This paper studies the bounded parallel-batching scheduling problem considering job rejection, deteriorating jobs, setup time, and non-identical job sizes. Each job will be either rejected with a certain penalty cost, or accepted and further processed in batches on a single machine. There is a setup time before processing each batch, and the objective is to minimize the sum of the makespan and the total penalty. Several useful preliminaries for arranging accepted job with identical size are proposed. Based on these preliminaries, we first investigate a special case where all the jobs are considered to have the identical size, and develop a dynamic programming algorithm to solve it. The preliminaries help to reduce the complexity of the dynamic programming algorithm from \( O\left( {n!n^{2} \sum\nolimits_{i = 1}^{n} {w_{j} } } \right) \) to \( O\left( {n^{2} \sum\nolimits_{i = 1}^{n} {w_{j} } } \right) \). For the general problem with non-identical job sizes, we propose a hybrid algorithm combining heuristic with dynamic programming algorithm (H-DP) to obtain satisfactory solutions within reasonable time. Finally, the effectiveness and efficiency of the H-DP algorithm are illustrated by a series of computational experiments.

中文翻译：

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尺寸不一样的恶化作业的并行批处理调度，并在一台机器上被拒绝

本文研究了考虑到工作拒绝，恶化的工作，准备时间和不相同的工作规模的有界并行批处理调度问题。每个作业将被拒绝，并承担一定的罚款，或者被接受并在单台机器上分批处理。在处理每一批之前有一个准备时间，其目的是最大程度地减少工期和总罚款。提出了一些有用的准备工作，以安排相同大小的已接受工作。基于这些初步知识，我们首先研究一种特殊情况，其中所有作业都被认为具有相同的大小，然后开发一种动态编程算法来解决该问题。初步帮助降低了动态编程算法的复杂性，从\（O \ left（{n！n ^ {2} \ sum \ nolimits_ {i = 1} ^ {n} {w_ {j}}} \ right）\）到\（O \ left（{n ^ { 2} \ sum \ nolimits_ {i = 1} ^ {n} {w_ {j}}} \ right）\）。针对工作量不相同的一般问题，提出了一种启发式算法与动态规划算法（H-DP）相结合的混合算法，以在合理的时间内获得满意的解决方案。最后，通过一系列计算实验说明了H-DP算法的有效性和效率。