The no-wait flow shop with rejection

We consider the no-wait flow shop scheduling problem with rejection. We first show how to augment the travelling salesman problem formulation for the no-wait flow shop minimum makespan problem to incorporate the rejection option. We then focus on polynomially solvable cases by considering problems with ordered jobs. We present a third-order polynomial-time dynamic programming algorithm to minimise the sum of makespan and total rejection cost and faster quadratic algorithms for two special cases. We also present a third-order polynomial-time dynamic programming algorithm to minimise the sum of total completion time and total rejection cost with ordered jobs. We also exploit the duality between no-wait and no-idle flow shops to determine which operations to outsource in a no-idle flow shop with an outsourcing option.