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July  2022, 15(7): 1777-1795. doi: 10.3934/dcdss.2022006

## Distributionally robust front distribution center inventory optimization with uncertain multi-item orders

 1 School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Intelligent Supply Chain, Beijing Jingdong Zhenshi Information Technology Co., Ltd., Beijing 100176, China

*Corresponding authors: Yuli Zhang and Xiaotian Zhuang

Received  November 2021 Revised  December 2021 Published  July 2022 Early access  January 2022

As a new retail model, the front distribution center (FDC) has been recognized as an effective instrument for timely order delivery. However, the high customer demand uncertainty, multi-item order pattern, and limited inventory capacity pose a challenging task for FDC managers to determine the optimal inventory level. To this end, this paper proposes a two-stage distributionally robust (DR) FDC inventory model and an efficient row-and-column generation (RCG) algorithm. The proposed DR model uses a Wasserstein distance-based distributional set to describe the uncertain demand and utilizes a robust conditional value at risk decision criterion to mitigate the risk of distribution ambiguity. The proposed RCG is able to solve the complex max-min-max DR model exactly by repeatedly solving relaxed master problems and feasibility subproblems. We show that the optimal solution of the non-convex feasibility subproblem can be obtained by solving two linear programming problems. Numerical experiments based on real-world data highlight the superior out-of-sample performance of the proposed DR model in comparison with an existing benchmark approach and validate the computational efficiency of the proposed algorithm.

Citation: Yuli Zhang, Lin Han, Xiaotian Zhuang. Distributionally robust front distribution center inventory optimization with uncertain multi-item orders. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1777-1795. doi: 10.3934/dcdss.2022006
##### References:
 [1] J. Acimovic and S. C. Graves, Making better fulfillment decisions on the fly in an online retail environment, Manufacturing & Service Operations Management, 17 (2015), 34-51.  doi: 10.1287/msom.2014.0505. [2] J. Acimovic and S. C. Graves, Mitigating spillover in online retailing via replenishment, Manufacturing & Service Operations Management, 19 (2017), 419-436. [3] B. Bebitoglu, A. Șen and P. Kaminsky, Multi-location assortment optimization under capacity constraints, Available at SSRN 3249175, 2018. doi: 10.2139/ssrn. 3249175. [4] A. Catalán and M. Fisher, Assortment allocation to distribution centers to minimize split customer orders, Available at SSRN 2166687. [5] B. Dai, H. Chen, Y. Li, Y. Zhang, X. Wang and Y. Deng, Inventory replenishment planning in a distribution system with safety stock policy and minimum and maximum joint replenishment quantity constraints, In 2019 International Conference on Industrial Engineering and Systems Management (IESM), (2019), 1–6. doi: 10.1109/IESM45758.2019.8948155. [6] L. Deng, W. Bi, H. Liu and K. L. Teo, A multi-stage method for joint pricing and inventory model with promotion constrains, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1653-1682.  doi: 10.3934/dcdss.2020097. [7] [8] D. K. Nayak, S. S. Routray, S. K. Paikray and H. Dutta, A fuzzy inventory model for weibull deteriorating items under completely backlogged shortages, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 2435-2453.  doi: 10.3934/dcdss.2020401. [9] R. T. Rockafellar and S. Uryasev et al., Optimization of conditional value-at-risk, Stochastic Optimization: Algorithms and Applications, 54 (2001), 411-435.  doi: 10.1007/978-1-4757-6594-6_17. [10] M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171. [11] Z. Wang, K. You, S. Song and Y. Zhang, Wasserstein distributionally robust shortest path problem, European J. Oper. Res., 284 (2020), 31-43.  doi: 10.1016/j.ejor.2020.01.009. [12] T. Wu, H. Mao, Y. Li and D. Chen, Assortment selection for a frontend warehouse: A robust data-driven approach, In 49th International Conference on Computers and Industrial Engineering (CIE 2019), (2019), 56–64. [13] P. J. Xu, Order Fulfillment in Online Retailing: What Goes Where, PhD thesis, Massachusetts Institute of Technology, 2005. [14] P. J. Xu, R. Allgor and S. C. Graves, Benefits of reevaluating real-time order fulfillment decisions, Manufacturing & Service Operations Management, 11 (2009), 340-355.  doi: 10.1287/msom.1080.0222. [15] B. Zeng and L. Zhao, Solving two-stage robust optimization problems using a column-and-constraint generation method, Oper. Res. Lett., 41 (2013), 457-461.  doi: 10.1016/j.orl.2013.05.003. [16] Y. Zhang, Z.-J. M. Shen and S. Song, Exact algorithms for distributionally $\beta$-robust machine scheduling with uncertain processing times, INFORMS J. Comput., 30 (2018), 662-676.  doi: 10.1287/ijoc.2018.0807. [17] Y. Zhang, S. Song, Z.-J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 19 (2017), 1080-1090. [18] S. Zhu, X. Hu, K. Huang and Y. Yuan, Optimization of product category allocation in multiple warehouses to minimize splitting of online supermarket customer orders, European J. Oper. Res., 290 (2021), 556-571.  doi: 10.1016/j.ejor.2020.08.024.

