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Dynamic lotsizing with a finite production rate

机译：具有有限生产率的动态批量

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The dynamic lotsizing problem concerns the determination of optimally produced/delivered batch quantities, when demand, which is to be satisfied, is distributed over time in different amounts at different times. The standard formulation assumes that these batches are provided instantaneously, i.e. that the production rate is infinite. Using a cumulative geometrical representation for demand and production, it has previously been demonstrated that the inner-corner condition for an optimal production plan reduces the number of possible optimal replenishment times to a finite set of given points, at which replenishments can be made. The problem is thereby turned into choosing from a set of zero/one decisions, whether or not to replenish each time there is a demand. If n is the number of demand events, this provides 2~(n-1) alternatives, of which at least one solution must be optimal. This condition applies, whether an Average Cost approach or the Net Present Value principle is applied, and the condition is valid in continuous time, and therefore in discrete time. In the current paper, the assumption of an infinite production rate is relaxed, and consequences for the inner-corner condition are investigated. It is then shown that the inner-corner condition needs to be modified to a tangency condition between cumulative requirements and cumulative production. Also, we have confirmed the additional restriction for feasibility in the finite production case (provided by Hill, 1997), namely the production rate restriction. Furthermore, in the NPV case, one further necessary condition for optimality, the distance restriction concerning the proximity between adjacent production intervals, has been derived. In an example this condition has shown to reduce the number of candidate solutions for optimality still further. An algorithm leading to the optimal solution is presented.
机译：动态批量问题涉及确定最佳生产/交付批次数量的情况，当要满足的需求随时间在不同时间以不同的数量分配时。标准配方假设这些批次是即时提供的，即生产率是无限的。使用用于需求和生产的累积几何表示，先前已证明最佳生产计划的内角条件将可能的最佳补货时间数量减少到给定点的有限集合，在该点上可以进行补货。问题由此变成从一组零/一个决策中进行选择，每次有需求时是否进行补充。如果n是需求事件的数量，则提供2〜（n-1）个替代方案，其中至少一个解决方案必须是最优的。无论应用“平均成本”方法还是“净现值”原理，都适用此条件，并且该条件在连续时间内有效，因此在离散时间内有效。在本文中，无限生产率的假设被放宽，并且研究了内部角条件的后果。然后表明，内部拐角条件需要修改为累积需求和累积生产之间的相切条件。同样，我们已经证实了有限产量情况下可行性的其他限制（Hill，1997年提供），即生产率限制。此外，在NPV情况下，已经得出了另一个优化的必要条件，即与相邻生产间隔之间的接近程度有关的距离限制。在示例中，该条件已显示出进一步减少了最优性候选解决方案的数量。提出了导致最优解的算法。

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