Optimal Control of ATO System with Individual Components and Product Demands Based on Markov Decision Process
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摘要: 研究多维组件, 单一产品的双需求型面向订单装配(Assemble-to-order, ATO)系统. 产品需求为延期交货型, 当其不被满足时将产生缺货等待成本; 而独立组件需求为销售损失型, 其不被满足时将产生缺货损失成本. 该问题可以抽象成一个动态马尔科夫决策过程(Markov decision process, MDP), 通过对双需求模型求解得到状态依赖型最优策略, 即任一组件的最优生产--库存策略由系统内其他组件的库存水平决定. 研究解决了多需求复杂ATO系统的生产和库存优化控制问题. 提出在一定条件下, 组件的基础库存值可以等价于最终产品需求的库存配给值. 组件的基础库存值与库存配给值随系统内其他组件库存的增加而增加, 而产品需求的库存配给值随系统组件库存和产品缺货量的增加而减少. 最后通过数值实验分析缺货量及组件库存对最优策略结构的影响, 并得到了相应的企业生产实践的管理启示.Abstract: An assemble-to-order (ATO) system is considered which produces n components to be assembled into a single product. Demand for the product is backlogged while demand for components is lost, if it is not immediately satisfied. The problem is to control component production and component inventory allocation. The Markov decision process (MDP) framework is used to formulate this problem. It is shown that for components, the optimal policy is characterized by two state-dependent thresholds: a production base-stock level and an inventory rationing level, and that for the assembled product, the optimal policy is characterized by a state-dependent rationing level. Under a certain condition, the base-stock level of component equals the rationing level for the product. The base-stock level and rationing level of one component are both increasing with the inventories of other components. The rationing level of the product is decreasing with the backlog level and other components' inventories. Finally, though some numerical examples, the influence of the backlog level and inventory level on the optimal policy is studied and some managerial insights for manufacturing practice are also provided.
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图 2 ATO系统产品需求的最优库存配给策略
Fig. 2 Optimal inventory allocation policy for product demand in ATO
图 3 ATO系统独立组件需求最优库存配给策略
Fig. 3 Optimal inventory allocation policy for individual component demand in ATO
表 1 最优策略vs.启发式算法策略
Table 1 Optimal policy versus heuristics
$\lambda_0$ $\lambda_1$ $ \lambda_2$ $\mu_1$ $\mu_2$ $h_1$ $h_2$ $b_0$ $c_1$ $c_2$ $\frac{({v^{H}-v^{\ast}})}{v^{\ast}}(\%)$ 1 1.05 0.43 1.71 2.43 3.06 5.90 4.63 82.17 117.63 276.33 1.574 2 0.60 1.03 2.45 2.98 3.42 5.59 4.96 69.88 227.82 454.58 5.734 3 1.32 2.60 1.07 3.84 3.22 4.82 4.60 47.74 182.11 199.48 1.371 4 1.70 2.46 2.37 4.95 3.97 5.76 6.74 92.02 310.57 405.30 2.429 5 0.26 5.19 1.66 4.55 3.36 3.51 4.47 86.85 483.11 384.78 3.099 6 0.79 1.81 1.98 4.57 3.18 2.02 5.49 66.18 209.95 332.79 0.686 7 0.69 1.65 1.96 2.63 4.95 6.84 6.69 52.93 427.26 386.80 6.115 8 0.26 3.79 2.59 3.46 3.20 2.16 8.58 69.10 480.45 197.10 0.243 9 1.97 1.04 0.42 4.81 4.01 1.69 5.66 54.73 200.01 164.37 5.052 10 1.59 2.79 1.45 3.80 3.97 6.80 6.55 83.89 277.30 173.66 1.627 11 0.46 1.84 1.63 2.67 1.92 8.07 7.27 49.97 352.29 277.87 1.169 12 0.78 1.88 3.19 4.20 3.99 1.49 7.46 93.58 348.68 465.15 0.185 13 1.01 1.07 1.94 2.57 3.80 3.84 3.57 51.64 297.04 144.26 2.873 14 0.30 0.24 1.31 0.72 2.68 9.42 9.15 40.28 142.12 356.00 0.604 15 1.14 2.47 0.69 4.69 3.54 4.65 7.08 108.09 388.54 305.73 4.842 16 1.42 0.77 6.51 2.44 9.82 8.56 3.66 74.49 318.02 219.48 1.798 17 0.04 4.73 2.18 5.82 2.16 1.47 9.61 210.47 458.20 143.43 2.829 18 1.30 1.38 7.43 4.35 9.46 4.92 2.30 178.88 155.86 188.07 5.380 19 2.16 1.91 4.31 4.96 7.81 5.03 5.83 93.49 365.78 443.31 1.373 20 1.00 7.47 8.12 9.57 8.10 5.47 1.47 155.48 220.94 353.92 4.670 21 2.67 4.58 2.88 8.30 8.02 6.85 5.54 135.60 198.65 492.38 1.830 22 2.36 7.22 2.05 9.95 4.97 7.44 2.72 83.86 100.10 296.27 1.096 23 0.34 7.99 7.54 7.64 7.75 2.13 7.06 206.66 421.96 331.71 1.182 24 1.23 2.41 9.73 6.79 9.50 2.21 3.22 204.06 369.70 195.20 6.379 25 1.20 2.77 3.61 5.97 7.51 9.05 4.40 213.99 474.80 100.03 2.233 26 1.56 6.87 6.54 9.02 8.31 8.58 6.56 166.86 227.97 371.99 11.512 27 0.54 8.36 2.54 9.54 5.38 1.26 1.68 119.78 305.63 255.31 0.126 28 1.50 4.37 9.17 6.43 9.95 2.05 1.90 127.34 393.21 390.50 2.209 29 1.71 6.67 2.80 7.16 8.61 9.97 3.28 103.01 277.92 410.01 1.778 30 1.09 9.29 6.05 9.81 7.01 3.85 9.17 160.18 493.08 439.47 1.207 (注: $\lambda_0 \sim {\rm U}(0,10)$, $\lambda_k \sim {\rm U}(0,10)$, $\mu_k \sim {\rm U}(1,10)$, $0.5 \le \rho _k \le 1.2$, $h_k \sim {\rm U}(1,10)$, $b_0 \sim {\rm U}(5,15) \times \sum\nolimits_{k = 1}^2 {h_k }$, $c_k \sim {\rm U}(100,500)$, }\\ \multicolumn{12}{l}{$\rho _k =\left( {\lambda _0 + \lambda _k } \right) / {\mu _k }$, $k = 1,2$.) 
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