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##### References:
 [1] J. Acimovic and S. C. Graves, Making better fulfillment decisions on the fly in an online retail environment, Manufacturing & Service Operations Management, 17 (2015), 34-51.  doi: 10.1287/msom.2014.0505. [2] J. Acimovic and S. C. Graves, Mitigating spillover in online retailing via replenishment, Manufacturing & Service Operations Management, 19 (2017), 419-436. [3] B. Bebitoglu, A. Șen and P. Kaminsky, Multi-location assortment optimization under capacity constraints, Available at SSRN 3249175, 2018. doi: 10.2139/ssrn. 3249175. [4] A. Catalán and M. Fisher, Assortment allocation to distribution centers to minimize split customer orders, Available at SSRN 2166687. [5] B. Dai, H. Chen, Y. Li, Y. Zhang, X. Wang and Y. Deng, Inventory replenishment planning in a distribution system with safety stock policy and minimum and maximum joint replenishment quantity constraints, In 2019 International Conference on Industrial Engineering and Systems Management (IESM), (2019), 1–6. doi: 10.1109/IESM45758.2019.8948155. [6] L. Deng, W. Bi, H. Liu and K. L. Teo, A multi-stage method for joint pricing and inventory model with promotion constrains, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1653-1682.  doi: 10.3934/dcdss.2020097. [7] [8] D. K. Nayak, S. S. Routray, S. K. Paikray and H. Dutta, A fuzzy inventory model for weibull deteriorating items under completely backlogged shortages, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 2435-2453.  doi: 10.3934/dcdss.2020401. [9] R. T. Rockafellar and S. Uryasev et al., Optimization of conditional value-at-risk, Stochastic Optimization: Algorithms and Applications, 54 (2001), 411-435.  doi: 10.1007/978-1-4757-6594-6_17. [10] M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171. [11] Z. Wang, K. You, S. Song and Y. Zhang, Wasserstein distributionally robust shortest path problem, European J. Oper. Res., 284 (2020), 31-43.  doi: 10.1016/j.ejor.2020.01.009. [12] T. Wu, H. Mao, Y. Li and D. Chen, Assortment selection for a frontend warehouse: A robust data-driven approach, In 49th International Conference on Computers and Industrial Engineering (CIE 2019), (2019), 56–64. [13] P. J. Xu, Order Fulfillment in Online Retailing: What Goes Where, PhD thesis, Massachusetts Institute of Technology, 2005. [14] P. J. Xu, R. Allgor and S. C. Graves, Benefits of reevaluating real-time order fulfillment decisions, Manufacturing & Service Operations Management, 11 (2009), 340-355.  doi: 10.1287/msom.1080.0222. [15] B. Zeng and L. Zhao, Solving two-stage robust optimization problems using a column-and-constraint generation method, Oper. Res. Lett., 41 (2013), 457-461.  doi: 10.1016/j.orl.2013.05.003. [16] Y. Zhang, Z.-J. M. Shen and S. Song, Exact algorithms for distributionally $\beta$-robust machine scheduling with uncertain processing times, INFORMS J. Comput., 30 (2018), 662-676.  doi: 10.1287/ijoc.2018.0807. [17] Y. Zhang, S. Song, Z.-J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 19 (2017), 1080-1090. [18] S. Zhu, X. Hu, K. Huang and Y. Yuan, Optimization of product category allocation in multiple warehouses to minimize splitting of online supermarket customer orders, European J. Oper. Res., 290 (2021), 556-571.  doi: 10.1016/j.ejor.2020.08.024.
