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doi: 10.3934/jimo.2021180
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A new data-driven robust optimization approach to multi-item newsboy problems

School of Mathematics and Statistics, Central South University, Changsha 410083, China

* Corresponding authors: Zhong Wan

Received  May 2021 Revised  August 2021 Early access October 2021

Fund Project: This research is supported by the National Social Science Foundation of China (Grant No. 21BGL122)

A newsboy problem is a typical stochastic inventory management problem and has extensive applications in the fields of operational research, management sciences and marketing sciences. One of the challenges underlying such problems is to handle the uncertainty of demands. In the existing results, it is often to assume that the demand distribution is given to facilitate solution of the problems. In this paper, a novel data-driven robust optimization model for solving multi-item newsboy problems is proposed by combining the absolute robust optimization with a data-driven uncertainty set, and the latter is leveraged to address the uncertainty of demands. For the single-item situation, a closed-form solution is obtained and influences of parameters on the optimal solutions are analyzed. Owing to complexity of the multi-item situation, a uniform smoothing function is leveraged to smooth the proposed model. Then, an algorithm, called a modified Frank-Wolfe feasible direction algorithm, is developed to solve a series of smooth subproblems. Numerical simulation demonstrates that the proposed model in this paper can reduce over-conservation of robust optimization methods and is more robust than other similar well-established methods in the literature. By numerical simulation and sensitivity analysis, it is concluded that: (1) The proposed method can provide more stable optimal order policy and profits than the existing ones; (2) For a product with a higher unit purchase price, the optimal order quantities are more sensitive to its change; (3) In view of profitability, the newsboy should not to be too risk-averse.

Citation: Ying Kou, Zhong Wan. A new data-driven robust optimization approach to multi-item newsboy problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021180
References:
[1]

M. A. Abdel-Aal and S. Z. Selim, Robust optimization for selective newsvendor problem with uncertain demand, Computers & Industrial Engineering, 135 (2019), 838-854.  doi: 10.1016/j.cie.2019.06.047.  Google Scholar

[2]

L. L. Abdel-Malek and N. Areeratchakul, A quadratic programming approach to the multi-product newsboy problem with side constraints, European J. Oper. Res., 176 (2007), 1607-1619.  doi: 10.1016/j.ejor.2005.11.002.  Google Scholar

[3]

K. J. ArrowT. Harris and J. Marschak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813.  Google Scholar

[4]

A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Math. Program., 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar

[5]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Oper. Res. Lett., 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[6]

Y. Cao and Z. J. M. Shen, Quantile forecasting and data-driven inventory management under nonstationary demand, Oper. Res. Lett., 47 (2019), 465-472.  doi: 10.1016/j.orl.2019.08.008.  Google Scholar

[7]

E. CarrizosaA. V. Olivares-Nadal and P. Ram$\acute{i}$rez-Cobo, Robust newsvendor problem with autoregressive demand, Comput. Oper. Res., 68 (2016), 123-133.  doi: 10.1016/j.cor.2015.11.002.  Google Scholar

[8]

L. H. Chen and Y. C. Chen, A multiple-item budget-constraint newsboy problem with a reservation policy, Omega, 38 (2010), 431-439.  doi: 10.1016/j.omega.2009.10.007.  Google Scholar

[9]

Z. ChenS. Peng and J. Liu, Data-driven robust chance constrained problems: A mixture model approach, J. Optim. Theory Appli., 179 (2018), 1065-1085.  doi: 10.1007/s10957-018-1376-4.  Google Scholar

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K. W. DingN. J. Huang and Y. B. Xiao, Distributionally robust chance constrained problems under general moments information, J. Ind. Manag. Optim., 16 (2020), 2923-2942.  doi: 10.3934/jimo.2019087.  Google Scholar

[13]

F. FangT. D. Nguyen and C. S. Currie, Joint pricing and inventory decisions for substitutable and perishable products under demand uncertainty, European J. Oper. Res., 293 (2021), 594-602.  doi: 10.1016/j.ejor.2020.08.002.  Google Scholar

