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doi: 10.3934/jimo.2019109

Multi-stage distributionally robust optimization with risk aversion

Ripeng Huang 1,2, , Shaojian Qu 1,, , Xiaoguang Yang 3, and  Zhimin Liu 1,

1. 

Business School, University of Shanghai for Science and Technology, Shanghai, China
2. 

School of Mathematics and Finance, Chuzhou University, Anhui, China
3. 

Academy of Mathematics and Systems Science CAS, Beijing, China

* Corresponding author: Shaojian Qu

Received  November 2018 Revised  April 2019 Published  September 2019

Fund Project: The first author is supported by National Natural Science Foundation of China (71571055)

Two-stage risk-neutral stochastic optimization problem has been widely studied recently. The goals of our research are to construct a two-stage distributionally robust optimization model with risk aversion and to extend it to multi-stage case. We use a coherent risk measure, Conditional Value-at-Risk, to describe risk. Due to the computational complexity of the nonlinear objective function of the proposed model, two decomposition methods based on cutting planes algorithm are proposed to solve the two-stage and multi-stage distributional robust optimization problems, respectively. To verify the validity of the two models, we give two applications on multi-product assembly problem and portfolio selection problem, respectively. Compared with the risk-neutral stochastic optimization models, the proposed models are more robust.

Keywords: Distributionally robust optimization, multi-stage, ambiguity set, risk aversion, decomposition algorithm.
Mathematics Subject Classification: Primary: 90C15; Secondary: 90C90.
Citation: Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019109
References:
[1]

S. Ahmed, Convexity and decomposition of mean-risk stochastic programs, Mathematical Programming, 106 (2006), 433-446.  doi: 10.1007/s10107-005-0638-8.  Google Scholar

[2]

P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[3]

A. Ben-Tal, D. D. Hertog, A. De Waegenaere, B. Melenberg and G. Rennen, Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59 (2013), 341-357.   Google Scholar

[4]

A. Ben-Tal, T. Margalit and A. Nemirovski, Robust modeling of multi-stage portfolio problems, High Performance Optimization, 33 (2000), 303-328.  doi: 10.1007/978-1-4757-3216-0_12.  Google Scholar

[5]

D. Victor, G. Lorenzo and U. Raman, Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.   Google Scholar

[6]

D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, Belmont, MA, 2015.  Google Scholar

[7]

G. Bayraksan and D. K. Love, Data-driven stochastic programming using phi-divergences, The Operations Research Revolution, INFORMS TutORials in Operations Research, (2015), 1–19. doi: 10.1287/educ.2015.0134.  Google Scholar

[8]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2011. doi: 10.1007/978-1-4614-0237-4.  Google Scholar

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H. H. Chen and C.-B. Yang, Multiperiod portfolio investment using stochastic programming with conditional value at risk, Computers and Operations Research, 81 (2017), 305-321.  doi: 10.1016/j.cor.2016.11.011.  Google Scholar

[11]

E. Delage and Y. Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.  Google Scholar

[12]

C. I. Fábián, C. Wolf, A. Koberstein and L. Suhl, Risk-averse optimization in two-stage stochastic models: Computational aspects and a study, SIAM Journal on Optimization, 25 (2015), 28-52.  doi: 10.1137/130918216.  Google Scholar

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K. Fan, Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 39 (1953), 42-47.  doi: 10.1073/pnas.39.1.42.  Google Scholar

[14]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Applications of Mathematics (New York), 53. Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2004.  Google Scholar

[15]

R. Henrion, C. Küchler and W. Römisch, Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming, Journal of Industrial and Management Optimization, 4 (2008), 363-384.  doi: 10.3934/jimo.2008.4.363.  Google Scholar

[16]

J. C. Hull, Options, Futures, and Other Derivatives (Seventh Edition), Pearson Education International, 2009. Google Scholar

[17]

H. T. Huynh and I. Soumare, Stochastic Simulation and Applications in Finance with MATLAB Programs, John Wiley and Sons, 2012. doi: 10.1002/9781118467374.  Google Scholar

