Theoretical analysis of scheduling of a new batching machine

The processing time of the steel coils,which are to be processed in the bell type annealing furnace to improve quality,is composed of three operations steps,that is,heating,keeping and lowering temperature.The three-step processing of the jobs cannot be regarded as a whole process or three independent processes for technical reasons,and is different from the classical batching machine.An integer nonlinear programming is proposed and a heuristic algorithm based on dynamic programming is applied to the total weighted completion time for the new batching machine.The worst case performance of the heuristic algorithm is proved to be at most 3.If any two steps' processing times are the same,the heuristic algorithm can obtain the optimal solution.If any one step's processing time of all the jobs is the same,the worst performance of the heuristic algorithm is proved to be at most 2 and the bound is tight.We also analyze the worst case of the heuristic algorithm for the general case where jobs processing are composed of any step-processing.