Lot streaming in three-stage production processes

Lot streaming with detached setups in three-machine flow shops

“…The presence of setups makes the structure of the optimal solution substantially more complex in most cases. Nevertheless, it is possible to extend most of the results from [6] to our case. For the three-machine ft.ow shop, this leads to an algorithm which finds the optimal schedule in O(log s) time.…”

“…The structure of the optimal network for such a problem was described in detail in [6]. Of this, we need only the fact that every sublot is critical in an optimal solution of the relaxed problem.…”

“…If s; < p1xf' then the setup does not affect the length of the critical path, thus it can be ignored and the reduced problem is equivalent to the relaxed problem. Ifs; > P1Xf' then the segment (1,0)-(2,0)-(2,1) is always at least as long as (1, 0) -(1, 1) -(2, 1) in any optimal solution x for the reduced problem, otherwise, H(l, 2, 1, x) = M (x) and we could reduce M(x) by reducing x1 [6], contradicting the optimality of x. D The above proof indicates that the reduced problem is really different from the relaxed problem only ifs; > P1X f .…”

“…al. [6] point out, however, the linear programming approach provides little insight into the structure of the solution which would enable more general models to be solved. Potts and Baker [14] show that for a single job, it is sufficient to consider identical sublot sizes on the first two machines, and on the last two machines.…”

“…al. [6] develop the solution to minimize the makespan for a single job in a three stage production process without setup times. Their algorithms compute the minimum makespan in O(log s) time for both the ft.ow shop and job shop problem.…”