In this paper, we introduce a new problem in the field of production planning, called the production leveling problem. The task is to assign orders to production periods such that the load in each period and for each product type 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. We provide a formal model of the problem and study its computational complexity. As an exact method for solving moderately sized instances, we introduce a mixed integer programming (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 simulated annealing. Furthermore, three possible extensions that arise from the application in practice are described and implemented, both within the MIP model and within simulated annealing. We make publicly available a set of realistic problem instances from the industry as well as from random instance generators. The experimental evaluation on our test sets shows that the proposed MIP model is well suited for solving instances with up to 250 orders. Simulated annealing produces solutions with less than \(3\%\) average optimality gap on small instances, and scales well up to thousands of orders and dozens of periods and product types. The metaheuristic method presented herein is already being successfully used in the industry.

在本文中，我们介绍了生产计划领域的一个新问题，称为生产均衡问题。任务是将订单分配到生产周期，以便平衡每个周期和每种产品类型的负载，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。我们提供问题的正式模型并研究其计算复杂性。作为解决中等大小实例的精确方法，我们引入了混合整数规划 (MIP) 公式。为了解决大型问题实例，研究了元启发式局部搜索。提出了一种用于局部搜索的贪心启发式和两个邻域结构，以便使用模拟退火来应用它们。此外，在 MIP 模型和模拟退火中描述和实现了三种可能的扩展，这些扩展来自实践中的应用。我们公开了一组来自行业以及随机实例生成器的现实问题实例。对我们的测试集的实验评估表明，所提出的 MIP 模型非常适合解决多达 250 个订单的实例。模拟退火产生的溶液小于 在 MIP 模型和模拟退火中。我们公开了一组来自行业以及随机实例生成器的现实问题实例。对我们的测试集的实验评估表明，所提出的 MIP 模型非常适合解决多达 250 个订单的实例。模拟退火产生的溶液小于 在 MIP 模型和模拟退火中。我们公开了一组来自行业以及随机实例生成器的现实问题实例。对我们的测试集的实验评估表明，所提出的 MIP 模型非常适合解决多达 250 个订单的实例。模拟退火产生的溶液小于\(3\%\)小实例上的平均最优性差距，并且可以扩展到数千个订单和数十个周期和产品类型。本文介绍的元启发式方法已经在工业中成功使用。