Flexible job shop scheduling problem (FJSP) has been extensively considered; however, multiobjective FJSP with energy consumption threshold is seldom investigated, the goal of which is to minimize makespan and total tardiness under the constraint that total energy consumption does not exceed a given threshold. Energy constraint is not always met and the threshold is difficult to be decided in advance. These features make it more difficult to solve the problem. In this paper, a two-phase meta-heuristic (TPM) based on imperialist competitive algorithm (ICA) and variable neighborhood search (VNS) is proposed. In the first phase, the problem is converted into FJSP with makespan, total tardiness and total energy consumption and the new FJSP is solved by an ICA, which uses some new methods to build initial empires and do imperialist competition. In the second phase, new strategies are provided for comparing solutions and updating the nondominated set of the first phase and a VNS is used for the original problem. The current solution of VNS is periodically replaced with member of the set $\Omega $ to improve solution quality. An energy consumption threshold is obtained by optimization. Extensive experiments are conducted to test the performance of TPM finally. The computational results show that TPM is a very competitive algorithm for the considered FJSP.

中文翻译：

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总能耗阈值的多目标柔性作业车间调度问题的两阶段元启发式

灵活的车间调度问题（FJSP）已得到广泛考虑；但是，很少研究具有能耗阈值的多目标FJSP，其目标是在总能耗不超过给定阈值的约束下，最小化制造时间和总延迟。能量约束并不总是满足，并且阈值很难预先确定。这些功能使解决问题变得更加困难。本文提出了一种基于帝国主义竞争算法（ICA）和可变邻域搜索（VNS）的两阶段元启发式算法（TPM）。在第一阶段，问题被转化为具有制造期，总拖延时间和总能耗的FJSP，而新的FJSP由ICA解决，该ICA使用了一些新方法来建立初始帝国并进行帝国主义竞争。在第二阶段，提供了用于比较解决方案和更新第一阶段的非支配集合的新策略，并且VNS用于原始问题。VNS的当前解决方案会定期替换为集合\\ Omega $的成员，以提高解决方案质量。通过优化获得能耗阈值。最后进行了广泛的测试以测试TPM的性能。计算结果表明，对于考虑的FJSP，TPM是一种非常有竞争力的算法。最后进行了广泛的测试以测试TPM的性能。计算结果表明，对于考虑的FJSP，TPM是一种非常有竞争力的算法。最后进行了广泛的测试以测试TPM的性能。计算结果表明，对于考虑的FJSP，TPM是一种非常有竞争力的算法。