On the average-cost optimality equations and convergence of discounted-cost relative value functions for inventory control problems with quasiconvex cost functions
On the average-cost optimality equations and convergence of discounted-cost relative value functions for inventory control problems with quasiconvex cost functions
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DOI：
10.1109/cdc.2017.8263733
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发表时间：
2017-12
期刊：
影响因子：
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通讯作者：
E. Feinberg;Yan Liang
中科院分区：
文献类型：
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作者：
E. Feinberg;Yan Liang
平均成本最优不等式意味着马尔可夫决策过程存在单位时间平均成本的固定最优策略，并且这些不等式在广泛的自然条件下成立。平均成本最优方程的有效性需要附加条件。最近，Feinberg 和 Liang [10，定理 3.2] 表明，对于具有弱连续转移概率和可能无界一步成本的问题，贴现成本的价值函数的等连续性是平均成本最优方程有效性的充分附加条件，并且这一条件适用于具有延期交货和凸持有/积压成本的设置成本库存控制问题。本文研究了带延期交货、拟凸成本函数和一般需求的定期审查设置成本库存控制问题。结果表明此类问题满足等连续性条件。因此，对于这个问题，最优不等式以具有连续平均成本相对值函数的等式的形式成立。此外，这意味着库存控制问题的平均成本最优（s，S）策略可以从平均成本最优方程导出。在成本函数单调性的附加假设下，当折扣因子收敛到1时，我们建立了贴现成本最优排序阈值sa的收敛性和贴现成本相对值函数的收敛性，即平均成本问题的相应最优阈值和最优相对值函数。
Average-cost optimality inequalities imply the existence of stationary optimal policies for Markov Decision Processes with average costs per unit time, and these inequalities hold under broad natural conditions. Additional conditions are required for the validity of the average-cost optimality equations. Recently Feinberg and Liang [10, Theorem 3.2] showed that the equicontinuity of value functions for discounted costs is sufficient additional condition for the validity of average-cost optimality equations for problems with weakly continuous transition probabilities and with possibly unbounded one-step costs, and this condition holds for setup-cost inventory control problems with backorders and convex holding/backlog costs. This paper studies periodic-review setup-cost inventory control problem with backorders and with quasiconvex cost functions and general demands. It is shown that such problems satisfy the equicontinuity condition. Therefore, optimality inequalities hold in the form of equalities with a continuous average-cost relative value function for this problem. In addition, this implies that average-cost optimal (s, S) policies for the inventory control problem can be derived from the average-cost optimality equation. With the additional assumption on the monotonicity of the cost function, we establish the convergence of discounted-cost optimal ordering threshold sa and convergence of discounted-cost relative value functions, when the discount factor converges to 1, to the corresponding optimal threshold and optimal relative value function for the average-cost problem.