[14]

A. FuduliM. Gaudioso and G. Giallombardo, Minimizing nonconvex nonsmooth functions via cutting planes and proximity control, SIAM J. Optim., 14 (2003), 743-756.  doi: 10.1137/S1052623402411459.  Google Scholar

[15]

R. HuangS. QuX. Yang and Z. Liu, Multi-stage distributionally robust optimization with risk aversion, J. Ind. Manag. Optim., 17 (2021), 233-259.  doi: 10.3934/jimo.2019109.  Google Scholar

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[17]

O. JadidiM. Y. JaberS. ZolfaghriR. Pinto and F. Firouzi, Dynamic pricing and lot sizing for a newsvendor problem with supplier selection, quantity discounts, and limited supply capacity, Computers & Industrial Engineering, 154 (2021), 107113.  doi: 10.1016/j.cie.2021.107113.  Google Scholar

[18]

A. Jalilvand-NejadR. Shafaei and H. Shahriari, Robust optimization under correlated polyhedral uncertainty set, Computers & Industrial Engineering, 92 (2016), 82-94.  doi: 10.1016/j.cie.2015.12.006.  Google Scholar

[19]

G. J. Kyparisis and C. Koulamas, The price-setting newsvendor problem with nonnegative linear additive demand, European J. Oper. Res., 269 (2018), 695-698.  doi: 10.1016/j.ejor.2018.02.019.  Google Scholar

[20]

R. LeviG. Perakis and J. Uichanco, The data-driven newsvendor problem: New bounds and insights, Oper. Res., 63 (2015), 1294-1306.  doi: 10.1287/opre.2015.1422.  Google Scholar

[21]

C. Ning and F. You, Data-driven decision making under uncertainty integrating robust optimization with principal component analysis and kernel smoothing methods, Computers & Chemical Engineering, 112 (2018), 190-210.  doi: 10.1016/j.compchemeng.2018.02.007.  Google Scholar

[22]

D. Noll, Cutting plane oracles to minimize non-smooth non-convex functions, Set-Valued Var. Anal., 18 (2010), 531-568.  doi: 10.1007/s11228-010-0159-3.  Google Scholar

[23]

S. PuniaS. P. Singh and J. K. Madaan, From predictive to prescriptive analytics: A data-driven multi-item newsvendor model, Decision Support Systems, 136 (2020), 113340.  doi: 10.1016/j.dss.2020.113340.  Google Scholar

[24]

Y. QinR. WangA. J. VakhariaY. Chen and M. M. Seref, The newsvendor problem: Review and directions for future research, European J. Oper. Res., 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.  Google Scholar

[25]

R. QiuY. Sun and M. Sun, A distributionally robust optimization approach for multi-product inventory decisions with budget constraint and demand and yield uncertainties, Comput. Oper. Res., 126 (2021), 105081.  doi: 10.1016/j.cor.2020.105081.  Google Scholar

[26]

R. QiuY. SunZ. P. Fan and M. Sun, Robust multi-product inventory optimization under support vector clustering-based data-driven demand uncertainty set, Soft Computing, 24 (2019), 6259-6275.  doi: 10.1007/s00500-019-03927-2.  Google Scholar

[27]

A. L. Sachs and S. Minner, The data-driven newsvendor with censored demand observations, International Journal of Production Economics, 149 (2014), 28-36.   Google Scholar

[28]

H. E. Scarf, A Min-Max Solution of an Inventory Problem, Rand Corp Santa Monica Calif, 1957. Google Scholar

[29]

N. TurkenY. TanA. J. VakhariaL. WangR. Wang and A. Yenipazarli, The multi-product newsvendor problem: Review, extensions, and directions for future research, Handbook of Newsboy Problems, 176 (2012), 3-39.  doi: 10.1007/978-1-4614-3600-3_1.  Google Scholar

