In this paper the classical one-dimensional bin packing problem is integrated with scheduling elements: a due date is assigned to each item and the time required to process each bin depends on the pattern being used. The objective is to minimize a convex combination of the material waste and the delay costs, both significant in many real-world contexts. We present a novel pattern-based mixed integer linear formulation suitable for different classical scheduling objective functions, and focus on the specific case where the delay cost corresponds to the maximum tardiness. The formulation is tackled by a branch-and-price algorithm where the pricing of the column generation scheme is a quadratic problem solved by dynamic programming. A sequential value correction heuristic (SVC) is used to feed with warm starting solutions the column generation which, in turn, feeds the SVC with optimal prices so as to compute refined feasible solutions during the enumeration. Computational tests show that both column generation and branch-and-price substantially outperform standard methods in computing dual bounds and exact solutions. Additional tests are presented to analyze the sensitivity to parameters’ changes.

中文翻译：

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具有与模式相关的处理时间的一维料箱包装


在本文中，经典的一维箱子包装问题与调度元素相结合：为每个项目分配一个截止日期，处理每个箱子所需的时间取决于所使用的模式。目标是最大限度地减少材料浪费和延迟成本的凸组合，这在许多实际环境中都很重要。我们提出了一种适用于不同经典调度目标函数的新型基于模式的混合整数线性公式，并专注于延迟成本对应于最大延迟的特定情况。该公式由 branch-and-price 算法处理，其中列生成方案的定价是一个由动态规划解决的二次问题。顺序值校正启发式 （SVC） 用于为热启动解决方案提供色谱柱生成，这反过来又为 SVC 提供最佳价格，以便在枚举期间计算出精细的可行解决方案。计算测试表明，在计算对偶边界和精确解方面，列生成和 branch-and-price 的性能都大大优于标准方法。提出了其他测试来分析对参数变化的敏感性。