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A multi-objective distributionally robust model for sustainable last mile relief network design problem

  • S.I. : Data-Driven OR in Transportation and Logistics
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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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References

  • Akbari, V., & Salman, F. S. (2017). Multi-vehicle synchronized arc routing problem to restore post-disaster network connectivity. European Journal of Operational Research, 257(2), 625–640.

    Google Scholar 

  • Anparasan, A., & Lejeune, M. (2019). Resource deployment and donation allocation for epidemic out-breaks. Annals of Operations Research, 283(1), 9–32.

    Google Scholar 

  • Balcik, B., Beamon, B. M., & Smilowitz, K. (2008). Last mile distribution in humanitarian relief. Journal of Intelligent Transportation Systems, 12(2), 51–63.

    Google Scholar 

  • Balcik, B., Beamon, B. M., Krejci, C. C., Muramatsu, K. M., & Ramirez, M. (2010). Coordination in humanitarian relief chains: Practices, challenges and opportunities. International Journal of Production Economics, 126(1), 22–34.

    Google Scholar 

  • Beamon, B., & Balcik, B. (2008). Performance measurement in humanitarian relief chains. International Journal of Public Sector Management, 21(1), 4–25.

    Google Scholar 

  • Behl, A., & Dutta, P. (2019). Humanitarian supply chain management: A thematic literature review and future directions of research. Annals of Operations Research, 283(1), 1001–1044.

    Google Scholar 

  • Ben-Tal, A., & Hochman, E. (1972). More bounds on the expectation of a convex function of a random variable. Journal of Applied Probability, 9(4), 803–812.

    Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (2008). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88, 411–424.

    Google Scholar 

  • Ben-Tal, A., Ghaoui, E. L., & Nemirovski, A. (2009). Robust optimization. Princeton: Princeton University Press.

    Google Scholar 

  • Berke, P. R., Kartez, J., & Wenger, D. (1993). Recovery after disaster: Achieving sustainable development, mitigation and equity. Disaster, 17(2), 93–109.

    Google Scholar 

  • Bertsimas, D., & Sim, D. (2004). The price of robustness. Operations Research, 52(1), 1–22.

    Google Scholar 

  • Bozorgi-Amiri, A., Jabalameli, M. S., & Al-e-Hashem, S. M. J. M. (2013). A multi-objective robust stochastic programming model for disaster relief logistics under uncertainty. OR Spectrum, 35(4), 905–933.

    Google Scholar 

  • Cao, C., Li, C., Yang, Q., Liu, Y., & Qu, T. (2018). A novel multi-objective programming model of relief distribution for sustainable disaster supply chain in large-scale natural disasters. Journal of Cleaner Production, 174, 1422–1435.

    Google Scholar 

  • Chakravarty, A. K. (2014). Humanitarian relief chain: Rapid response under uncertainty. International Journal of Production Economics, 151, 146–157.

    Google Scholar 

  • Chalmardi, M. K., & Camacho-Vallejo, J. F. (2019). A bi-level programming model for sustainable supply chain network design that considers incentives for using cleaner technologies. Journal of Cleaner Production, 213, 1035–1050.

    Google Scholar 

  • Chang, C. T. (2007). Multi-choice goal programming. Omega, 35(4), 389–396.

    Google Scholar 

  • Chang, C. T. (2008). Revised multi-choice goal programming. Applied Mathematical Modelling, 32(12), 2587–2595.

    Google Scholar 

  • Charnes, A., & Cooper, W. W. (1957). Management models and industrial applications of linear programming. Management Science, 4(1), 38–91.

    Google Scholar 

  • Dubey, R., & Gunasekaran, A. (2016). The sustainable humanitarian supply chain design: Agility, Adaptability and Alignment. International Journal of Logistics Research and Applications, 19(1), 62–82.

    Google Scholar 

  • Dubey, R., Gunasekaran, A., Childe, S. J., Roubaud, D., Wamba, S. F., Giannakis, M., et al. (2019). Big data analytics and organizational culture as complements to swift trust and collaborative performance in the humanitarian supply chain. International Journal of Production Economics, 210, 120–136.

    Google Scholar 

  • Dubey, R., Gunasekaran, A., & Papadopoulos, T. (2019). Disaster relief operations: Past, Present and Future. Annals of Operations Research, 283(1–2), 1–8.