[30]

G. L. Vairaktarakis, Robust multi-item newsboy models with a budget constraint, International Journal of Production Economics, 66 (2000), 213-226.  doi: 10.1016/S0925-5273(99)00129-2.  Google Scholar

[31]

Z. WanS. Zhu and Z. Wan, An integrated stochastic model and algorithm for multi-product newsvendor problems, International Journal of Modeling, Simulation, and Scientific Computing, 11 (2020), 2050027.  doi: 10.1142/S1793962320500270.  Google Scholar

[32]

Z. WanJ. Liu and J. Zhang, Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees, J. Ind. Manag. Optim., 16 (2020), 2117-2139.  doi: 10.3934/jimo.2019046.  Google Scholar

[33]

Z. WangF. LiuC. ZhaoZ. Ma and W. Wei, Distributed optimal load frequency control considering nonsmooth cost functions, Systems Control Lett., 136 (2020), 104607.  doi: 10.1016/j.sysconle.2019.104607.  Google Scholar

[34]

X. XuH. WangC. Dang and P. Ji, The loss-averse newsvendor model with backordering, International Journal of Production Economics, 188 (2017), 1-10.  doi: 10.1016/j.ijpe.2017.03.005.  Google Scholar

[35]

L. Xu, Y. Zheng and L. Jiang, A robust data-driven approach for the newsvendor problem with nonparametric information, Manufacturing & Service Operations Management, 2021. doi: 10.1287/msom.2020.0961.  Google Scholar

[36]

Y. YangM. PesaventoZ. Q. Luo and B. Ottersten, Inexact block coordinate descent algorithms for nonsmooth nonconvex optimization, IEEE Trans. Signal Process., 68 (2019), 947-961.  doi: 10.1109/TSP.2019.2959240.  Google Scholar

[37]

L. Yong, Some uniform smooth approximating functions and their properties, Journal of Shaanxi University of Technology (Natural Science Edition), 43 (2018), 74–79. (in Chinese) Google Scholar

[38]

H. Yu and J. Sun, Robust stochastic optimization with convex risk measures: A discretized subgradient scheme, J. Ind. Manag. Optim., 17 (2021), 81-99.  doi: 10.3934/jimo.2019100.  Google Scholar

[39]

L. ZhangG. Zhang and Z. Yao, Analysis of two substitute products newsvendor problem with a budget constraint, Computers & Industrial Engineering, 140 (2020), 106235.  doi: 10.1016/j.cie.2019.106235.  Google Scholar

[40]

G. Zhang, The multi-product newsboy problem with supplier quantity discounts and a budget constraint, European J. Oper. Res, 206 (2010), 350-360.  doi: 10.1016/j.ejor.2010.02.038.  Google Scholar

[41]

J. ZhangW. Xie and S. C. Sarin, Robust multi-product newsvendor model with uncertain demand and substitution, European J. Oper. Res., 293 (2021), 190-202.  doi: 10.1016/j.ejor.2020.12.023.  Google Scholar

[42]

X. ZhangS. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Appl. Math. Model., 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

show all references

References:
[1]

M. A. Abdel-Aal and S. Z. Selim, Robust optimization for selective newsvendor problem with uncertain demand, Computers & Industrial Engineering, 135 (2019), 838-854.  doi: 10.1016/j.cie.2019.06.047.  Google Scholar

[2]

L. L. Abdel-Malek and N. Areeratchakul, A quadratic programming approach to the multi-product newsboy problem with side constraints, European J. Oper. Res., 176 (2007), 1607-1619.  doi: 10.1016/j.ejor.2005.11.002.  Google Scholar

[3]

K. J. ArrowT. Harris and J. Marschak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813.  Google Scholar

[4]

A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Math. Program., 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar

[5]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Oper. Res. Lett., 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[6]

