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Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design

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Abstract

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features multivariate stochastic benchmarking constraints based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the decision-based random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in the context of pre-disaster humanitarian logistics and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. The numerical results provide practical insights about our modeling approach and show that the proposed algorithm is computationally scalable.

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References

  1. Federal Emergency Management Agency. https://egateway.fema.gov/ESF6/DRCLocator#. Accessed 08 Dec 2017

  2. Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Progr. 106(3), 433–446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Artzner, P., Delbaen, F., Eber, J., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balcik, B., Beamon, B.: Facility location in humanitarian relief. Int. J. Logist. Res. Appl. 11(2), 101–121 (2008)

    Article  Google Scholar 

  5. Bodur, M., Luedtke, J.R.: Mixed-integer rounding enhanced Benders decomposition for multiclass service-system staffing and scheduling with arrival rate uncertainty. Manag. Sci. 63(7), 2073–2091 (2017)

    Article  Google Scholar 

  6. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)

    MATH  Google Scholar 

  7. Çelik, M., Ergun, Ö., Johnson, B., Keskinocak, P., Lorca, Á., Pekgün, P., Swann, J.: Humanitarian logistics. In: New Directions in Informatics, Optimization, Logistics, and Production, pp. 18–49. INFORMS (2012)

  8. Dentcheva, D., Martinez, G.: Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse. Eur. J. Oper. Res. 219(1), 1–8 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dentcheva, D., Ruszczyński, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14(2), 548–566 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dentcheva, D., Ruszczyński, A.: Optimization with multivariate stochastic dominance constraints. Math. Program. 117(1), 111–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dentcheva, D., Wolfhagen, E.: Optimization with multivariate stochastic dominance constraints. SIAM J. Optim. 25(1), 564–588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dentcheva, D., Wolfhagen, E.: Two-stage optimization problems with multivariate stochastic order constraints. Math. Oper. Res. 41(1), 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Döyen, A., Aras, N., Barbarosoğlu, G.: A two-echelon stochastic facility location model for humanitarian relief logistics. Optim. Lett. 6(6), 1123–1145 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dupǎcová, J.: Risk objectives in two-stage stochastic programming models. Kybernetika 44(2), 227–242 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Elçi, Ö., Noyan, N.: A chance-constrained two-stage stochastic programming model for humanitarian relief network design. Transp. Res. Part B: Methodol. 108, 55–83 (2018)

    Article  MATH  Google Scholar 

  16. Fábián, C.I.: Handling CVaR objectives and constraints in two-stage stochastic models. Eur. J. Oper. Res. 191(3), 888–911 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gollmer, R., Gotzes, U., Schultz, R.: A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse. Math. Program. 216(1), 179–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gutjahr, W.J., Nolz, P.C.: Multicriteria optimization in humanitarian aid. Eur. J. Oper. Res. 252(2), 351–366 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Homem-de-Mello, T., Mehrotra, S.: A cutting surface method for uncertain linear programs with linear stochastic dominance constraints. SIAM J. Optim. 20(3), 1250–1273 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hong, X., Lejeune, M.A., Noyan, N.: Stochastic network design for disaster preparedness. IIE Trans. 47(4), 329–357 (2015)

    Article  Google Scholar 

  21. Hu, J., Mehrotra, S.: Robust and stochastically weighted multiobjective optimization models and reformulations. Oper. Res. 60(4), 936–953 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu, J., Homem-de Mello, T., Mehrotra, S.: Sample average approximation of stochastic dominance constrained programs. Math. Program. 133(1–2), 171–201 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huang, M., Smilowitz, K., Balcik, B.: Models for relief routing: equity, efficiency and efficacy. Transp. Res. Part E 48(1), 2–18 (2012)

    Article  Google Scholar 

  24. Küçükyavuz, S., Noyan, N.: Cut generation for optimization problems with multivariate risk constraints. Math. Program. 159(1), 165–199 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liberatore, F., Pizarro, C., de Blas, C.S., Ortuño, M.: Uncertainty in humanitarian logistics for disaster management. A review. In: Vitoriano, B., Montero, J., Ruan, D. (eds.) Decision aid Models for Disaster Management and Emergencies, pp. 45–74. Springer, Berlin (2013)

    Chapter  Google Scholar 

  27. Liu, X., Küçükyavuz, S., Noyan, N.: Robust multicriteria risk-averse stochastic programming models. Ann. Oper. Res. 259(1), 259–294 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Noyan, N.: Risk-averse two-stage stochastic programming with an application to disaster management. Comput. Oper. Res. 39(3), 541–559 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Noyan, N., Balcik, B., Atakan, S.: A stochastic optimization model for designing last mile relief networks. Transp. Sci. 50(3), 1092–1113 (2016)

    Article  Google Scholar 

  30. Noyan, N., Rudolf, G.: Optimization with multivariate conditional value-at-risk-constraints. Oper. Res. 61(4), 990–1013 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Noyan, N., Rudolf, G.: Optimization with stochastic preferences based on a general class of scalarization functions. Oper. Res. 66(2), 463–486 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rawls, C.G., Turnquist, M.A.: Pre-positioning of emergency supplies for disaster response. Transp. Res. Part B: Methodol. 44(4), 521–534 (2010)

    Article  Google Scholar 

  33. Rawls, C.G., Turnquist, M.A.: Pre-positioning planning for emergency response with service quality constraints. OR Spectr. 33(3), 481–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rockafellar, R., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)

    Article  Google Scholar 

  35. Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming, Handbooks in Operations Research and Management Science 10. Elsevier, Amsterdam (2003)

    Google Scholar 

  36. Saaty, T.L.: Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World. RWS Publications, Pittsburgh (1990)

    Google Scholar 

  37. Salmerón, J., Apte, A.: Stochastic optimization for natural disaster asset prepositioning. Prod. Oper. Manag. 19(5), 561–574 (2010)

    Article  Google Scholar 

  38. Van Slyke, R., Wets, R.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17(4), 638–663 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vitoriano, B., Ortuño, M.T., Tirado, G., Montero, J.: A multi-criteria optimization model for humanitarian aid distribution. J. Glob. Optim. 51(2), 189–208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the guest editors and reviewers for their valuable comments that improved the paper. Nilay Noyan acknowledges the support from The Scientific and Technological Research Council of Turkey under Grant #115M560. Simge Küçükyavuz and Merve Meraklı are supported, in part, by National Science Foundation Grant 1907463.

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Authors and Affiliations

  1. Industrial Engineering Program, Sabancı University, Tuzla, Istanbul, Turkey

    Nilay Noyan

  2. Industrial Engineering and Management Sciences, Northwestern University, Evanston, USA

    Merve Meraklı & Simge Küçükyavuz

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  1. Nilay Noyan
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  2. Merve Meraklı
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  3. Simge Küçükyavuz
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Correspondence to Nilay Noyan.

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Noyan, N., Meraklı, M. & Küçükyavuz, S. Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design. Math. Program. 191, 7–45 (2022). https://doi.org/10.1007/s10107-019-01373-4

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