    Google Scholar 

  • Fahimnia, B., Jabbarzadeh, A., Ghavamifar, A., & Bell, M. (2017). Supply chain design for efficient and effective blood supply in disasters. International Journal of Production Economics, 183, 700–709.

    Google Scholar 

  • Goh, J., & Sim, M. (2010). Distributionally robust optimization and its tractable approximations. Operations Research, 58, 902–917.

    Google Scholar 

  • Goldschmidt, K. H., & Kumar, S. (2019). Reducing the cost of humanitarian operations through disaster preparation and preparedness. Annals of Operations Research, 283(1–2), 1139–1152.

    Google Scholar 

  • Gu, J., Zhou, Y., & Das, A. (2018). Medical relief shelter location problem with patient severity under a limited relief budget. Computers & Industrial Engineering, 125, 720–728.

    Google Scholar 

  • Gupta, S., Altay, N., & Luo, Z. (2019). Big data in humanitarian supply chain management: A review and further research directions. Annals of Operations Research, 283(1–2), 1153–1173.

    Google Scholar 

  • Haavisto, I., & Kovács, G. (2013). Sustainability in humanitarian operations. Sustainable Value Chain Management Analyzing, Designing, Implementing, and Monitoring for Social and Environmental Responsibility.

  • Hu, S., Han, C., Dong, Z. S., & Meng, L. (2019). A multi-stage stochastic programming model for relief distribution considering the state of road network. Transportation Research Part B: Methodological, 123, 64–87.

    Google Scholar 

  • Huang, M., Smilowitz, K., & Balcik, B. (2012). Models for relief routing: Equity, efficiency and efficacy. Transportation Research Part E: Logistics and Transportation Review, 48(1), 2–18.

    Google Scholar 

  • Jabbour, C. J., Sobreiro, V. A., Jabbour, A. B., Campos, L. M., Mariano, E. B., & Renwick, D. W. (2019). An analysis of the literature on humanitarian logistics and supply chain management: Paving the way for future studies. Annals of Operations Research, 283(1), 289–307.

    Google Scholar 

  • Johnson, C., PENNING-ROWSELL, E., & Parker, D. (2007). Natural and imposed injustices: the challenges in implementing “fair” flood risk management policy in England. Geographical Journal, 173(4), 374–390.

    Google Scholar 

  • Kaur, H., & Singh, S. P. (2019). Sustainable procurement and logistics for disaster resilient supply chain. Annals of Operations Research, 283(1), 309–354.

    Google Scholar 

  • Kaya, O., & Urek, B. (2016). A mixed integer nonlinear programming model and heuristic solutions for location, inventory and pricing decisions in a closed loop supply chain. Computers & Operations Research, 65, 93–103.

    Google Scholar 

  • Khorram-Manesh, A. (2017). Handbook of Disaster and Emergency Management. Gothenburg, İsvec: Kompendiet. Kasım, 15, 2018.

  • Kovács, G., & Spens, K. M. (2007). Humanitarian logistics in disaster relief operations. International Journal of Physical Distribution & Logistics Management, 37(2), 99–114.

    Google Scholar 

  • Lagunasalvado, L., Lauras, M., Okongwu, U., & Comes, T. (2019). A multicriteria Master Planning DSS for a sustainable humanitarian supply chain. Annals of Operations Research, 283(1), 1303–1343.

    Google Scholar 

  • Li, L., Jin, M., & Zhang, L. (2011). Sheltering network planning and management with a case in the gulf coast region. International Journal of Production Economics, 131(2), 431–440.

    Google Scholar 

  • Liu, Y. J., Lei, H. T., Zhang, D. Z., & Wu, Z. Y. (2018). Robust optimization for relief logistics planning under uncertainties in demand and transportation time. Applied Mathematical Modelling, 55, 262–280.

    Google Scholar 

  • Liu, Y. K., Chen, Y., & Yang, G. (2019). Developing multi-objective equilibrium optimization method for sustainable uncertain supply chain planning problems. IEEE Transactions on Fuzzy Systems, 27(5), 1037–1051.

    Google Scholar 

  • Liu, K., Li, Q., & Zhang, Z. H. (2019). Distributionally robust optimization of an emergency medical service station location and sizing problem with joint chance constraints. Transportation Research Part B: Methodological, 119, 79–101.

    Google Scholar 

  • Mavrotas, G. (2009). Effective implementation of the \(\varepsilon \)-constraint method in multi-objective mathematical programming problems. Applied Mathematics and Computation, 213(2), 455–465.

