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The inventory replenishment policy in an uncertain production-inventory-routing system

This work was supported in part by the National Natural Science Foundation of China under Grant 71471038, in part by the Program for Huiyuan Distinguished Young Scholars, University of International Business and Economics (UIBE) under Grant 17JQ09, in part by the Fundamental Research Funds for the Central Universities in UIBE under Grant CXTD10-05, in part by the Youth Innovative Talent Projects of Guangdong's Universities Provincial Key Platforms and Significant Scientific Research Projects under Grant 2018KQNCX362, in part by the Educational Science Research Projects for 13th Five-Year Plan in Guangdong under Grant 2020GXJK163, and in part by the Key Discipline Project of Guangzhou Xinhua University under Grant 2020XZD02. The authors are grateful to the editor and the referees for their valuable comments and suggestions

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  • This study introduces an uncertain programming model for the integrated production routing problem (PRP) in an uncertain production-inventory-routing system. Based on uncertainty theory, an uncertain programming model is proposed firstly and then transformed into a deterministic and equivalent model. The study further probes into different types of replenishment policies under the condition of uncertain demands, mainly the uncertain maximum level (UML) policy and the uncertain order-up to level (UOU) policy. Some inequalities are put forward to define the UML policy and the UOU policy under the uncertain environments, and the influences brought by uncertain demands are highlighted. The overall costs with optimal solution of the uncertain decision model grow with the increase of the confidence levels. And they are simultaneously affected by the variances of uncertain variables but rely on the value of confidence levels. Results show that when the confidence levels are not less than 0.5, the cost difference between the two policies begins to narrow along with the increase of the confidence levels and the variances of uncertain variables, eventually being trending to zero. When there are higher confidence levels and relatively large uncertainty in realistic applications, in which the solution scale is escalated, being conducive to its efficiency advantage, the comprehensive advantages of the UOU policy is obvious.

    Mathematics Subject Classification: Primary: 90B05, 90C70; Secondary: 90B99.

    Citation:

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  • Figure 1.  The changes of cost in the UMLI policy

    Figure 2.  The changes of cost in the UML policy

    Figure 3.  The changes of cost in the UOU policy

    Figure 4.  The change of the total cost with cost uncertainty

    Table 1.  Indices and sets

    $ i,j $: Indices for retailers, where $ 0 $ corresponds to the plant.
    $ t $: Index for periods or days, $ |T|=\tau $.
    $ N $: Set of retailers, $ N_{0}=N\bigcup\{0\} $.
    $ K $: Set of vehicles, $ K=\{1,2,...,m\} $.
     | Show Table
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    Table 2.  Parameters

    $ \tilde{d}_{it} $: Uncertain demand at retailer $ i $ in period $ t $.
    $ \tilde{f} $: Uncertain fixed production setup cost.
    $ \tilde{u} $: Uncertain unit production cost.
    $ \tilde{h}_{i} $: Uncertain unit inventory holding cost at the plant or retailer.
    $ \tilde{c}_{ij} $: Uncertain transportation cost from node $ i $ to node $ j $.
    $ C $: Production capacity of the plant.
    $ m $: The number of vehicles.
    $ Q_{k} $: Capacity of vehicle $ k $.
    $ B_{i} $: Initial inventory at retailer $ i $, where $ 0 $ corresponds to the plant.
    $ L_{i} $: Maximum inventory level at the plant and retailers.
    $ \alpha $: Confidence level about uncertain costs.
    $ \beta _{i} $: Confidence level of node $ i $ (satisfaction degree of uncertain demands).
    $ \gamma_{i} $: Confidence level of node $ i $ (satisfaction degree of uncertain demands).
     | Show Table
    DownLoad: CSV

    Table 3.  Decision variables

    $ z_{t} $: Equal to 1 if there is production at the plant in period $ t $, 0 otherwise.
    $ p_{t} $: Production quantity in period $ t $.
    $ x_{ijkt} $: Equal to 1 if vehicle $ k $ travels directly from node $ i $ to node $ j $ in period $ t $, 0 otherwise.
    $ w_{ikt} $: Load of vehicle $ k $ immediately before making a delivery to retailer $ i $ in period $ t $.
    $ q_{ikt} $: Quantity delivered to retailer $ i $ by vehicle $ k $ in period $ t $.
    $ y_{ikt} $: Equal to 1 if node $ i $ is visited by vehicle $ k $ in period $ t $, 0 otherwise.
     | Show Table
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    Table 4.  The lower and upper limit, and the interval in the MLI and UMLI policy

    Situation Policy Lower Limit Upper Limit Interval
    Deterministic MLI 0 +$ \infty $ [0, $ +\infty $]
    Linear UMLI $ R_{L}(\beta_{i}) $ +$ \infty $ [$ R_{L}(\beta_{i}) $, $ +\infty $]
    Normal UMLI $ R_{N}(\beta_{i}) $ +$ \infty $ [$ R_{N}(\beta_{i}) $, $ +\infty $]
     | Show Table
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    Table 5.  The lower and upper limit, and the interval in the ML and UML policy