[18]

R. W. Jiang and Y. P. Guan, Risk-averse two-stage stochastic program with distributional ambiguity, Operations Research, 66 (2018), 1390-1405.  doi: 10.1287/opre.2018.1729.  Google Scholar

[19]

R. W. Jiang and Y. P. Guan, Data-driven chance constrained stochastic program, Mathematical Programming, 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7.  Google Scholar

[20]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.  Google Scholar

[21]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust minimum variance beam former design, IEEE Signal Processing Letters, 25 (2018), 105-109.   Google Scholar

[22]

B. Li, X. Qian, J. Sun, K. L. Teo and C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039.  Google Scholar

[23]

B. Li, J. Sun, H. L. Xu and M. Zhang, A class of two-stage distributionally robust games, Journal of Industrial and Management Optimization, 15 (2019), 387-400.   Google Scholar

[24]

A. Ling, J. Sun, N. H. Xiu and X. G. Yang, Robust two-stage stochastic linear optimization with risk aversion, European Journal of Operational Research, 256 (2017), 215-229.  doi: 10.1016/j.ejor.2016.06.017.  Google Scholar

[25]

Z. M. Liu, S. J. Qu, M. Goh, R. P. Huang and S. L. Wang, Optimization of fuzzy demand distribution supply chain using modified sequence quadratic programming approach, Journal of Intelligent & Fuzzy Systems, (2019). Google Scholar

[26]

D. Love and G. Bayraksan, Phi-divergence constrained ambiguous stochastic programs for data-driven optimization, Optimization Online, (2016). Available from: http://www.optimization-online.org/DB_HTML/2016/03/5350.html. Google Scholar

[27]

F. W. Meng, R. Tan and G. Y. Zhao, A superlinearly convergent algorithm for large scale multi-stage stochastic nonlinear programming, International Journal of Computational Engineering Science, 5 (2012), 327-344.  doi: 10.1142/9781860949524_0156.  Google Scholar

[28]

N. Miller and A. Ruszczyński, Risk-averse two-stage stochastic linear programming: Modeling and decomposition, Operations Research, 59 (2011), 125-132.  doi: 10.1287/opre.1100.0847.  Google Scholar

[29]

J. M. Mulvey and B. Shetty, Financial planning via multi-stage stochastic optimization, Computers and Operations Research, 31 (2004), 1-20.  doi: 10.1016/S0305-0548(02)00141-7.  Google Scholar

[30]

J. M. Mulvey, R. J. Vanderbei and S. A. Zenios, Robust optimization of large-scale systems, Operations Research, 43 (1995), 264-281.  doi: 10.1287/opre.43.2.264.  Google Scholar

[31]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996.  doi: 10.1137/050622328.  Google Scholar

[32]

S. Nickel, F. Saldanha-Da-Gama and H.-P. Ziegler, A multi-stage stochastic supply network design problem with financial decisions and risk management, Omega, 40 (2012), 511-524.  doi: 10.1016/j.omega.2011.09.006.  Google Scholar

[33]

N. Noyan, Risk-averse two-stage stochastic programming with an application to disaster management, Computers and Operations Research, 39 (2012), 541-559.  doi: 10.1016/j.cor.2011.03.017.  Google Scholar

[34]

W. de Oliveira and C. Sagastizabal, Level bundle methods for oracles with on demand accuracy, Optimization Methods and Software, 29 (2014), 1180-1209.  doi: 10.1080/10556788.2013.871282.  Google Scholar

[35]

L. Pardo, Statistical Inference Based on Divergence Measures, Textbooks and Monographs, 185. Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[36]

A. Parisio and C. N. Jones, A two-stage stochastic programming approach to employee scheduling in retail outlets with uncertain demand, Omega, 53 (2015), 97-103.   Google Scholar

[37]