    Google Scholar 

  • Mete, H. O., & Zabinsky, Z. B. (2010). Stochastic optimization of medical supply location and distribution in disaster management. International Journal of Production Economics., 126(1), 76–84.

    Google Scholar 

  • Najafi, M., Eshghi, K., & Dullaert, W. (2013). A multi-objective robust optimization model for logistics planning in the earthquake response phase. Transportation Research Part E: Logistics and Transportation Review, 49(1), 217–249.

    Google Scholar 

  • Nelson, T. (2010). When disaster strikes: on the relationship between natural disaster and interstate conflict. Global Change, Peace & Security, 22(2), 155–174.

    Google Scholar 

  • Noyan, N., Balcik, B., & Ataman, S. (2015). A stachastic optimization model for designing last mile relief networks. Transportation Science, 50(3), 1–22.

    Google Scholar 

  • Oliveira, C., De Mello, A., Bandeira, R., Vasconcelos Goes, G., & D’Agosto, M. (2017). Sustainable vehicles-based alternatives in last mile distribution of urban freight transport: A systematic literature review. Sustainability, 9(8), 1324.

    Google Scholar 

  • Ouhimmou, M., Nourelfath, M., Bouchard, M., & Bricha, N. (2019). Design of robust distribution network under demand uncertainty: A case study in the pulp and paper. International Journal of Production Economics, 218, 96–105.

    Google Scholar 

  • Ozdamar, L., Ekinci, E., & Kucukyazici, B. (2004). Emergency logistics planning in natural disasters. Annals of Operations Research, 129, 217–245.

    Google Scholar 

  • Papadopoulos, T., Gunasekaran, A., Dubey, R., Altay, N., Childe, S. J., & Fosso-Wamba, S. (2017). The role of Big Data in explaining disaster resilience in supply chains for sustainability. Journal of Cleaner Production, 142, 1108–1118.

    Google Scholar 

  • Postek, K., Ben-Tal, A., Hertog, D. D., & Melenberg, B. (2018). Robust optimization with ambiguous stochastic constraints under mean and dispersion information. Operations Research, 66(3), 814–833.

    Google Scholar 

  • Prékopa, A. (2013). Stochastic programming. Berlin: Springer.

    Google Scholar 

  • Ransikarbum, K., & Mason, S. J. (2016). Goal programming-based post-disaster decision making for integrated relief distribution and early-stage network restoration. International Journal of Production Economics, 182, 324–341.

    Google Scholar 

  • Rawls, C. G., & Turnquist, M. A. (2010). Pre-positioning of emergency supplies for disaster response. Transportation Tesearch Part B: Methodological, 44(4), 521–534.

    Google Scholar 

  • Rezaei-Malek, M., Tavakkoli-Moghaddam, R., Zahiri, B., & Bozorgi-Amiri, A. (2016). An interactive approach for designing a robust disaster relief logistics network with perishable commodities. Computers & Industrial Engineering, 94, 201–215.

    Google Scholar 

  • Saadatseresht, M., Mansourian, A., & Taleai, M. (2009). Evacuation planning using multi-objective evolutionary optimization approach. European Journal of Operational Research, 198(1), 305–314.

    Google Scholar 

  • Selim, H., Araz, C., & Ozkarahan, I. (2009). Collaborative production distribution planning in supply chain: A fuzzy goal programming approach. Transportation Research Part E: Logistics and Transportation Review, 44(3), 396–419.

    Google Scholar 

  • Sheu, J. B. (2014). Post-disaster relief-service centralized logistics distribution with survivor resilience maximization. Transportation Research Part B: Methodological, 68, 288–314.

    Google Scholar 

  • Slettebak, R. T. (2012). Don’t blame the weather! Climate-related natural disasters and civil conflict. Journal of Peace Research, 163–176.

  • Sun, G., Yang, B., Yang, Z., & Xu, G. (2019). An adaptive differential evolution with combined strategy for global numerical optimization. Soft Computing,. https://doi.org/10.1007/s00500-019-03934-3.

    Article  Google Scholar 

  • Tofighi, S., Torabi, S. A., & Mansouri, S. A. (2016). Humanitarian logistics network design under mixed uncertainty. European Journal of Operational Research, 250(1), 239–250.