    Situation Policy Lower Limit Upper Limit Interval
    Deterministic ML 0 $ L[i] $ [0, $ L[i] $]
    Linear UML $ R_{L}(\beta_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $ [$ R_{L}(\beta_{i}) $, $ L_{i}-R_{L}(\gamma_{i}) $]
    Normal UML $ R_{N}(\beta_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $ [$ R_{N}(\beta_{i}) $, $ L_{i}-R_{N}(\gamma_{i}) $]
     | Show Table
    DownLoad: CSV

    Table 6.  The lower and upper limit, and the interval in the OU and UOU policies

    Situation Policy Lower Limit Upper Limit Interval
    Deterministic OU 0 $ L[i] $ $ L[i] $
    Linear UOU $ R_{L}(\beta_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $
    Normal UOU $ R_{N}(\beta_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $
     | Show Table
    DownLoad: CSV

    Table 7.  The uncertain variables of the PRP in linear uncertain environment

    Parameters Values
    $ \tilde{d_{it}} $ $ \mathcal{L}(d_{it}(1-\epsilon^{ld}),d_{it}(1+\epsilon^{ld})) $
    $ \tilde{f} $ $ \mathcal{L}(f(1-\epsilon^{lf}),f(1+\epsilon^{lf})) $
    $ \tilde{p} $ $ \mathcal{L}(p(1-\epsilon^{lp}),p(1+\epsilon^{lp})) $
    $ \tilde{h_{0}} $ $ \mathcal{L}(h_{0}(1-\epsilon^{lh}),h_{0}(1+\epsilon^{lh})) $
    $ \tilde{h_{i}} $ $ \mathcal{L}(h_{i}(1-\epsilon^{lh}),h_{i}(1+\epsilon^{lh})) $
    $ \tilde{c_{ij}} $ $ \mathcal{L}(c_{ij}(1-\epsilon^{lc}),c_{ij}(1+\epsilon^{lc})) $
     | Show Table
    DownLoad: CSV

    Table 8.  The change trend of lower and upper limit, the interval, and the cost in the UML policy

    ($ \beta_{i}, \gamma_{i}) $ Lower Limit Upper limit Gap $ Cost_{M} $
    ($<0.5, <0.5) $ $ \downarrow $ $ \uparrow $ $ \uparrow $ $ \downarrow $
    ($<0.5, =0.5) $ $ \downarrow $ = $ \downarrow $ $ \downarrow $
    ($<0.5,>0.5) $ $ \downarrow $ $ \downarrow $ $ \ast $ $ \ast $
    ($ =0.5,<0.5) $ = $ \uparrow $ $ \uparrow $ $ \downarrow $
    ($ =0.5, =0.5) $ = = = =
    ($ =0.5,>0.5) $ = $ \downarrow $ $ \downarrow $ $ \uparrow $
    ($>0.5,<0.5) $ $ \uparrow $ $ \uparrow $ $ \ast $ $ \ast $
    ($>0.5, =0.5) $ $ \uparrow $ = $ \downarrow $ $ \uparrow $
    ($>0.5,>0.5) $ $ \uparrow $ $ \downarrow $ $ \downarrow $ $ \uparrow $
     | Show Table
    DownLoad: CSV

    Table 9.  The change trend of lower and upper limit, the interval, and the cost in the UOU policies

    ($ \beta_{i}, \gamma_{i}) $ Lower Limit Upper limit Gap $ Cost_{U} $
    ($<0.5,<0.5) $ $ \downarrow $ $ \uparrow $ $ \uparrow $ $ \uparrow $
    ($<0.5, =0.5) $ $ \downarrow $ = $ \downarrow $ =
    ($<0.5,>0.5) $ $ \downarrow $ $ \downarrow $ $ \ast $ $ \downarrow $
    ($ =0.5,<0.5) $ = $ \uparrow $ $ \uparrow $ $ \uparrow $
    ($ =0.5, =0.5) $ = = = =
    ($ =0.5,>0.5) $ = $ \downarrow $ $ \downarrow $ $ \downarrow $
    ($>0.5,<0.5) $ $ \uparrow $ $ \uparrow $ $ \ast $ $ \uparrow $
    ($>0.5, =0.5) $ $ \uparrow $ = $ \downarrow $ =
    ($>0.5,>0.5) $ $ \uparrow $ $ \downarrow $ $ \downarrow $ $ \downarrow $
     | Show Table
    DownLoad: CSV

    Table 10.  The cost relative difference value between the UML and UOU policies with $ Lx = 1 $

