Two-machine flow-shop scheduling with rejection

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP - hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP - hard due to the NP - hardness of the same problem variation on a single machine 20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most ( 1 + ? ) R and a makespan value of at most ( 1 + ? ) K . This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them. Highlights? We study a two-machine flow-shop scheduling problem with rejection. ? We measure the quality of a schedule by two criteria: the makespan and the total rejection cost. ? We show that the problem of minimizing the sum of the two criteria (problem P1) is ordinary NP - hard . ? We provide an FPTAS and two 2-approximation algorithms for solving the P1 problem. ? The bicriteria problem is shown to be ordinary NP - hard and a two-dimensional FPTAS is provided.

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