A solution robustness approach applied to network optimization problems

Solution robustness focuses on structural similarities between the nominal solution and the scenario solutions. Most other robust optimization approaches focus on the quality robustness and only evaluate the relevance of their solutions through the objective function value. However, it can be more important to optimize the solution robustness and, once the uncertainty is revealed, find an alternative scenario solution $x^s$ which is as similar as possible to the nominal solution $x^{nom}$. This for example occurs when the robust solution is implemented on a regular basis or when the uncertainty is revealed late. We call this distance between $x^{nom}$ and $x^s$ the solution cost. We consider the proactive problem which minimizes the average solution cost over a discrete set of scenarios while ensuring the optimality of the nominal objective of $x^{nom}$. We show for two different solution distances $d_{val}$ and $d_{struct}$ that the proactive problem is NP-hard for both the integer min-cost flow problem with uncertain arc demands and for the integer max-flow problem with uncertain arc capacities. For these two problems, we prove that once the uncertainty is revealed, even identifying a reactive solution $x^r$ with a minimal distance to a given solution $x^{nom}$ is NP-hard for $d_{struct}$, and that it is polynomial for $d_{val}$. We highlight the benefits of solution robustness in a case study on a railroad planning problem. First, we compare our proactive approach to the anchored and the $k$-distance approaches. Then, we show the efficiency of the proactive solution over reactive solutions. Finally, we illustrate the solution cost reduction when relaxing the optimality constraint on the nominal objective of the proactive solution $x^{nom}$.