Abstract
Natural disasters not only inflict massive life and economic losses but also result in psychological damage to survivors, at times even causing social unrest. It is necessary to design a sustainable last mile relief network for distributing relief supplies regarding social factors, disaster relief efficiency as well as the economic cost of three perspectives in terms of sustainability. We establish a multi-objective distributionally robust optimization model for a sustainable last mile relief network problem that maximizes the equitable distribution of relief supplies and simultaneously minimizes the transportation time and operation cost. Under the partial probability information of uncertainties, such as the disaster situation, transportation time, freight, road capacity, and demand, we characterize the uncertain variables in an ambiguity set incorporating the bounds, means and the mean absolute deviations. Then, the bounds on the objective values and the safe approximations of the chance constraints are deduced under the ambiguity sets. Based on a revised multi-choice goal programming approach, we obtain a computationally tractable framework of the multi-objective model. To verify the effectiveness of the model and methods, a case study of the Banten tsunami is illustrated. The results demonstrate our proposed model can obtain a trade-off between the equitability, timeliness and economics for relief distribution in a relief network.
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Acknowledgements
This work are supported by the National Natural Science Foundation of China (Grant No. 61773150 and Grant No. 71801077), the Top-notch talents of Hebei province (Grant No. 702800118009), Social Science Foundation of Hebei Province (HB17GL012), the High-Level Innovative Talent Foundation of Hebei University (Grant No. 521000981073), and Natural Sciences and Engineering Research Council of Canada discovery grant (Grant No. RGPIN-2014-03594, RGPIN-2019-07115).
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Appendices
Appendix 1: Proofs of Propositions
Proof of Proposition 1
Proof
Depending on the nature of the expectation, we can have the following form:
We assume that \(f(\rho )=\frac{1}{\rho }\) is convex and solve the upper bound of the expectation as the worse-case expectation. By the Eq. (23), we provide the worse-case expression of first objective:
\(\square \)
Proof of Proposition 2
Proof
We have the equivalent expression:
We assume that \(g(s)=e^{(\delta +b\mathbf{s} )}\) and the function g(s) is convex. Besides, we use the upper bound (23) of the expectation as the worse-case expectation:
\(\square \)
Proof of Proposition 3
Proof
Because the formulas \(\sum _{j \in J}f_jy_{j}\), \(\sum _{j \in J}\sqrt{cs^0_j}r_{j}\) and \(\sum _{j \in J }\sum _{i \in M_{j}}\sqrt{cs_{ji}}q_{ji}\) are linear, we have the form:
Among that \(h(s)=\sqrt{cs}\) and function h(s) is concave. By the expression (22), we provide the upper bound of expectation as the worse-case expectation and minimize the expectation:
\(\square \)
Proof of Proposition 4
Proof
Assume that the distribution \({\mathbb {P}}\) of the uncertain parameter \(z_i\) is such that
Among \(w_i\) for some convex function \(\Phi (.)\) that is finite everywhere and satisfies \(\Phi (0)=0\). Then, the any (\(w_0\),\(w_i\)) feasible for
is feasible for the chance constraint
We assume function \(\Phi (w_i)\):
satisfies the conditions (D.1) and (D.2), and (\(\mu ,d,\beta \)) ambiguity set \({\mathcal {P}}\) degenerates into (\(\mu ,d\)) ambiguity set. In addition, we know the function \(\Psi (w)\) has a tight upper bound on \({\mathbb {E}}_{\mathbb {P}}(w_iz_i)\), and the function \(\Phi (w_i)\) is convex. We have
Bring (D.4) and (D.5) into (D.2), we can induce
\(\square \)
Appendix 2: Table
See Table 11.
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Zhang, P., Liu, Y., Yang, G. et al. A multi-objective distributionally robust model for sustainable last mile relief network design problem. Ann Oper Res 309, 689–730 (2022). https://doi.org/10.1007/s10479-020-03813-3
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DOI: https://doi.org/10.1007/s10479-020-03813-3