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 Ω 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：

---

具有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


柔性作业车间调度问题（FJSP）已被广泛考虑；然而，带有能耗阈值的多目标FJSP却很少被研究，其目标是在总能耗不超过给定阈值的约束下最小化完工时间和总拖期。能量约束并不总是满足，阈值也很难提前确定。这些特征使得问题的解决变得更加困难。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（TPM）。第一阶段将问题转化为具有完工时间、总拖期和总能耗的FJSP，新的FJSP通过ICA来解决，ICA使用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，提供了新的策略来比较解决方案并更新第一阶段的非支配集，并且将 VNS 用于原始问题。 VNS的当前解定期用集合Ω的成员替换，以提高解的质量。通过优化得到能耗阈值。最后进行了大量的实验来测试TPM的性能。计算结果表明，对于所考虑的 FJSP，TPM 是一种非常有竞争力的算法。