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Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming

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  • In recent years, numerous studies have been conducted regarding inventory control of deteriorating items. However, due to the complexity of the solution methods, various real assumptions such as discrete variables and capacity constraints were neglected. In this study, we presented a multi-item inventory model for deteriorating items with limited carrier capacity. The proposed research considered the carrier, which transports the order has limited capacity and the quantity of orders cannot be infinite. Dynamic programming is used for problem optimization. The results show that the proposed solution method can solve the mixed-integer problem, and it can provide the global optimum solution.

    Mathematics Subject Classification: Primary: 90B05, 90C26; Secondary: 90C39.

    Citation:

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  • Figure 1.  Inventory level of each item vs. time

    Figure 2.  The flowchart of the proposed solution method

    Figure 3.  The inventory level of each item over time

    Table 1.  A. Review of previous works

    Paper Multi Demand Constraints Variables Solution Shortages
    Item Function type method
    [7] No Constant Logical Continuous Soft Allowed
    constraints computing
    [2] Yes Stock- Capacity Continuous Soft Allowed
    dependent constraint computing
    [8] No Constant No Continuous Mathematical Not
    derivation allowed
    [15] No Time- Logical Continuous Soft Not
    dependent constraints computing allowed
    [9] No Trade No Continuous Soft Not
    credit- computing allowed
    dependent
    [14] No Time-price No Continuous Mathematical Allowed
    backlog derivation
    dependent
    [6] No Time- No Continuous Mathematical Allowed
    dependent derivation
    [11] No Stock and Capacity Continuous Mathematical Not
    price constraint derivation allowed
    dependent
    This Yes Time Capacity Discrete and Dynamic Allowed
    Paper -dependent constraint continuous Programming
     | Show Table
    DownLoad: CSV

    Table 2.  The required and remaining space for each action in the stage 1

    $ k^{'}_{1} $ 0 1 2 3 4 5
    $ k_{1} $ 0 110.7 221.7 330.7 435.4 533.5
    $ k_{1}v_{1} $ 0 166.05 332.55 496.05 653.1 800.25
    $ j_{1} $ 800 633.95 467.45 303.95 146.9 -0.25
    (infeasible)
     | Show Table
    DownLoad: CSV

    Table 3.  Different values of the state in the stage 1

    $ i_{1} $ $ 0\leq i_{1} $ $ 166.05\leq i_{1} $ $ 332.555\leq i_{1} $ $ 496.05\leq i_{1} $ $ 653.1\leq i_{1} $
    $<166.05 $ $<332.55 $ $<496.05 $ $<653.1 $ $ \leq800 $
    $ i^{'}_{1} $ {0} {0, 1} {0, 1, 2} {0, 1, 2, 3} {0, 1, 2, 3, 4}
     | Show Table
    DownLoad: CSV

    Table 4.  The required space for each action in the stage 2

    $ k^{'}_{2} $ 0 1 2 3 4 5
    $ k_{2} $ 0 36.9 77.5 127.1 200.4 338.3
    $ k_{2}v_{1} $ 0 73.8 155 254.2 400.8 676.6
    $ j_{2} $ 800 726.2 645 545.8 399.8 123.4
     | Show Table
    DownLoad: CSV

    Table 5.  Different values of the state in the stage n = 2

    $ i_{2} $ $ 0\leq i_{2} $ $ 73.8\leq i_{2} $ $ 155\leq i_{2} $ $ 166.05\leq i_{2} $ $ 240.3\leq i_{2} $
    $<73.8 $ $<155 $ $<166.05 $ $<240.3 $ $<254.2 $
    $ i^{'}_2 $ {0} {0, 1} {0, 1, 2} {0, 1, 2} {0, 1, 2}
    $ i_{2} $ $ 254.2\leq i_{2} $ $ 321.5\leq i_{2} $ $ 332.55\leq i_{2} $ $ 400.8\leq i_{2} $ $ 406.3\leq i_{2} $
    $<321.5 $ $<332.55 $ $<400.8 $ $<406.3 $ $<420.7 $
    $ i^{'}_2 $ {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4}
    $ i_{2} $ $ 420.7\leq i_{2} $ $ 487.5\leq i_{2} $ $ 496.05\leq i_{2} $ $ 567.3\leq i_{2} $ $ 569.7\leq i_{2} $
    $<487.5 $ $<496.05 $ $<567.3 $ $<569.7 $ $<586.7 $
    $ i^{'}_2 $ {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4}
    $ i_{2} $ $ 586.7\leq i_{2} $ $ 650.9\leq i_{2} $ $ 653.1\leq i_{2} $ $ 676.6\leq i_{2} $ $ 726.9\leq i_{2} $
    $<650.9 $ $<653.1 $ $<676.6 $ $<726.9 $ $<733.3 $
    $ i^{'}_2 $ {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, ..., 5} {0, 1, ..., 5}
    $ i_{2} $} $ 733.3\leq i_{2} $ $ 750.1\leq i_{2} $
    $<750.1 $ $ \leq800 $
    $ i^{'}_2 $ {0, 1, ..., 5} {0, 1, ..., 5}
     | Show Table
    DownLoad: CSV

