A note on valid inequalities for minimizing the total tardiness in a two-machine flow shop

This note reconsiders the valid inequalities designed by Kharbeche and Haouari (2013) for mixed-integer programming formulations of minimizing the total tardiness in a two-machine flow shop scheduling problem. While the majority of their proposed valid inequalities are able to substantially improve the computational time to find an optimal solution, we show that one of them is incorrect, due to the misinterpretation of a dominance criterion proven in Pan and Fan (1997). As a consequence, in some instances, all optimal solutions are excluded from the solution space, resulting in suboptimal solutions. To resolve this issue, we first demonstrate the suboptimality of the proposed valid inequality with a small illustrative example. Second, we formulate a new valid inequality, which correctly captures the dominance criteria of Pan and Fan (1997). Finally, we show the effect of this correction on the solution quality in a numerical study with scheduling instances that have 10 to 20 jobs. Depending on the number of jobs, between 10% and 22% of the supposedly optimal solutions with the incorrect valid inequality are in fact suboptimal. In terms of the objective function value, the error results in a loss of up to 12%.