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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 and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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)

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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)

[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.

[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.

[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.

[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.

[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.

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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