    Google Scholar 

  • Tzeng, G. H., Cheng, H. J., & Huang, T. D. (2007). Multi-objective optimal planning for designing relief delivery systems. Transportation Research Part E: Logistics and Transportation Review, 43(6), 673–686.

    Google Scholar 

  • Uria, M. V. R., Caballero, R., & Ruiz, F. (2002). Meta-goal programming. European Journal of Operational Research, 136(2), 422–429.

    Google Scholar 

  • Wang, Y., Zhang, Y., & Tang, J. (2019). A distributionally robust optimization approach for surgery block allocation. European Journal of Operational Research, 273(2), 740–753.

    Google Scholar 

  • Wiesemann, W., Kuhn, D., & Sim, M. (2014). Distributionally robust convex optimization. Operations Research, 62(6), 1358–1376.

    Google Scholar 

  • Yahyaei, M., & Bozorgi-Amiri, A. (2018). Robust reliable humanitarian relief network design: An integration of shelter and supply facility location. Annals of Operations Research, 1–20.

  • Yao, T., Mandala, S. R., & Chung, B. D. (2009). Evacuation transportation planning under uncertainty: A robust optimization approach. Networks and Spatial Economics, 9(2), 171–189.

    Google Scholar 

  • Yushimito, W. F., Jaller, M., & Ukkusuri, S. (2012). A voronoi-based heuristic algorithm for locating distribution centers in disasters. Networks & Spatial Economics, 12(1), 21–39.

    Google Scholar 

  • Zhang, Z. H., & Jiang, H. (2014). A robust counterpart approach to the bi-objective emergency medical service design problem. Applied Mathematical Modelling, 38(3), 1033–1040.

    Google Scholar 

  • Zhang, P. Y., Liu, Y. K., Yang, G. Q., & Zhang, G. Q. (2019). A distributionally robust optimization model for designing humanitarian relief network with resource reallocation. Soft Computing, 24(4), 2749–2767.

    Google Scholar 

  • Zokaee, S., Bozorgiamiri, A., & Sadjadi, S. J. (2016). A robust optimization model for humanitarian relief chain design under uncertainty. Applied Mathematical Modelling, 40(17), 7996–8016.

    Google Scholar 

Download references

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

$$\begin{aligned} \max _{{\mathbb {P}}_1 \in {\mathcal {P}}_1 }&E_{\rho \sim {\mathbb {P}}_1}\left[ \sum _{j \in J}\frac{1}{\rho _{j}}r_{j}+\sum _{j \in J }\sum _{i \in M_{j}}\frac{1}{\rho ^{'}_{i}}q_{ji}\right] \end{aligned}$$
(A.1)

Depending on the nature of the expectation, we can have the following form:

$$\begin{aligned}&\max _{{\mathbb {P}}_1 \in {\mathcal {P}}_1}E_{\rho \sim {\mathbb {P}}_1}\left[ \sum _{j \in J}\frac{1}{\rho _{j}}r_{j}\right] +\max _{{\mathbb {P}}_1 \in {\mathcal {P}}}E_{\rho \sim {\mathbb {P}}_1}\left[ \sum _{j \in J }\sum _{i \in M_{j}}\frac{1}{\rho ^{'}_{i}}q_{ji}\right] \end{aligned}$$
(A.2)
$$\begin{aligned} \Rightarrow&\sum _{j \in J}r_{j}\max _{{\mathbb {P}}_1 \in {\mathcal {P}}}E_{\rho \sim {\mathbb {P}}_1}f(\rho _{j})+\sum _{j \in J }\sum _{i \in M_{j}}q_{ji}\max _{{\mathbb {P}}_1 \in {\mathcal {P}}_1}E_{\rho \sim {\mathbb {P}}_1}f(\rho ^{'}_{i}) \end{aligned}$$
(A.3)