Y. Cao and Z. J. M. Shen, Quantile forecasting and data-driven inventory management under nonstationary demand, Oper. Res. Lett., 47 (2019), 465-472.  doi: 10.1016/j.orl.2019.08.008.  Google Scholar

[7]

E. CarrizosaA. V. Olivares-Nadal and P. Ram$\acute{i}$rez-Cobo, Robust newsvendor problem with autoregressive demand, Comput. Oper. Res., 68 (2016), 123-133.  doi: 10.1016/j.cor.2015.11.002.  Google Scholar

[8]

L. H. Chen and Y. C. Chen, A multiple-item budget-constraint newsboy problem with a reservation policy, Omega, 38 (2010), 431-439.  doi: 10.1016/j.omega.2009.10.007.  Google Scholar

[9]

Z. ChenS. Peng and J. Liu, Data-driven robust chance constrained problems: A mixture model approach, J. Optim. Theory Appli., 179 (2018), 1065-1085.  doi: 10.1007/s10957-018-1376-4.  Google Scholar

[10]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[11]

S. DengZ. Wan and Y. Zhou, Optimization model and solution method for dynamically correlated two-product newsboy problems based on Copula, Discrete Contin. Dyn. Syst. Ser. S, 13 (2018), 1637-1652.  doi: 10.3934/dcdss.2020096.  Google Scholar

[12]

K. W. DingN. J. Huang and Y. B. Xiao, Distributionally robust chance constrained problems under general moments information, J. Ind. Manag. Optim., 16 (2020), 2923-2942.  doi: 10.3934/jimo.2019087.  Google Scholar

[13]

F. FangT. D. Nguyen and C. S. Currie, Joint pricing and inventory decisions for substitutable and perishable products under demand uncertainty, European J. Oper. Res., 293 (2021), 594-602.  doi: 10.1016/j.ejor.2020.08.002.  Google Scholar

[14]

A. FuduliM. Gaudioso and G. Giallombardo, Minimizing nonconvex nonsmooth functions via cutting planes and proximity control, SIAM J. Optim., 14 (2003), 743-756.  doi: 10.1137/S1052623402411459.  Google Scholar

[15]

R. HuangS. QuX. Yang and Z. Liu, Multi-stage distributionally robust optimization with risk aversion, J. Ind. Manag. Optim., 17 (2021), 233-259.  doi: 10.3934/jimo.2019109.  Google Scholar

[16]

J. HuberS. M$\ddot{u}$llerM. Fleischmann and H. Stuckenschmidt, A data-driven newsvendor problem: From data to decision, European J. Oper. Res., 278 (2019), 904-915.  doi: 10.1016/j.ejor.2019.04.043.  Google Scholar

[17]

O. JadidiM. Y. JaberS. ZolfaghriR. Pinto and F. Firouzi, Dynamic pricing and lot sizing for a newsvendor problem with supplier selection, quantity discounts, and limited supply capacity, Computers & Industrial Engineering, 154 (2021), 107113.  doi: 10.1016/j.cie.2021.107113.  Google Scholar

[18]

A. Jalilvand-NejadR. Shafaei and H. Shahriari, Robust optimization under correlated polyhedral uncertainty set, Computers & Industrial Engineering, 92 (2016), 82-94.  doi: 10.1016/j.cie.2015.12.006.  Google Scholar

[19]

G. J. Kyparisis and C. Koulamas, The price-setting newsvendor problem with nonnegative linear additive demand, European J. Oper. Res., 269 (2018), 695-698.  doi: 10.1016/j.ejor.2018.02.019.  Google Scholar

[20]

R. LeviG. Perakis and J. Uichanco, The data-driven newsvendor problem: New bounds and insights, Oper. Res., 63 (2015), 1294-1306.  doi: 10.1287/opre.2015.1422.  Google Scholar

[21]

C. Ning and F. You, Data-driven decision making under uncertainty integrating robust optimization with principal component analysis and kernel smoothing methods, Computers & Chemical Engineering, 112 (2018), 190-210.  doi: 10.1016/j.compchemeng.2018.02.007.  Google Scholar

