Reliable design of humanitarian supply chain under correlated disruptions: a two-stage distributionally robust approach

Abstract

This paper considers a humanitarian supply chain (SC) under uncertain correlated disruptions of facilities caused by unpredictable disasters. Prepositioned relief items in these facilities might be partly damaged. Hence, it is of great importance to design a reliable SC to mitigate the disruptions. However, the challenge is that the joint disruption distribution is usually unavailable due to the scarcity of disaster data. To address this issue, our study considers that only the disruption probability of each relief item storage point is known, and develops a two-stage distributionally robust optimization (DRO) model based on the ambiguity set under the given marginal failure probabilities. Multiple resilience strategies are optimized in an integrated way, including inventory prepositioning strategy and often-used relief activities, e.g., relief item replenishment, transshipment, and demand transfer. We uncover the relationship between the worst-case joint distribution and supermodular order by exploring the structure–property of the recourse model, which is a generic relief logistics problem, and then identify the closed-form worst-case disruption distribution that lies within the ambiguity set for the robust counterpart. Thus, an equivalent stochastic program is formulated to address the intractability of the two-stage DRO. To validate its applicability, the proposed approach is applied to address the reliable design of a blood SC, in which pre-stocked human blood in hospitals may be partly ruined in disasters. Finally, a real-world case study of the Longmenshan Fault in China is performed. The sensitivity analysis and out-of-sample test are also implemented to give findings and insights. Our study reveals how to strategically preposition inventory with the consideration of disruptive facilities (uncertain correlations and geographic positions), relief items (perishability and shortage penalty), other anti-disaster strategies, as well as the two-stage decision-making way. Thus, our work addresses the reliable SC design under uncertain correlation by efficiently utilizing the minimal disruption information and developing a computationally tractable way.

Appendix A:

For the recourse model (2) and (3-1) ~ (3-6), introduce multipliers

$${\alpha }_{i}, {\beta }_{k}, {\lambda }_{k}, {\gamma }_{k}$$

to relax constraints (3-1) ~ (3-4). The Lagrange function can be given as below:

$${\sum }_{k\in K}{\sum }_{i\in I}{o}_{i}{X}_{ik}^{\omega }+{\sum }_{k\in K}{\sum }_{i\in I}{c}_{ik}{X}_{ik}^{\omega }+{\sum }_{k\in K}{\sum }_{{k}^{{^{\prime}}}\in K}{b}_{k{k}^{{^{\prime}}}}{Z}_{kk{^{\prime}}}^{\omega }+{\sum }_{k\in K}{\sum }_{{k}^{{^{\prime}}}\in K}{c}_{k{k}^{{^{\prime}}}}{Y}_{kk{^{\prime}}}^{\omega }+h{\sum }_{k\in K}{S}_{k}^{\omega }+{\sum }_{i\in I}{\alpha }_{i}\left({\sum }_{k\in K}{X}_{ik}^{\omega }- {s}_{i}\right)+{\sum }_{k\in K}{\beta }_{k}\left({d}_{k}^{\omega }-{S}_{k}^{\omega }-{\sum }_{{k}^{{^{\prime}}}\in K}{Z}_{k{k}^{{^{\prime}}}}^{\omega }+{\sum }_{{k}^{{^{\prime}}}\in K}{Z}_{{k}^{{^{\prime}}}k}^{\omega } -{\sum }_{i\in I}{X}_{ik}^{\omega }-{\sum }_{{k}^{{^{\prime}}}\in K}{Y}_{{k}^{{^{\prime}}}k}^{\omega }+{\sum }_{{k}^{{^{\prime}}}\in K}{Y}_{k{k}^{{^{\prime}}}}^{\omega }-{Q}_{k}\left(1-\left(1-{\xi }_{k}^{\omega }\right)\rho \right)\right)+{\sum }_{k\in K}{\lambda }_{k}\left({\sum }_{{k}^{{^{\prime}}}\in K}{Y}_{kk{^{\prime}}}^{\omega }-{Q}_{k}\left(1-\left(1-{\xi }_{k}^{\omega }\right)\rho \right)\right)+{\sum }_{k\in K}{ \gamma }_{k}\left({\sum }_{{k}^{{^{\prime}}}\in K}{Z}_{kk{^{\prime}}}^{\omega }-{d}_{k}^{\omega }\right)$$

$$=\left({\sum }_{k\in K}{d}_{k}{(\beta }_{k}-{\gamma }_{k})+ {\sum }_{k\in K}{Q}_{k}\left(1-\rho +\rho {\xi }_{k}\right)\left(-{\beta }_{k}-{\lambda }_{k}\right)-{\sum }_{i\in I}{s}_{i}{\alpha }_{i}\right)+{\sum }_{i\in I}{\sum }_{k\in K}\left({o}_{i}{+c}_{ik}+{\alpha }_{i}-{\beta }_{k}\right){X}_{ik}^{\omega }+{\sum }_{k\in K}{\sum }_{{k}^{{^{\prime}}}\in K}\left({b}_{k{k}^{\prime}}+{\beta }_{{k}^{\prime}}-{\beta }_{k}+{\gamma }_{k}\right){Z}_{kk{^{\prime}}}^{\omega }+{\sum }_{k\in K}{\sum }_{{k}^{{^{\prime}}}\in K}\left({c}_{k{k}^{\prime}}-{\beta }_{{k}^{\prime}}+{\beta }_{k}{+\lambda }_{k}\right){Y}_{kk{^{\prime}}}^{\omega }$$

Then, omit the superscript $\omega$ in ${\xi }_{k}$ and ${d}_{k}$, and it is easy to obtain the dual model (9).