In this thesis, we investigated a makespan minimization scheduling problem in a three-machine two-stage flowshop, known as differentiation flowshop which has a critical ma-chine at stage one and two independent dedicated mama-chines at stage two. The first-stage machine process the jobs in batches subject to the assumption of continuous process and batch availability.
For the batch scheduling problem with compatible items, it is known that the problem is NP-hard in the strong sense even if the batch setup time is 0. While the problem is computationally intractable, we proposed an algorithm to derive a lower bound of the problem in O(n2) time. The lower bound can be used for the design of branch and bound algorithms or the evaluation of heuristic approaches. Moreover, we investigated one special case that is polynomially solvable. With two given fixed sequences of two types of jobs, a polynomial time algorithm is devised to produce the optimal solutions.
On the other hand, for the batch problem with incompatible items under the assump-tion of fixed sequences, we developed an O(n41n42n3(n1 + n2) × (max{max{n21, n22}, n})) dynamic programming algorithm for solving the problem in a recursive way. Although the recursive program can solve this problem in polynomial time, its time complexity may be unaffordable for larger instances. The development of faster algorithms is clearly required for practical significance.
For further research, it may be interesting to generalize the studied problem to the setting with a variable number of parallel dedicated machines, i.e., the number m of
parallel dedicated machines is part of the input. Even though this problem is clearly NP-hard in the strong sense, it may be interesting to determine the complexity status of F (1, m) model in such special cases as (1) all jobs have the same processing time on the critical machine, or (2) sequences of different types of jobs are fixed and given. Developing lower bounds and dominance properties for designing branch-and-bound algorithms can be another worthy direction.
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