S. J. Qu, Y. Y. Zhou, Y. L. Zhang, M. Wahab, G. Zhang and Y. Y. Ye, Optimal strategy for a green supply chain considering shipping policy and default risk, Computers and Industrial Engineering, 131 (2019), 172-186.  doi: 10.1016/j.cie.2019.03.042.  Google Scholar

[38]

M. A. Quddus, S. Chowdhury, M. Marufuzzaman, F. Yu and L. Bian, A two-stage chance-constrained stochastic programming model for a bio-fuel supply chain network, International Journal of Production Economics, 195 (2018), 27-44.  doi: 10.1016/j.ijpe.2017.09.019.  Google Scholar

[39]

C. G. Rawls and M. A. Turnquist, Pre-positioning of emergency supplies for disaster response, 2006 IEEE International Symposium on Technology and Society, (2006). doi: 10.1109/ISTAS.2006.4375894.  Google Scholar

[40]

M. I. Restrepo, B. Gendron and L.-M. Rousseau, A two-stage stochastic programming approach for multi-activity tour scheduling, European Journal of Oerational Research, 262 (2017), 620-635.  doi: 10.1016/j.ejor.2017.04.055.  Google Scholar

[41]

A. Rezaee, F. Dehghanian, B. Fahimnia and B. Beamon, Green supply chain network design with stochastic demand and carbon price, Annals of Operations Research, 250 (2017), 463-485.  doi: 10.1007/s10479-015-1936-z.  Google Scholar

[42]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41.  doi: 10.21314/JOR.2000.038.  Google Scholar

[43]

A. Ruszczyński, Decomposition methods, Handbooks in Operations Research and Management Science, 10 (2003), 141-211.  doi: 10.1016/S0927-0507(03)10003-5.  Google Scholar

[44]

A. Ruszczyński and A. Shapiro, Optimality and duality in stochastic programming, Handbooks in Operations Research and Management Science, 10 (2003), 65-139.  doi: 10.1016/S0927-0507(03)10002-3.  Google Scholar

[45]

T. Santoso, S. Ahmed, M. Goetschalckx and A. Shapiro, A stochastic programming approach for supply chain network design under uncertainty, European Journal of Operational Research, 167 (2005), 96-115.  doi: 10.1016/j.ejor.2004.01.046.  Google Scholar

[46]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, , MPS/SIAM Series on Optimization, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2009. doi: 10.1137/1.9780898718751.  Google Scholar

[47]

J. Shu and J. Sun, Designing the distribution network for an integrated supply chain, Journal of Industrial and Management Optimization, 2 (2006), 339-349. doi: 10.3934/jimo.2006.2.339.  Google Scholar

[48]

H. L. Sun and H. F. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Mathematics of Operations Research, 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.  Google Scholar

[49]

S. Zymler, D. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.  Google Scholar

[50]

W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.  Google Scholar

[51]

S. S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684.  Google Scholar

[52]

W. N. Zhang, H. Rahimian and G. Bayraksan, Decomposition algorithms for risk-averse multistage stochastic programs with application to water allocation under uncertainty, Informs Journal on Computing, 28 (2016), 385-404.  doi: 10.1287/ijoc.2015.0684.  Google Scholar

show all references

References:
[1]

S. Ahmed, Convexity and decomposition of mean-risk stochastic programs, Mathematical Programming, 106 (2006), 433-446.  doi: 10.1007/s10107-005-0638-8.  Google Scholar

[2]

P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[3]

A. Ben-Tal, D. D. Hertog, A. De Waegenaere, B. Melenberg and G. Rennen, Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59 (2013), 341-357.   Google Scholar

[4]

A. Ben-Tal, T. Margalit and A. Nemirovski, Robust modeling of multi-stage portfolio problems, High Performance Optimization, 33 (2000), 303-328.  doi: 10.1007/978-1-4757-3216-0_12.  Google Scholar

[5]

D. Victor, G. Lorenzo and U. Raman, Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.   Google Scholar

[6]

D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, Belmont, MA, 2015.  Google Scholar

[7]