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:

$$\begin{aligned}&\sum _{j \in J}r_{j}\left[ \frac{d^{\rho }_j}{2(\mu ^{\rho }_j-a^{\rho }_j)}\frac{1}{a^{\rho }_{j}}+\left( 1-\frac{d^{\rho }_j}{2(\mu ^{\rho }_j-a^{\rho }_j)}-\frac{d^{\rho }_j}{2(b^{\rho }_j-\mu ^{\rho }_j)}\right) \frac{1}{\mu _{j}^{\rho }}+\left( \frac{d^{\rho }_j}{2(b^{\rho }_j-\mu ^{\rho }_j)}\right) \frac{1}{b_{j}^{\rho }}\right] \end{aligned}$$
(A.4)
$$\begin{aligned}&\quad +\sum _{j \in J }\sum _{i \in M_{j}}q_{ji}\left[ \frac{d^{\rho '}_{i}}{2(\mu ^{\rho '}_{i}-a^{\rho '}_{i})}\frac{1}{a_{i}^{\rho '}}+\left( 1-\frac{d^{\rho '}_{i}}{2(\mu ^{\rho '}_{i}-a^{\rho '}_{i})}-\frac{d^{\rho '}_{i}}{2(b^{\rho '}_{i} -\mu ^{\rho '}_{i})}\right) \frac{1}{\mu _{i}^{\rho '}}\right. \nonumber \\&\qquad \left. +\left( \frac{d^{\rho '}_{i}}{2(b^{\rho '}_{i}-\mu ^{\rho '}_{i})}\right) \frac{1}{b^{\rho '}_{i}} \right] \end{aligned}$$
(A.5)

\(\square \)

Proof of Proposition 2

Proof

$$\begin{aligned}&\max _{{\mathbb {P}}_2 \in {\mathcal {P}}} E_\mathbf{s \sim {\mathbb {P}}_2}\left[ \sum _{j \in J}e^{(\delta +\tau \mathbf{s} ^0_{j})}r_{j}+\sum _{j \in J }\sum _{i \in M_{j}}e^{(\delta +\tau \mathbf{s} _{ji})}q_{ji}\right] \end{aligned}$$
(B.1)

We have the equivalent expression:

$$\begin{aligned}&\max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2} E_\mathbf{s \sim {\mathbb {P}}_2}\left[ \sum _{j \in J}e^{(\delta +\tau \mathbf{s} ^0_{j})}r_{j}\right] +\max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2} E_\mathbf{s \sim {\mathbb {P}}_2}\left[ \sum _{j \in J }\sum _{i \in M_{j}}e^{(\delta +\tau \mathbf{s} _{ji})}q_{ji}\right] \end{aligned}$$
(B.2)
$$\begin{aligned}&\sum _{j \in J}r_{j}\max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2} E_\mathbf{s \sim {\mathbb {P}}_2}(e^{(\delta +\tau \mathbf{s} ^0_{j})})+\sum _{j \in J }\sum _{i \in M_{j}}q_{ji}\max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2} E_\mathbf{s \sim {\mathbb {P}}_2}(e^{(\delta +\tau \mathbf{s} _{ji})}) \end{aligned}$$
(B.3)

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:

$$\begin{aligned}&\sum _{j \in J}r_{j}\left[ \frac{d^{s^0}_j}{2(\mu ^{s^0}_j-a^{s^0}_j)}e^{(\delta +\tau a^{s^0}_{j})}+\left( 1-\frac{d^{s^0}_j}{2(\mu ^{s^0}_j-a^{s^0}_j)}-\frac{d^{s^0}_j}{2(b^{s^0}_j-\mu ^{s^0}_j)}\right) e^{(\delta +\tau \mu ^{s^0}_{j})}\right. \nonumber \\&\quad \left. +\left( \frac{d^{s^0}_j}{2(b^{s^0}_j-\mu ^{s^0}_j)}\right) e^{(\delta +\tau b^{s^0}_{j})}\right] \end{aligned}$$
(B.4)
$$\begin{aligned}&\quad +\sum _{j \in J }\sum _{i \in M_{j}}q_{ji}\left[ \frac{d^s_{ji}}{2(\mu ^s_{ji}-a^s_{ji})}e^{(\delta +\tau a^s_{ji})}+\left( 1-\frac{d^s_{ji}}{2(\mu ^s_{ji}-a^s_{ji})}-\frac{d^s_{ji}}{2(b^s_{ji}-\mu ^s_{ji})}\right) e^{(\delta +\tau \mu ^s_{ji})}\right. \nonumber \\&\quad \left. +\left( \frac{d^s_{ji}}{2(b^s_{ji}-\mu ^s_{ji})}\right) e^{(\delta +\tau b^s_{ji})} \right] \end{aligned}$$
(B.5)

\(\square \)

Proof of Proposition 3

Proof

$$\begin{aligned} \max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2} E_{s \sim {\mathbb {P}}_2}\left[ \sum _{j \in J}f_jy_{j}+\sum _{j \in J}\sqrt{cs^0_j}r_{j}+\sum _{j \in J }\sum _{i \in M_{j}}\sqrt{cs_{ji}}q_{ji}\right] \end{aligned}$$
(C.1)

