While almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness tardiness penalties around a common due-date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis are proposed to assess the practical relevance of this mixed integer programming based methods.

中文翻译：

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使用自然变量的混合整数公式，用于围绕共同到期日的单机调度

虽然几乎所有现有的最佳解决即时调度问题的工作都提出了专用的算法方法，但我们在这项工作中提出了混合整数公式。我们考虑一个单机调度问题，该问题旨在最小化一个共同到期日附近的提前延迟惩罚的加权和。使用自然变量，我们为非限制性情况提供了一个紧凑的公式，对于一般情况，我们提供了一个基于非重叠不等式的非紧凑公式。我们表明与后一个公式相关的分离问题是多项式解决的。在此公式中，解决方案仅由极值点编码。我们建立了一个理论框架来证明这种使用非重叠不等式的公式的有效性，它可以用于其他调度问题。