In this paper we introduce a new problem in the field of production planning which we call the Production Leveling Problem. The task is to assign orders to production periods such that the load in each period and on each production resource is balanced, capacity limits are not exceeded and the orders' priorities are taken into account. Production Leveling is an important intermediate step between long-term planning and the final scheduling of orders within a production period, as it is responsible for selecting good subsets of orders to be scheduled within each period. A formal model of the problem is proposed and NP-hardness is shown by reduction from Bin Backing. As an exact method for solving moderately sized instances we introduce a MIP formulation. For solving large problem instances, metaheuristic local search is investigated. A greedy heuristic and two neighborhood structures for local search are proposed, in order to apply them using Variable Neighborhood Descent and Simulated Annealing. Regarding exact techniques, the main question of research is, up to which size instances are solvable within a fixed amount of time. For the metaheuristic approaches the aim is to show that they produce near-optimal solutions for smaller instances, but also scale well to very large instances. A set of realistic problem instances from an industrial partner is contributed to the literature, as well as random instance generators. The experimental evaluation conveys that the proposed MIP model works well for instances with up to 250 orders. Out of the investigated metaheuristic approaches, Simulated Annealing achieves the best results. It is shown to produce solutions with less than 3% average optimality gap on small instances and to scale well up to thousands of orders and dozens of periods and products. The presented metaheuristic methods are already being used in the industry.

中文翻译：

---

生产均衡问题的精确和元启发式方法

在本文中，我们介绍了生产计划领域的一个新问题，我们称之为生产平衡问题。任务是将订单分配到生产期间，以便平衡每个期间和每个生产资源上的负载，不超过产能限制并考虑订单的优先级。生产平衡是长期计划和生产周期内订单最终调度之间的重要中间步骤，因为它负责选择要在每个周期内调度的良好订单子集。提出了该问题的正式模型，并通过 Bin Backing 的减少来显示 NP-hardness。作为解决中等大小实例的精确方法，我们引入了 MIP 公式。为了解决大型问题实例，研究了元启发式局部搜索。提出了用于局部搜索的贪婪启发式和两个邻域结构，以便使用可变邻域下降和模拟退火来应用它们。关于精确技术，研究的主要问题是，在固定的时间内可以解决多大的实例。对于元启发式方法，目的是表明它们为较小的实例生成近乎最优的解决方案，但也能很好地扩展到非常大的实例。来自工业合作伙伴的一组现实问题实例以及随机实例生成器都贡献给了文献。实验评估表明，所提出的 MIP 模型适用于多达 250 个订单的实例。在研究的元启发式方法中，模拟退火取得了最好的结果。它被证明可以在小实例上生成平均最优性差距小于 3% 的解决方案，并且可以很好地扩展到数千个订单和数十个时期和产品。所提出的元启发式方法已经在行业中使用。