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:

$$\begin{aligned}&\sum _{j \in J}f_jy_{j}+ \max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2}E_{s \sim {\mathbb {P}}_2}\left( \sum _{j \in J}\sqrt{cs^0_j}r_{j}\right) +\max _{{\mathbb {P}}_2 \in {\mathcal {P}}_2}E_{s \sim {\mathbb {P}}_2}\left( \sum _{j \in J }\sum _{i \in M_{j}}\sqrt{cs_{ji}}q_{ji}\right) \end{aligned}$$
(C.2)
$$\begin{aligned} \Rightarrow&\sum _{j \in J}f_jy_{j}+\sum _{j \in J}r_{j}\max _{{\mathbb {P}}_2 \in {\mathcal {P}}}E_{s \sim {\mathbb {P}}_2} \left( \sqrt{cs^0_j}\right) +\sum _{j \in J }\sum _{i \in M_{j}}q_{ji}\max _{{\mathbb {P}}_2 \in {\mathcal {P}}}E_{s \sim {\mathbb {P}}_2}\left( \sqrt{cs_{ji}}\right) \end{aligned}$$
(C.3)

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:

$$\begin{aligned}&\sum _{j \in J}f_{j}y_{j}+\sum _{j \in J}r_{j}\left[ \beta ^{s^0}_j\sqrt{c(\mu ^{s^0}_j+\frac{d^{s^0}_j}{2\beta ^{s^0}_j})}+(1-\beta ^{s^0}_j)\sqrt{c(\mu ^{s^0}_j-\frac{d^{s^0}_j}{2(1-\beta ^{s^0}_j)})} \right] \end{aligned}$$
(C.4)
$$\begin{aligned}&\quad +\sum _{j \in J }\sum _{i \in M_{j}}q_{ji}\left[ \beta ^s_{ji}\sqrt{c(\mu ^s_{ji}+\frac{d^s_{ji}}{2\beta ^s_{ji}})}+(1-\beta ^s_{ji})\sqrt{c(\mu ^s_{ji}-\frac{d^s_{ji}}{2(1-\beta ^s_{ji}) })} \right] \end{aligned}$$
(C.5)

\(\square \)

Proof of Proposition 4

Proof

Assume that the distribution \({\mathbb {P}}\) of the uncertain parameter \(z_i\) is such that

$$\begin{aligned} \log \left\{ {\mathbb {E}}\exp (w_iz_i)\right\} \le \Phi (w_i) \end{aligned}$$
(D.1)

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

$$\begin{aligned} \inf _{\eta \ge 0}\left\{ w_0+\eta \Phi (\eta ^{-1}w_i)+\eta \log (1/\alpha )\right\} \le 0 \end{aligned}$$
(D.2)

is feasible for the chance constraint

$$\begin{aligned} {\mathbb {P}}\left( w_0+z_iw_i\ge 0\right) \le \alpha \end{aligned}$$
(D.3)

We assume function \(\Phi (w_i)\):

$$\begin{aligned} \Phi (w_i)=\log (\Psi (w_i)), \ \ \Psi (w_i)=\sup _{{\mathbb {P}}\in {\mathcal {P}}}{\mathbb {E}}_{\mathbb {P}}(w_iz_i)=(d_i\cosh (w_i)+1-d_i) \end{aligned}$$
(D.4)

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

$$\begin{aligned}&o_i\ge q_{ji} \ \ \Rightarrow o_i-q_{ji}\ge 0 \ \ \Rightarrow o_i+z_io_i^l-q_{ji}\ge 0 \ \Rightarrow \underbrace{o_i-q_{ji}}_{w_0}+z_i\underbrace{o_i^l}_{w_i} \end{aligned}$$
(D.5)

Bring (D.4) and (D.5) into (D.2), we can induce

$$\begin{aligned} -q_{ji}+o_i^0+\eta \log \left( d^o_i\cosh \left( \frac{o_i^l}{\eta }\right) +1-d^0_i\right) +\eta \log \left( \frac{1}{\alpha }\right) \le 0 \end{aligned}$$
(D.6)

\(\square \)

Appendix 2: Table

See Table 11.

Table 11 The rode conditions from the PODs to demand nodes (in kilometer)

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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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