The Value of Flexibility in Robust Location-Transportation Problems

This article studies a multi-period capacitated fixed-charge location-transportation problem in which, while the location and capacity of each facility need to be determined immediately, the determination of final production and distribution of products can be delayed until actual orders are received in each period. In contexts where little is known about future demand, robust optimization, namely using a budgeted uncertainty set, becomes a natural method to employ in order to identify meaningful decisions. Unfortunately, it is well known that these types of multi-period robust decision problems are computationally intractable. To overcome this difficulty, we propose a set of tractable conservative approximations to the problem that each exploits to a different extent the idea of reducing the flexibility of the delayed decisions. While all of these approximation models outperform previous approximation models that have been proposed for this problem, each of them also has the potential to reach a different level of compromise between efficiency of resolution and quality of the solution. A row generation algorithm is also presented in order to address problem instances of realistic size. We also demonstrate that full flexibility is often unnecessary to reach nearly, or even exact, optimal robust locations and capacities for the facilities. Finally, we illustrate our findings with an extensive numerical study where we evaluate the effect of the amount of uncertainty on the performance and structure of each approximate solutions that can be obtained.