A matrix cube-based estimation of distribution algorithm for the energy-efficient distributed assembly permutation flow-shop scheduling problem.

• The sequence model of energy-efficient DAPFSP is first established. • A heuristic and a hybrid method are presented to generate initial solutions. • A multidimensional probabilistic model is designed to learn the promising patterns. • A critical path-based local search is used to improve the quality of solutions. • Two speed adjustment strategies are proposed to enhance the performance of MCEDA. In this paper, a matrix-cube-based estimation of distribution algorithm (MCEDA) is proposed to solve the energy-efficient distributed assembly permutation flow-shop scheduling problem (EE_DAPFSP) that minimizes both the maximum completion time (C max) and the total carbon emission (TCE) simultaneously. Firstly, a high-quality and diverse initial population is constructed via a hybrid initialization method. Secondly, a matrix-cube-based probabilistic model and its update mechanism are designed to appropriately accumulate the valuable pattern information from superior solutions. Thirdly, a suitable sampling strategy is developed to sample the probabilistic model to generate a new population per generation, so as to guide the search direction toward promising regions in solution space. Fourthly, a problem-dependent neighborhood search based on critical path is provided to perform an in-depth local search around the promising regions found by the global search. Fifthly, two types of speed adjustment strategies based on problem properties are also embedded to further improve the quality of the obtained solutions. Sixthly, the influence of the parameters is investigated based on the multi-factor analysis of variance of Design-of-Experiments. Finally, extensive experiments and comprehensive comparisons with several recent state-of-the-art multi-objective algorithms are carried out based on the well-known benchmark instances, and the statistical results demonstrate the efficiency and effectiveness of the proposed MCEDA in addressing the EE_DAPFSP. [ABSTRACT FROM AUTHOR]