G. Bayraksan and D. K. Love, Data-driven stochastic programming using phi-divergences, The Operations Research Revolution, INFORMS TutORials in Operations Research, (2015), 1–19. doi: 10.1287/educ.2015.0134.  Google Scholar

[8]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2011. doi: 10.1007/978-1-4614-0237-4.  Google Scholar

[9] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar
[10]

H. H. Chen and C.-B. Yang, Multiperiod portfolio investment using stochastic programming with conditional value at risk, Computers and Operations Research, 81 (2017), 305-321.  doi: 10.1016/j.cor.2016.11.011.  Google Scholar

[11]

E. Delage and Y. Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.  Google Scholar

[12]

C. I. Fábián, C. Wolf, A. Koberstein and L. Suhl, Risk-averse optimization in two-stage stochastic models: Computational aspects and a study, SIAM Journal on Optimization, 25 (2015), 28-52.  doi: 10.1137/130918216.  Google Scholar

[13]

K. Fan, Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 39 (1953), 42-47.  doi: 10.1073/pnas.39.1.42.  Google Scholar

[14]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Applications of Mathematics (New York), 53. Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2004.  Google Scholar

[15]

R. Henrion, C. Küchler and W. Römisch, Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming, Journal of Industrial and Management Optimization, 4 (2008), 363-384.  doi: 10.3934/jimo.2008.4.363.  Google Scholar

[16]

J. C. Hull, Options, Futures, and Other Derivatives (Seventh Edition), Pearson Education International, 2009. Google Scholar

[17]

H. T. Huynh and I. Soumare, Stochastic Simulation and Applications in Finance with MATLAB Programs, John Wiley and Sons, 2012. doi: 10.1002/9781118467374.  Google Scholar

[18]

R. W. Jiang and Y. P. Guan, Risk-averse two-stage stochastic program with distributional ambiguity, Operations Research, 66 (2018), 1390-1405.  doi: 10.1287/opre.2018.1729.  Google Scholar

[19]

R. W. Jiang and Y. P. Guan, Data-driven chance constrained stochastic program, Mathematical Programming, 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7.  Google Scholar

[20]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.  Google Scholar

[21]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust minimum variance beam former design, IEEE Signal Processing Letters, 25 (2018), 105-109.   Google Scholar

[22]

B. Li, X. Qian, J. Sun, K. L. Teo and C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039.  Google Scholar

[23]

B. Li, J. Sun, H. L. Xu and M. Zhang, A class of two-stage distributionally robust games, Journal of Industrial and Management Optimization, 15 (2019), 387-400.   Google Scholar

[24]

A. Ling, J. Sun, N. H. Xiu and X. G. Yang, Robust two-stage stochastic linear optimization with risk aversion, European Journal of Operational Research, 256 (2017), 215-229.  doi: 10.1016/j.ejor.2016.06.017.  Google Scholar

[25]

Z. M. Liu, S. J. Qu, M. Goh, R. P. Huang and S. L. Wang, Optimization of fuzzy demand distribution supply chain using modified sequence quadratic programming approach, Journal of Intelligent & Fuzzy Systems, (2019). Google Scholar

[26]

D. Love and G. Bayraksan, Phi-divergence constrained ambiguous stochastic programs for data-driven optimization, Optimization Online, (2016). Available from: http://www.optimization-online.org/DB_HTML/2016/03/5350.html. Google Scholar

[27]

F. W. Meng, R. Tan and G. Y. Zhao, A superlinearly convergent algorithm for large scale multi-stage stochastic nonlinear programming, International Journal of Computational Engineering Science, 5 (2012), 327-344.  doi: 10.1142/9781860949524_0156.  Google Scholar

[28]

N. Miller and A. Ruszczyński, Risk-averse two-stage stochastic linear programming: Modeling and decomposition, Operations Research, 59 (2011), 125-132.  doi: 10.1287/opre.1100.0847.  Google Scholar

[29]

J. M. Mulvey and B. Shetty, Financial planning via multi-stage stochastic optimization, Computers and Operations Research, 31 (2004), 1-20.  doi: 10.1016/S0305-0548(02)00141-7.  Google Scholar

