Two-machine flow-shop scheduling with rejection

  • Authors:

  • Dvir Shabtay;Nufar Gasper

  • Affiliations:

  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel;Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

  • Venue:
  • Computers and Operations Research
  • Year:
  • 2012

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Abstract

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.