[22]

D. Noll, Cutting plane oracles to minimize non-smooth non-convex functions, Set-Valued Var. Anal., 18 (2010), 531-568.  doi: 10.1007/s11228-010-0159-3.  Google Scholar

[23]

S. PuniaS. P. Singh and J. K. Madaan, From predictive to prescriptive analytics: A data-driven multi-item newsvendor model, Decision Support Systems, 136 (2020), 113340.  doi: 10.1016/j.dss.2020.113340.  Google Scholar

[24]

Y. QinR. WangA. J. VakhariaY. Chen and M. M. Seref, The newsvendor problem: Review and directions for future research, European J. Oper. Res., 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.  Google Scholar

[25]

R. QiuY. Sun and M. Sun, A distributionally robust optimization approach for multi-product inventory decisions with budget constraint and demand and yield uncertainties, Comput. Oper. Res., 126 (2021), 105081.  doi: 10.1016/j.cor.2020.105081.  Google Scholar

[26]

R. QiuY. SunZ. P. Fan and M. Sun, Robust multi-product inventory optimization under support vector clustering-based data-driven demand uncertainty set, Soft Computing, 24 (2019), 6259-6275.  doi: 10.1007/s00500-019-03927-2.  Google Scholar

[27]

A. L. Sachs and S. Minner, The data-driven newsvendor with censored demand observations, International Journal of Production Economics, 149 (2014), 28-36.   Google Scholar

[28]

H. E. Scarf, A Min-Max Solution of an Inventory Problem, Rand Corp Santa Monica Calif, 1957. Google Scholar

[29]

N. TurkenY. TanA. J. VakhariaL. WangR. Wang and A. Yenipazarli, The multi-product newsvendor problem: Review, extensions, and directions for future research, Handbook of Newsboy Problems, 176 (2012), 3-39.  doi: 10.1007/978-1-4614-3600-3_1.  Google Scholar

[30]

G. L. Vairaktarakis, Robust multi-item newsboy models with a budget constraint, International Journal of Production Economics, 66 (2000), 213-226.  doi: 10.1016/S0925-5273(99)00129-2.  Google Scholar

[31]

Z. WanS. Zhu and Z. Wan, An integrated stochastic model and algorithm for multi-product newsvendor problems, International Journal of Modeling, Simulation, and Scientific Computing, 11 (2020), 2050027.  doi: 10.1142/S1793962320500270.  Google Scholar

[32]

Z. WanJ. Liu and J. Zhang, Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees, J. Ind. Manag. Optim., 16 (2020), 2117-2139.  doi: 10.3934/jimo.2019046.  Google Scholar

[33]

Z. WangF. LiuC. ZhaoZ. Ma and W. Wei, Distributed optimal load frequency control considering nonsmooth cost functions, Systems Control Lett., 136 (2020), 104607.  doi: 10.1016/j.sysconle.2019.104607.  Google Scholar

[34]

X. XuH. WangC. Dang and P. Ji, The loss-averse newsvendor model with backordering, International Journal of Production Economics, 188 (2017), 1-10.  doi: 10.1016/j.ijpe.2017.03.005.  Google Scholar

[35]

L. Xu, Y. Zheng and L. Jiang, A robust data-driven approach for the newsvendor problem with nonparametric information, Manufacturing & Service Operations Management, 2021. doi: 10.1287/msom.2020.0961.  Google Scholar

[36]

Y. YangM. PesaventoZ. Q. Luo and B. Ottersten, Inexact block coordinate descent algorithms for nonsmooth nonconvex optimization, IEEE Trans. Signal Process., 68 (2019), 947-961.  doi: 10.1109/TSP.2019.2959240.  Google Scholar

[37]

L. Yong, Some uniform smooth approximating functions and their properties, Journal of Shaanxi University of Technology (Natural Science Edition), 43 (2018), 74–79. (in Chinese) Google Scholar