[30]

J. M. Mulvey, R. J. Vanderbei and S. A. Zenios, Robust optimization of large-scale systems, Operations Research, 43 (1995), 264-281.  doi: 10.1287/opre.43.2.264.  Google Scholar

[31]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996.  doi: 10.1137/050622328.  Google Scholar

[32]

S. Nickel, F. Saldanha-Da-Gama and H.-P. Ziegler, A multi-stage stochastic supply network design problem with financial decisions and risk management, Omega, 40 (2012), 511-524.  doi: 10.1016/j.omega.2011.09.006.  Google Scholar

[33]

N. Noyan, Risk-averse two-stage stochastic programming with an application to disaster management, Computers and Operations Research, 39 (2012), 541-559.  doi: 10.1016/j.cor.2011.03.017.  Google Scholar

[34]

W. de Oliveira and C. Sagastizabal, Level bundle methods for oracles with on demand accuracy, Optimization Methods and Software, 29 (2014), 1180-1209.  doi: 10.1080/10556788.2013.871282.  Google Scholar

[35]

L. Pardo, Statistical Inference Based on Divergence Measures, Textbooks and Monographs, 185. Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[36]

A. Parisio and C. N. Jones, A two-stage stochastic programming approach to employee scheduling in retail outlets with uncertain demand, Omega, 53 (2015), 97-103.   Google Scholar

[37]

S. J. Qu, Y. Y. Zhou, Y. L. Zhang, M. Wahab, G. Zhang and Y. Y. Ye, Optimal strategy for a green supply chain considering shipping policy and default risk, Computers and Industrial Engineering, 131 (2019), 172-186.  doi: 10.1016/j.cie.2019.03.042.  Google Scholar

[38]

M. A. Quddus, S. Chowdhury, M. Marufuzzaman, F. Yu and L. Bian, A two-stage chance-constrained stochastic programming model for a bio-fuel supply chain network, International Journal of Production Economics, 195 (2018), 27-44.  doi: 10.1016/j.ijpe.2017.09.019.  Google Scholar

[39]

C. G. Rawls and M. A. Turnquist, Pre-positioning of emergency supplies for disaster response, 2006 IEEE International Symposium on Technology and Society, (2006). doi: 10.1109/ISTAS.2006.4375894.  Google Scholar

[40]

M. I. Restrepo, B. Gendron and L.-M. Rousseau, A two-stage stochastic programming approach for multi-activity tour scheduling, European Journal of Oerational Research, 262 (2017), 620-635.  doi: 10.1016/j.ejor.2017.04.055.  Google Scholar

[41]

A. Rezaee, F. Dehghanian, B. Fahimnia and B. Beamon, Green supply chain network design with stochastic demand and carbon price, Annals of Operations Research, 250 (2017), 463-485.  doi: 10.1007/s10479-015-1936-z.  Google Scholar

[42]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41.  doi: 10.21314/JOR.2000.038.  Google Scholar

[43]

A. Ruszczyński, Decomposition methods, Handbooks in Operations Research and Management Science, 10 (2003), 141-211.  doi: 10.1016/S0927-0507(03)10003-5.  Google Scholar

[44]

A. Ruszczyński and A. Shapiro, Optimality and duality in stochastic programming, Handbooks in Operations Research and Management Science, 10 (2003), 65-139.  doi: 10.1016/S0927-0507(03)10002-3.  Google Scholar

[45]

T. Santoso, S. Ahmed, M. Goetschalckx and A. Shapiro, A stochastic programming approach for supply chain network design under uncertainty, European Journal of Operational Research, 167 (2005), 96-115.  doi: 10.1016/j.ejor.2004.01.046.  Google Scholar

[46]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, , MPS/SIAM Series on Optimization, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2009. doi: 10.1137/1.9780898718751.  Google Scholar

[47]

J. Shu and J. Sun, Designing the distribution network for an integrated supply chain, Journal of Industrial and Management Optimization, 2 (2006), 339-349. doi: 10.3934/jimo.2006.2.339.  Google Scholar

