Exact method to optimize the total electricity cost in two-machine permutation flow shop scheduling problem under Time-of-use tariff

This study investigates the minimization of total electricity cost in a two-machine permutation flow shop scheduling problem under the most common electricity tariff, Time-of-Use (ToU). We provide a new property and solution approach to enhance existing methods in the literature. First, we provide an overview of the optimal cases for some specific ToU pricing structures that consist of only two pricing intervals. When the electricity price decreases, Johnson's rule and dynamic programming give rise to an optimal solution. On the other hand, when the electricity price increases, we provide a condition of optimality for Johnson's rule. Second, we develop a property based on Johnson's rule to determine the optimal sequence for general ToU pricing structures. Third, we propose a new mixed-integer linear programming. Then, we design an exact method based on "Logic-based Benders decomposition"to solve the problem. Finally, the numerical tests show that our proposed approach significantly improves the quality of existing results in the literature.