[38]

H. Yu and J. Sun, Robust stochastic optimization with convex risk measures: A discretized subgradient scheme, J. Ind. Manag. Optim., 17 (2021), 81-99.  doi: 10.3934/jimo.2019100.  Google Scholar

[39]

L. ZhangG. Zhang and Z. Yao, Analysis of two substitute products newsvendor problem with a budget constraint, Computers & Industrial Engineering, 140 (2020), 106235.  doi: 10.1016/j.cie.2019.106235.  Google Scholar

[40]

G. Zhang, The multi-product newsboy problem with supplier quantity discounts and a budget constraint, European J. Oper. Res, 206 (2010), 350-360.  doi: 10.1016/j.ejor.2010.02.038.  Google Scholar

[41]

J. ZhangW. Xie and S. C. Sarin, Robust multi-product newsvendor model with uncertain demand and substitution, European J. Oper. Res., 293 (2021), 190-202.  doi: 10.1016/j.ejor.2020.12.023.  Google Scholar

[42]

X. ZhangS. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Appl. Math. Model., 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

Figure 1.  Impacts of confidence level on order quantities and profits
Figure 2.  Impacts of demands on optimal order quantities
Figure 3.  Impacts of uniform approximation levels
Figure 4.  Impacts of confidence level and deviation on order quantities
Figure 5.  Impacts of confidence levels and data volatility on profits
Figure 6.  Impacts of unit purchase prices on optimal order quantities
Table 1.  Values of model parameters
Item $ s $ (Yuan RMB) $ v $ (Yuan RMB) $ p $ (Yuan RMB) $ c $ (Yuan RMB)
A 80 3 6 12
B 97 10 16 32
Item $ s $ (Yuan RMB) $ v $ (Yuan RMB) $ p $ (Yuan RMB) $ c $ (Yuan RMB)
A 80 3 6 12
B 97 10 16 32
Table 2.  Profits of different robust methods with changing demands
($ \mu_1 $, $ \sigma_1 $) ($ \mu_2 $, $ \sigma _2 $) $ R_U $ $ R_A $ $ R_D $ $ R_R $
(239.55, 5.80) (82.34, 3.87) 20824.79 20523.29 20205.75 20214.74
(+10.82, 0) (0, 0) +767.31 +737.58 +738.20 +738.10
(-8.85, 0) (0, 0) -625.43 -601.57 -601.61 -601.57
(0, 0) (+10.20, 0) +691.21 +663.47 +663.91 +662.99
(0, 0) (-5.13, 0) -350.61 -331.46 -330.69 -330.23
(0, +9.61) (0, 0) -702.95 -1096.27 -1329.40 -1320.59
(0, -1.42) (0, 0) +101.46 +161.14 +195.23 +194.66
(0, 0) (0, +5.55) -608.36 -656.60 -910.03 -869.37
(0, 0) (0, -0.48) +42.60 +57.51 +79.44 +77.63
($ \mu_1 $, $ \sigma_1 $) ($ \mu_2 $, $ \sigma _2 $) $ R_U $ $ R_A $ $ R_D $ $ R_R $
(239.55, 5.80) (82.34, 3.87) 20824.79 20523.29 20205.75 20214.74
(+10.82, 0) (0, 0) +767.31 +737.58 +738.20 +738.10
(-8.85, 0) (0, 0) -625.43 -601.57 -601.61 -601.57
(0, 0) (+10.20, 0) +691.21 +663.47 +663.91 +662.99
(0, 0) (-5.13, 0) -350.61 -331.46 -330.69 -330.23
(0, +9.61) (0, 0) -702.95 -1096.27 -1329.40 -1320.59
(0, -1.42) (0, 0) +101.46 +161.14 +195.23 +194.66
(0, 0) (0, +5.55) -608.36 -656.60 -910.03 -869.37
(0, 0) (0, -0.48) +42.60 +57.51 +79.44 +77.63
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