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

  • *Corresponding authors: Yuli Zhang and Xiaotian Zhuang

    *Corresponding authors: Yuli Zhang and Xiaotian Zhuang 
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  • 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.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  Histogram of random demand for a product over two periods

    Figure 2.  Proportion of single-item order and multi-item orders in two FDCs of JD.com

    Table 1.  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 $.
     | Show Table
    DownLoad: CSV

    Table 2.  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
     | Show Table
    DownLoad: CSV

    Table 3.  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
     | Show Table
    DownLoad: CSV

    Table 4.  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
     | Show Table
    DownLoad: CSV

    Table 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
     | Show Table
    DownLoad: CSV
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