Mnich and Wiese (Math Program 154:533–562, 2015) proved that the \(\mathcal {NP}\)-hard single-machine scheduling problem with rejection and the total weighted completion time plus total rejection cost as objective is fixed parameter tractable with respect to the number of different processing times and weights in the instance. This result is obtained by reducing the problem to an integer convex programming problem where the number of variables depends solely on the parameter (number of different processing times and weights). We show that this method can be incorporated into a more general scheme for solving a large set of scheduling problems with rejection which share common properties. These problems include various different machine environments, namely single machine, identical machines in parallel, and flow-shops.

Mnich 和 Wiese (Math Program 154:533–562, 2015) 证明了\(\mathcal {NP}\)- 以拒绝和总加权完成时间加上总拒绝成本作为目标的硬单机调度问题是相对于实例中不同处理时间和权重的数量易于处理的固定参数。该结果是通过将问题简化为整数凸规划问题而获得的，其中变量的数量仅取决于参数（不同处理时间和权重的数量）。我们表明，这种方法可以结合到一个更通用的方案中，以解决具有共同属性的拒绝的大量调度问题。这些问题包括各种不同的机器环境，即单机、并行相同的机器和流水线车间。