This paper addresses the multi-product, multi-period capacitated lot sizing problem. In particular, this work determines the optimal lot size allowing for shortages (imposed by budget restrictions), but with a penalty cost. The developed models are well suited to the usually rather inflexible production resources found in retail industries. Two models are proposed based on mixed-integer formulations: (i) one that allows shortage and (ii) one that forces fulfilling the demand. Both models are implemented over test instances and a case study of a real industry. By investigating the properties of the obtained solutions, we can determine whether the shortage allowance will benefit the company. The experimental results indicate that, for the test instances, the fact of allowing shortages produces savings up to 17% in comparison with the model without shortages, whereas concerning the current situation of the company, these savings represent 33% of the total costs while preserving the revenue.

中文翻译：

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批处理允许短缺的批量订货问题的新公式

本文解决了多产品，多时期的容量批量问题。尤其是，这项工作确定了允许短缺（由预算限制所致）的最佳手数，但要付出一定的代价。所开发的模型非常适合零售行业中通常较不灵活的生产资源。基于混合整数公式，提出了两种模型：（i）一种允许短缺的模型，（ii）一种强制满足需求的模型。两种模型都是在测试实例和实际行业的案例研究上实现的。通过调查获得的解决方案的属性，我们可以确定短缺津贴是否会使公司受益。实验结果表明，对于测试实例，