[48]

H. L. Sun and H. F. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Mathematics of Operations Research, 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.  Google Scholar

[49]

S. Zymler, D. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.  Google Scholar

[50]

W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.  Google Scholar

[51]

S. S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684.  Google Scholar

[52]

W. N. Zhang, H. Rahimian and G. Bayraksan, Decomposition algorithms for risk-averse multistage stochastic programs with application to water allocation under uncertainty, Informs Journal on Computing, 28 (2016), 385-404.  doi: 10.1287/ijoc.2015.0684.  Google Scholar

Figure 1.  The revenue of multi-product assembly problem under uncertain demanding
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Figure 2.  The revenue of multi-product assembly problem with different parameters
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Figure 3.  Three-stage scenario tree with 10 different scenarios at each ancestor node
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Figure 4.  Comparison of different methods
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Figure 5.  Comparison of cumulative wealth with different parameters
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Table 1.  The Probability of Different Demand in 10 Scenarios
Scenario s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
Product1 1253 1069 1407 1125 1293 1377 1265 1235 1155 1327
Product2 1350 1074 1122 1308 1275 1190 1390 1005 1264 1345
Product3 1446 1129 1465 1237 1459 1284 1467 1168 1082 1374
Product4 1480 1421 1175 1176 1143 1038 1065 1081 1301 1225
Product5 1274 1127 1098 1416 1379 1027 1284 1397 1131 1041
Probability 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
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Scenario s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
Product1 1253 1069 1407 1125 1293 1377 1265 1235 1155 1327
Product2 1350 1074 1122 1308 1275 1190 1390 1005 1264 1345
Product3 1446 1129 1465 1237 1459 1284 1467 1168 1082 1374
Product4 1480 1421 1175 1176 1143 1038 1065 1081 1301 1225
Product5 1274 1127 1098 1416 1379 1027 1284 1397 1131 1041
Probability 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Table 2.  The Optimal Solution of Single-stage Multi-product Assembly Problem
NO. 1 2 3 4 5 6 7
Pre-order quantity 8671.6 9907.3 9975.8 8664.7 11265 11212 7439.3
Unused quantity 0 0 0 0 0 0 0
Units produced 1250.6 1232.3 1311.1 1210.5 1217.4 - -
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NO. 1 2 3 4 5 6 7
Pre-order quantity 8671.6 9907.3 9975.8 8664.7 11265 11212 7439.3
Unused quantity 0 0 0 0 0 0 0
Units produced 1250.6 1232.3 1311.1 1210.5 1217.4 - -
Table 3.  The Optimal Solution of Two-stage Stochastic Multi-product Assembly Problem
NO. 1 2 3 4 5 6 7
Pre-order quantity 8719.4 9976.1 10008 8691.5 11304 11265 7487.1
Unused quantity 0 0 0 0 0 0 0
Units produced 1250.6 1232.3 1316.9 1210.5 1238.4 - -
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NO. 1 2 3 4 5 6 7
Pre-order quantity 8719.4 9976.1 10008 8691.5 11304 11265 7487.1
Unused quantity 0 0 0 0 0 0 0
Units produced 1250.6 1232.3 1316.9 1210.5 1238.4 - -
