Two-machine flow-shop scheduling with rejection - Shabtay, Dvir; Gasper, Nufar (Computers & operations research, 2012)
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+ɛ)R and a makespan value of at most (1+ɛ)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them. ► 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.