The scheduling problem of distributed permutation flow shop with limited buffer aiming at production efficiency measures has attracted widespread attention due to its closer alignment with real manufacturing environments. However, the energy efficiency metric is often ignored. The Energy-Efficient scheduling of Distributed Permutation Flow Shop Problem with Limited Buffer (EEDPFSP-LB) with the objectives of Makespan ( $C_{max}$) and Total Energy Consumption (TEC) is studied, and a Cooperative Fruit fly Optimization Algorithm (CFOA) is proposed in this paper. First, the critical path of EEDPFSP-LB is identified, and energy-efficient operation is applied to non-critical paths to reduce the system’s energy consumption. Second, five acceptance criteria for multi-objective optimization are introduced to enhance the diversity of the population. Third, to select a superior next-generation population, a new congestion calculation method is introduced to resolve the issue of indeterminate positional relationships among non-dominated solutions with identical crowding distances at the same dominance level. Finally, CFOA is extensively tested and compared with state-of-the-art algorithms across 360 instances, demonstrating CFOA’s strong competitiveness in solving EEDPFSP-LB.

In this paper, it aims to model wind speed time series at multiple sites. The five-parameter Johnson distribution is deployed to relate the wind speed at each site to a Gaussian time series, and the resultant $m$-dimensional Gaussian stochastic vector process $\mathit{Z}(t)$ is employed to model the temporal-spatial correlation of wind speeds at $m$ different sites. In general, it is computationally tedious to obtain the autocorrelation functions (ACFs) and cross-correlation functions (CCFs) of $\mathit{Z}(t)$, which are different to those of wind speed times series. In order to circumvent this correlation distortion problem, the rank ACF and rank CCF are introduced to characterize the temporal-spatial correlation of wind speeds, whereby the ACFs and CCFs of $\mathit{Z}(t)$ can be analytically obtained. Then, Fourier transformation is implemented to establish the cross-spectral density matrix of $\mathit{Z}(t)$, and an analytical approach is proposed to generate samples of wind speeds at $m$ different sites. Finally, simulation experiments are performed to check the proposed methods, and the results verify that the five-parameter Johnson distribution can accurately match distribution functions of wind speeds, and the spectral representation method can well reproduce the temporal-spatial correlation of wind speeds.