Robust facility location with structural complexity and demand uncertainty

Abstract

Building robust logistics networks against facility disruptions has been gaining more attention from both researchers and practitioners. However, in pursuit of high built-in network robustness, one usually has to increase the network redundancy and complexity, which adds to the difficulty of management and induces additional operating costs. In this paper, we aim to balance the trade-off between robustness and complexity in the logistics network design problem and propose a model that explicitly considers the demand uncertainty. Due to the non-convexity of our model, we present a linear reformulation method that transforms the model into a MILP. From our numerical studies, in some cases, we can simultaneously enhance robustness and reduce complexity. When the demand uncertainty increases, the network will become less robust. It will require more backup links to guarantee a given robustness requirement. Consequently, the network can exhibit higher complexity and incur additional costs. We also observe it might be possible to dramatically reduce the complexity at a reasonable cost. One can achieve a relatively simple and robust network by only maintaining backup links for customer zones with large demand volumes. Finally, we provide discussions on managerial implications.

Appendix: Proof of Proposition 1

Let S denote the set of scenarios (all realizations of demand), $\hat{D}_{is}$ denote the amount of unfilled demand in scenario s when DC i fails, and $d_{js}$ denote the demand of customer zone j in scenario s.

$$\begin{aligned} \beta = \min _{i,s} 1 - \frac{ \hat{D}_{is} }{ \sum _j d_{js}} = \min _{i,s} 1 - \frac{\sum _j F_{ij} d_{js}}{\sum _j d_{js}} = \min _{i,s} 1 - \frac{\sum _j F_{ij} d_{js}}{\sum _j F_{ij} d_{js} + \sum _j (1- F_{ij}) d_{js}} \end{aligned}$$

where $\sum _j F_{ij} d_{js}$ represents the amount of unfulfilled demand in scenario s after DC i fails, and $\sum _j (1- F_{ij}) d_{js}$ represents the amount of fulfilled demand in scenario s after DC i fails. Given $F_{ij}$, the above equation is minimized with respect to $s \in S$ only if $\sum _j F_{ij} d_{js}$ is maximized and $\sum _j (1- F_{ij}) d_{js}$ is minimized with respect to $s \in S$. Note that $\sum _j F_{ij} d^u_{j}= \max _s \sum _j F_{ij} d_{js}$ and $\sum _j (1-F_{ij}) d^l_{j} = \min _s \sum _j (1- F_{ij}) d_{js}$. Thus,

$$\begin{aligned} \beta = \min _i 1 - \frac{\sum _j F_{ij} d^u_{j}}{\sum _j F_{ij} d^u_{j} + \sum _j (1-F_{ij}) d^l_{j}} = \min _{i} 1- \frac{\sum _j F_{ij} d^u_{j}}{\sum _j F_{ij}(d^u_{j} -d^l_{j}) + \sum _{j} d^l_{j}} \end{aligned}$$