    $ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
    [0.0, 0.3) 0.1 0.936 0.858 0.780 0.702 0.624
    [0.0, 0.3) 0.3 0.809 0.736 0.663 0.590 0.457
    [0.0, 0.3) 0.5 0.698 0.630 0.561 0.438 0.372
    [0.0, 0.3) 0.7 0.560 0.472 0.409 0.347 0.264
    [0.0, 0.3) 0.9 0.443 0.384 0.325 0.347 0.190
    [0.3, 0.6) 0.1 2.028 1.726 1.424 1.122 0.803
    [0.3, 0.6) 0.3 1.364 1.284 0.893 0.651 0.354
    [0.3, 0.6) 0.5 0.948 0.754 0.561 0.294 0.000
    [0.3, 0.6) 0.7 0.593 0.421 0.233 0.000 0.000
    [0.3, 0.6) 0.9 0.361 0.193 0.000 0.000 0.000
    [0.6, 0.9) 0.1 4.432 3.646 2.860 2.055 1.205
    [0.6, 0.9) 0.3 2.118 1.664 1.211 0.736 0.218
    [0.6, 0.9) 0.5 1.197 0.879 0.561 0.154 $ \circ $
    [0.6, 0.9) 0.7 0.630 0.380 0.106 $ \circ $ $ \circ $
    [0.6, 0.9) 0.9 0.304 0.080 $ \circ $ $ \circ $ $ \circ $
     | Show Table
    DownLoad: CSV

    Table 11.  The cost relative difference value between the UML and UOU policies with $ Lx = 2 $

    $ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
    [0.0, 0.3) 0.1 1.448 1.369 1.278 1.199 1.120
    [0.0, 0.3) 0.3 1.268 1.195 1.215 1.048 0.975
    [0.0, 0.3) 0.5 1.128 1.060 0.991 0.923 0.854
    [0.0, 0.3) 0.7 0.991 0.927 0.862 0.798 0.734
    [0.0, 0.3) 0.9 0.870 0.809 0.749 0.689 0.629
    [0.3, 0.6) 0.1 2.843 2.531 2.219 1.906 1.570
    [0.3, 0.6) 0.3 1.959 1.718 1.478 1.194 0.938
    [0.3, 0.6) 0.5 1.378 1.184 0.991 0.798 0.590
    [0.3, 0.6) 0.7 0.989 0.828 0.666 0.487 0.299
    [0.3, 0.6) 0.9 0.710 0.571 0.405 0.267 0.000
    [0.6, 0.9) 0.1 5.495 4.709 3.923 3.118 2.253
    [0.6, 0.9) 0.3 2.850 2.384 1.918 1.441 0.928
    [0.6, 0.9) 0.5 1.627 1.309 0.991 0.666 0.278
    [0.6, 0.9) 0.7 0.989 0.748 0.507 0.234 $ \circ $
    [0.6, 0.9) 0.9 0.599 0.385 0.178 $ \circ $ $ \circ $
     | Show Table
    DownLoad: CSV

    Table 12.  The cost relative difference value between the UML and UOU policies with $ Lx = 4 $

    $ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
    [0.0, 0.3) 0.1 2.482 2.402 2.322 2.243 2.163
    [0.0, 0.3) 0.3 2.224 2.150 2.076 2.002 1.928
    [0.0, 0.3) 0.5 2.000 1.932 1.863 1.794 1.725
    [0.0, 0.3) 0.7 1.799 1.735 1.671 1.606 1.542
    [0.0, 0.3) 0.9 1.628 1.568 1.507 1.447 1.387
    [0.3, 0.6) 0.1 4.233 3.921 3.608 3.296 2.960
    [0.3, 0.6) 0.3 3.029 2.789 2.548 2.308 2.049
    [0.3, 0.6) 0.5 2.251 2.057 1.863 1.669 1.460
    [0.3, 0.6) 0.7 1.711 1.550 1.388 1.226 1.052
    [0.3, 0.6) 0.9 1.329 1.190 1.051 0.912 0.762
    [0.6, 0.9) 0.1 7.621 6.835 6.049 5.244 4.379
    [0.6, 0.9) 0.3 4.110 3.644 3.178 2.701 2.188
    [0.6, 0.9) 0.5 2.501 2.182 1.863 1.536 1.185
    [0.6, 0.9) 0.7 1.641 1.400 1.159 0.913 0.648
    [0.6, 0.9) 0.9 1.123 0.930 0.736 0.538 0.313
     | Show Table
    DownLoad: CSV

    Table 13.  The change trend of the cost relative difference between the UML and UOU policies

    $ \beta_{i} $}{$ \gamma_{i} $ $<0.5 $ $ =0.5 $ $>0.5 $
    $<0.5 $ $ \uparrow(\uparrow, \downarrow) $ $ \uparrow(=, \downarrow) $ $ \ast(\downarrow, \ast) $
    $ =0.5 $ $ \uparrow(\uparrow, \downarrow) $ $ =(=, =) $ $ \downarrow(\downarrow, \uparrow) $
    $>0.5 $ $ \ast(\uparrow, \ast) $ $ \downarrow(=, \uparrow) $ $ \downarrow(\downarrow, \uparrow) $
     | Show Table
    DownLoad: CSV
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