Histogram of random demand for a product over two periods
Proportion of single-item order and multi-item orders in two FDCs of JD.com
Notation list
 Sets $I$ Set of product SKUs. $I=\{1,\cdots,n\}.$ $J$ Set of multi-item orders. $J=\{1,\cdots,m\}.$ Parameters $a_{ij} \in Z_+$ Number of SKU $i\in I$ contained in order $j \in J$. $p'_{i} \in R_+$ Net profit for each SKU $i\in I$. $p_{j} \in R_+$ Net profit for each multi-item order $j\in J$, i.e., $p_j=\sum_{i\in I}a_{ij}p'_i$. $h'_{i} \in R_+$ Holding cost for each SKU $i\in I$. $h_{j} \in R_+$ Holding cost for each multi-item order $j\in J$, i.e., $h_j=\sum_{i\in I}a_{ij}h'_i$. $c_{i} \in R_+$ Occupied capacity for each SKU $i\in I$. $C \in R_+$ Capacity of the FDC. Random Parameters $d_j$ Random demand for the multi-item order $j\in J$. Decision Variables and Functions $s_i\in R_+$ First-stage decision; Inventory level for SKU $i\in I$. $y_j\in R_+$ Second-stage decision; Number of satisfied multi-item order $j\in J$. $g(s,d)$ Second-stage net profit function for given $s$ and $d$.
 Sets $I$ Set of product SKUs. $I=\{1,\cdots,n\}.$ $J$ Set of multi-item orders. $J=\{1,\cdots,m\}.$ Parameters $a_{ij} \in Z_+$ Number of SKU $i\in I$ contained in order $j \in J$. $p'_{i} \in R_+$ Net profit for each SKU $i\in I$. $p_{j} \in R_+$ Net profit for each multi-item order $j\in J$, i.e., $p_j=\sum_{i\in I}a_{ij}p'_i$. $h'_{i} \in R_+$ Holding cost for each SKU $i\in I$. $h_{j} \in R_+$ Holding cost for each multi-item order $j\in J$, i.e., $h_j=\sum_{i\in I}a_{ij}h'_i$. $c_{i} \in R_+$ Occupied capacity for each SKU $i\in I$. $C \in R_+$ Capacity of the FDC. Random Parameters $d_j$ Random demand for the multi-item order $j\in J$. Decision Variables and Functions $s_i\in R_+$ First-stage decision; Inventory level for SKU $i\in I$. $y_j\in R_+$ Second-stage decision; Number of satisfied multi-item order $j\in J$. $g(s,d)$ Second-stage net profit function for given $s$ and $d$.
Statistical Summary
 FDC A FDC B FDC C Number of orders 2632408 669161 1094012 Number of order types 588994 226448 319329 Number of product SKUs 684 659 684
 FDC A FDC B FDC C Number of orders 2632408 669161 1094012 Number of order types 588994 226448 319329 Number of product SKUs 684 659 684
Performance of the proposed DRM in comparison with SM under different $\alpha$-levels
 $\alpha$ CVaR Mean DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 0.85 440033.1 425087.0 3.5 463652.7 451409.4 2.7 0.9 448351.2 429470.7 4.4 465044.1 448640.9 3.7 0.95 456801.3 435469.0 4.9 465726.2 445833.8 4.5 1 465673.0 444738.3 4.7 465673.0 444738.4 4.7 $\alpha$ Std SL DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 0.85 96361.7 100889.8 4.5 0.855 0.831 2.9 0.9 100830.4 106131.3 5.0 0.851 0.819 3.9 0.95 104735.4 109443.4 4.3 0.845 0.811 4.2 1 106993.8 112404.5 4.8 0.841 0.803 4.7
