Dynamic programming for single batching machine with total weighted completion time

The problem of total weighted completion time for single batching machine is an open problem until now. Accordingly, heuristics is very useful to the problem. We construct a nonlinear mathematical model for the problem with unbounded batching machine. A dynamic programming is also carried for the problem, which is proved to be at most 2 times of the optimal solution later. With an example, we show that the bound is tight.