Abstract: The cast uncertain steelmaking continuous casting scheduling problem (CU_SCCSP) widely exists in the steel industry. This problem corresponds to three continuous production stages, i.e., ironmaking stage, refining stage and continuous casting stage. The ironmaking and refining stages can be modeled as a hybrid flow shop scheduling subproblem with transportation times, and the continuous casting stage can be regarded as a complex parallel machine scheduling subproblem with independent setting times. These two sub-problems are coupled with each other. To solve CU_SCCSP, a permutation-based model is established, and a co-evolution cross-entropy optimization algorithm (CCOA) is proposed to minimize the objective of weighted sum of the maximum completion time and the average waiting time. A two-stage encoding strategy and a bidirectional decoding strategy for the subproblems are designed，and a heuristic rule and a random method are adopted to initialize the population to ensure the quality and dispersion of the initial solutions. In the global search phase of CCOA, two probability matrices corresponding to the former and the latter subproblems are used to collaboratively learn and accumulate the information of high-quality solutions or individuals. Moreover, before generating new individuals by sampling the probability matrices, a fuzzy relation matrix considering the coupling between two subproblems is proposed to adjust the values in the probability matrices appropriately to enhance the ability of CCOA to reach the high-quality solution regions quickly. Meanwhile, a population splitting mechanism is designed to enhance CCOA's guiding ability and expand its search range. In order to improve the local search ability of CCOA, a cooperative search based on interchange neighborhood operation and insert neighborhood operation is executed on each individual in two splitting populations, and then a variable neighborhood search (VNS) method combined with the speedup evaluation of the swap neighborhood is performed on the current historical optimal solution, which can increase the search depth of the algorithm in multiple high-quality regions in solution space. Simulation experiments and algorithm comparisons verify the effectiveness of the proposed CCOA.

Key words: steelmaking-continuous casting, cross-entropy method, co-evolution, rapid evaluation method, fuzzy control