Bi-Objective Optimization of a Flow Shop Scheduling Problem Under Time-of-Use Tariffs

Time-of-use (ToU) tariffs flexibly offer markedly cheap electricity prices to industrial and residential users during off-peak periods, encouraging them to shift their peak electricity demands in valley periods. Although ToU tariffs play a crucial role in balancing electricity supply and demand, especially in energy-intensive industries, the best trade-off between industrial performance and energy costs has not been well explored. Manufacturing consumes a substantial amount of energy, primarily in the form of electricity, leading to imbalances in power consumption. The flow shop scheduling (FSS) model is one of the most prevalent models in manufacturing. To explore the significant role of ToU tariffs in manufacturing, this study addresses a bi-objective FSS problem under ToU tariffs. The objective is to find the optimal balance between customer satisfaction and total electricity cost. A tight mixed integer programming model is developed to solve this NP-hard problem using business optimizers. On the bases of the problem properties demonstrated in this study, valid inequalities are designed to reduce the solution space of the problem. For small-scale instances, an improved $\varepsilon $ -constraint method is presented to find the Pareto front. For medium and large-scale instances, a two-stage fruit fly optimization (TFFO) algorithm is developed to obtain the near Pareto front. Experimental results demonstrate the efficiency and effectiveness of the proposed model and algorithms.Note to Practitioners—Scheduling for complex systems remains a formidable challenge in manufacturing. Energy cost saving is a major objective for all energy-intensive industries. Effective scheduling is crucial for businesses achieving eco-friendly performance, especially under ToU tariffs. This study aims to provide efficient scheduling model and methods that can guide decision-makers in fostering ecological transitions. The $\varepsilon $ -constraint method can find globally optimal solutions within given constraints. This situation is particularly beneficial for small-scale production systems requiring high accuracy. The TFFO algorithm can handle complex industrial environments and enhance production efficiency. Additionally, the TFFO algorithm is flexible and extensible, enabling it to be generalized to other production scenarios. Overall, the proposed model and algorithms lay a solid foundation for achieving efficient scheduling under ToU tariffs.