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Inverse group 1-median problem on trees
Multi-stage distributionally robust optimization with risk aversion
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
A. Nemirovski and A. Shapiro,
Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996.
doi: 10.1137/050622328. |
[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. |
[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. |
[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. |
[35] |
L. Pardo, Statistical Inference Based on Divergence Measures, Textbooks and Monographs, 185. Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[43] |
A. Ruszczyński,
Decomposition methods, Handbooks in Operations Research and Management Science, 10 (2003), 141-211.
doi: 10.1016/S0927-0507(03)10003-5. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[50] |
W. Wiesemann, D. Kuhn and M. Sim,
Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.
doi: 10.1287/opre.2014.1314. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
A. Nemirovski and A. Shapiro,
Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996.
doi: 10.1137/050622328. |
[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. |
[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. |
[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. |
[35] |
L. Pardo, Statistical Inference Based on Divergence Measures, Textbooks and Monographs, 185. Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[43] |
A. Ruszczyński,
Decomposition methods, Handbooks in Operations Research and Management Science, 10 (2003), 141-211.
doi: 10.1016/S0927-0507(03)10003-5. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[50] |
W. Wiesemann, D. Kuhn and M. Sim,
Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.
doi: 10.1287/opre.2014.1314. |
[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. |
[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. |





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 |
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 |
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 | - | - |
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 | - | - |
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 | - | - |
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 | - | - |
NO. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Case 1 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 | - | - |
NO. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Case 1 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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 |
|||||||
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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