In the real-world production process, the firms need to determine the optimal production planning under minimum production quantity constraint in order to achieve economies of scale. However, the inventory cost will hugely increase when there is a very large amount of production in a period and also a large amount of total demands for the next few periods. This paper considers a single-item dynamic lot sizing problem with production-or-outsourcing decisions. In each period, the production level cannot be lower than a given quantity in order to make full use of resources, but the outsourcing is unrestricted. The demands in a period can be backlogged. The production and outsourcing costs are fixed-plus-linear, and the inventory and backlogging costs are linear. We establish a mathematical programming model according to the real problem in the firm. We explore some structural properties of the optimal solution and use them to develop a dynamic programming algorithm to solve the proposed problem. We further present a special case with stationary production and outsourcing costs which can be solved with reduced computational complexities. In the end, we use three numerical instances to show how to obtain the optimal solutions by using the dynamic programming algorithm. Furthermore, we show that the policy of backlogging or outsourcing can reduce the total cost.

中文翻译：

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在最小生产量下具有生产或外包决策的动态批量模型

在现实世界的生产过程中，企业需要确定在最小生产数量约束下的最佳生产计划，以实现规模经济。但是，当一个时期内的生产量非常大并且接下来的几个时期内的总需求量很大时，库存成本将大大增加。本文考虑具有生产或外包决策的单项动态批量确定问题。在每个时期中，为了充分利用资源，生产水平不能低于给定数量，但是外包是不受限制的。一段时间内的需求可以积压。生产和外包成本是固定加线性的，库存和积压成本是线性的。我们根据企业中的实际问题建立数学规划模型。我们探索了最优解的一些结构特性，并使用它们来开发动态规划算法来解决所提出的问题。我们还提出了一种固定生产和外包成本的特殊情况，可以通过降低计算复杂度来解决。最后，我们使用三个数值实例来说明如何使用动态规划算法来获得最优解。此外，我们证明了积压或外包的策略可以降低总成本。我们使用三个数值实例来说明如何使用动态规划算法来获得最佳解。此外，我们证明了积压或外包的策略可以降低总成本。我们使用三个数值实例来说明如何使用动态规划算法来获得最佳解。此外，我们证明了积压或外包的策略可以降低总成本。