 $\alpha$ CVaR Mean DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 0.85 440033.1 425087.0 3.5 463652.7 451409.4 2.7 0.9 448351.2 429470.7 4.4 465044.1 448640.9 3.7 0.95 456801.3 435469.0 4.9 465726.2 445833.8 4.5 1 465673.0 444738.3 4.7 465673.0 444738.4 4.7 $\alpha$ Std SL DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 0.85 96361.7 100889.8 4.5 0.855 0.831 2.9 0.9 100830.4 106131.3 5.0 0.851 0.819 3.9 0.95 104735.4 109443.4 4.3 0.845 0.811 4.2 1 106993.8 112404.5 4.8 0.841 0.803 4.7
Performance of the proposed DRM in comparison with SM under different $C$
 $C$ CVaR Mean DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 5000 382737.4 362668.6 5.5 385246.1 369465.5 4.3 6000 416751.5 389990.2 6.9 422162.8 398222.2 6.0 7000 440279.7 409294.0 7.6 447352.9 419146.7 6.7 8000 456801.3 435469.0 4.9 465726.2 445833.8 4.5 9000 468414.9 452002.1 3.6 478903.7 463654.4 3.3 10000 475970.4 463618.8 2.7 488072.9 476586.1 2.4 $C$ Std SL DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 5000 49968.8 78203.2 36.1 0.656 0.632 3.8 6000 74686.2 90576.8 17.5 0.727 0.703 3.4 7000 90135.3 102933.8 12.4 0.792 0.755 4.9 8000 104735.4 109443.4 4.3 0.845 0.811 4.2 9000 116587.0 119930.5 2.8 0.891 0.858 3.8 10000 126971.5 128840.1 1.5 0.928 0.905 2.5
 $C$ CVaR Mean DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 5000 382737.4 362668.6 5.5 385246.1 369465.5 4.3 6000 416751.5 389990.2 6.9 422162.8 398222.2 6.0 7000 440279.7 409294.0 7.6 447352.9 419146.7 6.7 8000 456801.3 435469.0 4.9 465726.2 445833.8 4.5 9000 468414.9 452002.1 3.6 478903.7 463654.4 3.3 10000 475970.4 463618.8 2.7 488072.9 476586.1 2.4 $C$ Std SL DRM SM $\Delta(\%)$ DRM SM $\Delta(\%)$ 5000 49968.8 78203.2 36.1 0.656 0.632 3.8 6000 74686.2 90576.8 17.5 0.727 0.703 3.4 7000 90135.3 102933.8 12.4 0.792 0.755 4.9 8000 104735.4 109443.4 4.3 0.845 0.811 4.2 9000 116587.0 119930.5 2.8 0.891 0.858 3.8 10000 126971.5 128840.1 1.5 0.928 0.905 2.5
Computational efficiency of the proposed RCG algorithm
 $C$ T(s) $\mathrm{T_{R}}$(s) $\mathrm{Iter_{R}}$ Var. Cons. $\mathrm{T_{F}(s)}$ $\mathrm{Iter_{F}}$ 5000 57.897 32.835 3 66606 36421 22.107 330 6000 67.840 32.131 3 67259 37625 34.127 330 7000 51.998 20.851 2 67259 31003 29.824 220 8000 59.025 20.980 2 67259 31003 36.657 220 9000 66.092 21.044 2 67259 31003 43.591 220 10000 72.998 21.054 2 67259 31003 50.402 220 Average 62.642 24.816 2.33 67150 33010 36.118 256.667
 $C$ T(s) $\mathrm{T_{R}}$(s) $\mathrm{Iter_{R}}$ Var. Cons. $\mathrm{T_{F}(s)}$ $\mathrm{Iter_{F}}$ 5000 57.897 32.835 3 66606 36421 22.107 330 6000 67.840 32.131 3 67259 37625 34.127 330 7000 51.998 20.851 2 67259 31003 29.824 220 8000 59.025 20.980 2 67259 31003 36.657 220 9000 66.092 21.044 2 67259 31003 43.591 220 10000 72.998 21.054 2 67259 31003 50.402 220 Average 62.642 24.816 2.33 67150 33010 36.118 256.667
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