This paper deals with the \(1|{p-\text {batch}, s_j\le b}|\sum C_j\) scheduling problem, where jobs are scheduled in batches on a single machine in order to minimize the total completion time. A size is given for each job, such that the total size of each batch cannot exceed a fixed capacity b. A graph-based model is proposed for computing a very effective lower bound based on linear programming; the model, with an exponential number of variables, is solved by column generation and embedded into both a heuristic price and branch algorithm and an exact branch and price algorithm. The same model is able to handle parallel-machine problems like \(Pm|{p-\text {batch}, s_j\le b}|\sum C_j\) very efficiently. Computational results show that the new lower bound strongly dominates the bounds currently available in the literature, and the proposed heuristic algorithm is able to achieve high-quality solutions on large problems in a reasonable computation time. For the single-machine case, the exact branch and price algorithm is able to solve all the tested instances with 30 jobs and a good amount of 40-job examples.

本文处理\(1|{p-\text {batch}, s_j\le b}|\sum C_j\)调度问题，其中作业在单机上分批调度，以最小化总完成时间. 每个作业都有一个大小，这样每个批次的总大小不能超过固定容量b。提出了一种基于图的模型，用于计算基于线性规划的非常有效的下界；该模型具有指数数量的变量，通过列生成求解，并嵌入到启发式价格和分支算法和精确分支和价格算法中。相同的模型能够处理并行机问题，如\(Pm|{p-\text {batch}, s_j\le b}|\sum C_j\)非常有效。计算结果表明，新的下界在很大程度上支配了目前文献中可用的界限，并且所提出的启发式算法能够在合理的计算时间内对大型问题实现高质量的解决方案。对于单机情况，精确分支和价格算法能够解决所有具有 30 个作业和大量 40 个作业示例的测试实例。