Table 4.  The Optimal Solution of Two-stage Distributionally Robust Optimization with Risk Aversion Multi-product Assembly Problem
NO. 1 2 3 4 5 6 7
Case 1 $ \lambda $=0.5 $ \rho $=0.5 $ \beta $=0.95
Pre-order quantity 8279 9568 9496 8281 10744 10785 7137
Unused quantity 0 0 0 0 0 0 0
Units produced 1258 1142.3 1214.5 1175.6 1173.1 - -
Case 2 $ \lambda $=0.2 $ \rho $=0.5 $ \beta $=0.95
Pre-order quantity 8556 9912 9831 8546 11184 11187 7402
Unused quantity 0 0 0 0 0 0 0
Units produced 1278.6 1154 1285.3 1221.5 1231.2 - -
Case 3 $ \lambda $=0.8 $ \rho $=0.5 $ \beta $=0.95
Pre-order quantity 8215 9487 9452 8224 10714 10723 7084
Unused quantity 0 0 0 0 0 0 0
Units produced 1245.5 1131.2 1227.3 1166.1 1157 - -
Case 4 $ \rho $=0.1 $ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity 8266 9550 9482 8268 10729 10766 7125
Unused quantity 0 0 0 0 0 0 0
Units produced 1252.5 1141 1213.7 1174.1 1172 - -
Case 5 $ \rho $=1 $ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity 8340 9625 9561 8336 10808 10846 7184
Unused quantity 0 0 0 0 0 0 0
Units produced 1258.3 1156.6 1225.5 1178 1182.7 - -
Case 6 $ \rho $=0.05 $ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity 8262 9543 9481 8263 10731 10762 7122
Unused quantity 0 0 0 0 0 0 0
Units produced 1249.2 1140.2 1218.1 1172.1 1171.3 - -
Case 7 $ \beta $=0.90 $ \lambda $=0.5 $ \rho $=0.5
Pre-order quantity 8375 9678 9637 8417 10952 10940 7234
Unused quantity 0 0 0 0 0 0 0
Units produced 1250.5 1141 1220.6 1235.1 1193.5 - -
Case 8 $ \beta $=0.99 $ \lambda $=0.5 $ \rho $=0.5
Pre-order quantity 8300 9593 9513 8299 10763 10806 7156
Unused quantity 0 0 0 0 0 0 0
Units produced 1255.6 1144 1213.7 1180.5 1181.1 - -
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NO. 1 2 3 4 5 6 7
Case 1 $ \lambda $=0.5 $ \rho $=0.5 $ \beta $=0.95
Pre-order quantity 8279 9568 9496 8281 10744 10785 7137
Unused quantity 0 0 0 0 0 0 0
Units produced 1258 1142.3 1214.5 1175.6 1173.1 - -
Case 2 $ \lambda $=0.2 $ \rho $=0.5 $ \beta $=0.95
Pre-order quantity 8556 9912 9831 8546 11184 11187 7402
Unused quantity 0 0 0 0 0 0 0
Units produced 1278.6 1154 1285.3 1221.5 1231.2 - -
Case 3 $ \lambda $=0.8 $ \rho $=0.5 $ \beta $=0.95
Pre-order quantity 8215 9487 9452 8224 10714 10723 7084
Unused quantity 0 0 0 0 0 0 0
Units produced 1245.5 1131.2 1227.3 1166.1 1157 - -
Case 4 $ \rho $=0.1 $ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity 8266 9550 9482 8268 10729 10766 7125
Unused quantity 0 0 0 0 0 0 0
Units produced 1252.5 1141 1213.7 1174.1 1172 - -
Case 5 $ \rho $=1 $ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity 8340 9625 9561 8336 10808 10846 7184
Unused quantity 0 0 0 0 0 0 0
Units produced 1258.3 1156.6 1225.5 1178 1182.7 - -
Case 6 $ \rho $=0.05 $ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity 8262 9543 9481 8263 10731 10762 7122
Unused quantity 0 0 0 0 0 0 0
Units produced 1249.2 1140.2 1218.1 1172.1 1171.3 - -
Case 7 $ \beta $=0.90 $ \lambda $=0.5 $ \rho $=0.5
Pre-order quantity 8375 9678 9637 8417 10952 10940 7234
Unused quantity 0 0 0 0 0 0 0
Units produced 1250.5 1141 1220.6 1235.1 1193.5 - -
Case 8 $ \beta $=0.99 $ \lambda $=0.5 $ \rho $=0.5
Pre-order quantity 8300 9593 9513 8299 10763 10806 7156
Unused quantity 0 0 0 0 0 0 0
Units produced 1255.6 1144 1213.7 1180.5 1181.1 - -
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