory model considered by [K. L. Helmes, R. H. Stockbridge, and C. Zhu, SIAM J. Control. Optim., 53 (2015), pp. 2100--2140] but with discontinuous quantity-dependent setup cost for each order. In particular, the setup cost is characterized by a two-step function and a higher cost would be charged once the order quantity exceeds a threshold $Q$. Unlike the optimality of the $(s,S)$-type policy obtained by Helmes, Stockbridge, and Zhu for continuous setup cost with the discounted cost criterion, we find that, in our model, although some $(s,S)$-type policy is indeed optimal in some cases, the $(s,S)$-type policy cannot always be optimal. In particular, we show that there exist cases in which an $(s,S)$ policy is optimal for some initial levels but it is strictly worse than a generalized $(s,\{S(x):x\leq s\})$ policy for the other initial levels. Under $(s,\{S(x):x\leq s\})$ policy, it orders nothing for $x>s$ and orders up to level $S(x)$ for $x\leq s$, where $S(x)$ is a nonconstant function of $x$. We further prove the optimality of such $(s,\{S(x):x\leq s\})$ policy in a large subset of admissible policies for those initial levels. Moreover, the optimality is obtained through establishing a more general lower bound theorem which will also be applicable in solving some other optimization problems by the lower bound approach.

中文翻译：

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具有不连续设置成本的脉冲控制：折扣成本准则

SIAM控制与优化杂志，第59卷，第1期，第267-295页，2021年1月。
本文研究了[KL Helmes，RH Stockbridge和C. Zhu，SIAM J. Control考虑的连续审查积压库存模型。Optim。，53（2015），pp。2100--2140]，但每个订单具有不连续的数量相关设置成本。特别地，建立成本的特征在于两步函数，并且一旦订单数量超过阈值$ Q $，就会收取更高的成本。与Helmes，Stockbridge和Zhu获得的$（s，S）$类型的策略对于采用折扣成本标准的连续设置成本的最优性不同，我们发现在我们的模型中，尽管有些$（s，S）$类型策略在某些情况下确实是最优的，但$（s，S）$类型策略不能总是最优的。特别是，我们表明在某些情况下，$（s，S）$策略在某些初始级别上是最佳的，但它比广义的$（s，其他初始级别的\ {S（x）：x \ leq s \}）$策略。根据$ {s，\ {{S（x）：x \ leq s \}）$$策略，它以$ x> s $的价格订购任何商品，以$ x \ leq s $的价格订购最高级别为$ S（x）$的商品，其中$ S（x）$是$ x $的非恒定函数。我们进一步证明了这些$（s，\ {S（x）：x \ leq s \}）$策略在这些初始级别的大量可接受策略中的最优性。此外，通过建立更通用的下界定理获得最优性，该定理也将适用于通过下界方法解决一些其他优化问题。这些初始级别的大部分可允许策略中的x \ leq s \}）$策略。而且，通过建立更通用的下界定理可以获得最优性，该定理也将适用于通过下界方法解决其他一些优化问题。这些初始级别的大部分可允许策略中的x \ leq s \}）$策略。而且，通过建立更通用的下界定理可以获得最优性，该定理也将适用于通过下界方法解决其他一些优化问题。