    Table 6.  The recursive function in the second stage

    $ i_{2} $ $ 0\leq i_{2} $
    $<73.8 $
    $ 73.8\leq i_{2} $
    $<155 $
    $ 155\leq i_{2} $
    $<166.05 $
    $ 166.05\leq i_{2} $
    $<240.3 $
    $ 240.3\leq i_{2} $
    $<254.2 $
    $ f(2, i_{2}) $ 24404 24262 24132 23991 23849
    $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
    :0 :0 :0 :1 :0 :2 :1 :0 :1 :1
    $ i_{2} $ $ 254.2\leq i_{2} $ $ 321.5\leq i_{2} $ $ 332.55\leq i_{2} $ $ 400.8\leq i_{2} $ $ 406.3\leq i_{2} $
    $<321.5 $ $<332.55 $ $<400.8 $ $<406.3 $ $<420.7 $
    $ f(2, i_{2}) $ 23849 23719 23651 23651 23509
    $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
    :1 :1 :1 :2 :2 :0 :2 :0 :2 :1
    $ i_{2} $ $ 420.7\leq i_{2} $ $ 487.5\leq i_{2} $ $ 496.05\leq i_{2} $ $ 567.3\leq i_{2} $ $ 596.7\leq i_{2} $
    $<487.5 $ $<496.05 $ $<567.3 $ $<569.7 $ $<586.7 $
    $ f(2, i_{2}) $ 23509 23379 23379 23379 23258
    $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
    :2 :1 :2 :2 :2 :2 :2 :2 :3 :1
    $ i_{2} $ $ 586.7\leq i_{2} $ $ 650.9\leq i_{2} $ $ 653.1\leq i_{2} $ $ 676.6\leq i_{2} $ $ 726.9\leq i_{2} $
    $<650.9 $ $<653.1 $ $<676.6 $ $<726.9 $ $<733.3 $
    $ f(2, i_{2}) $ 23256 23128 23128 23128 23127
    $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
    :2 :3 :3 :2 :3 :2 :3 :2 :4 :1
    $ i_{2} $ $ 733.3\leq i_{2} $ $ 750.1\leq i_{2} $
    $<750.1 $ $ \leq800 $
    $ f(2, i_{2}) $ 23127 23005
    $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
    :4 :1 :3 :3
     | Show Table
    DownLoad: CSV

    Table 7.  The required and remaining space for each action in stage n = 3

    $ k^{'}_{3} $ 0 1 2 3 4 5
    $ k_{3} $ 0 152.4 315.9 523.2 863.8 1517.5
    $ k_{3}v_{3} $ 0 152.4 315.9 523.2 863.8 1517.5
    $ j_{3} $ 800 647.6 484.1 276.8 -63.8 -717.5
    (infeasible) (infeasible)
     | Show Table
    DownLoad: CSV
  • [1] R. Bellman, Dynamic Programming, Science, 153 (1966), 34-37. 
    [2] D. ChakrabortyD. K. Jana and T. K. Roy, Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment, Soft Computing, 24 (2020), 14471-14490.  doi: 10.1007/s00500-020-04800-3.
    [3] C.-Y. DyeL.-Y. Ouyang and T.-P. Hsieh, Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate, European J. Oper. Res., 178 (2007), 789-807.  doi: 10.1016/j.ejor.2006.02.024.
    [4] S. K. GhoshT. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161.  doi: 10.1080/01966324.2014.980870.
    [5] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European J. Oper. Res., 134 (2001), 1-16.  doi: 10.1016/S0377-2217(00)00248-4.
    [6] M. KarimiS. J. Sadjadi and A. G. Bijaghini, An economic order quantity for deteriorating items with allowable rework of deteriorated products, J. Ind. Manag. Optim., 15 (2019), 1857-1879.  doi: 10.3934/jimo.2018126.
    [7] G. LiX. HeJ. Zhou and H. Wu, Pricing, replenishment and preservation technology investment decisions for non-instantaneous deteriorating items, Omega, 84 (2019), 114-126.  doi: 10.1016/j.omega.2018.05.001.
    [8] J.-J. Liao, K.-N. Huang, K.-J. Chung, S.-D. Lin, S.-T. Chuang and H. M. Srivastava, Optimal ordering policy in an economic order quantity (EOQ) model for non-instantaneous deteriorating items with defective quality and permissible delay in payments, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 41, 26 pp. doi: 10.1007/s13398-019-00777-3.
    [9] P. MahataG. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, International Journal of Systems Science: Operations and Logistics, 7 (2020), 1-17.  doi: 10.1080/23302674.2018.1473526.
    [10] A. K. Malik and A. Sharma, An inventory model for deteriorating items with multi-variate demand and partial backlogging under inflation, International Journal of Mathematical Sciences, 10 (2011), 315-321. 
    [11] M. Rezagholifam, S. J. Sadjadi, M. Heydari and M. Karimi, Optimal pricing and ordering strategy for non-instantaneous deteriorating items with price and stock sensitive demand and capacity constraint, International Journal of Systems Science: Operations and Logistics, (2020). doi: 10.1080/23302674.2020.1833259.
    [12] G. P. Samanta and A. Roy, A production inventory model with deteriorating items and shortages, Yugosl. J. Oper. Res., 14 (2004), 219-230.  doi: 10.2298/YJOR0402219S.
    [13] S. SanaS. K. Goyal and K. S. Chaudhuri, A production-inventory model for a deteriorating item with trended demand and shortages, European J. Oper. Res., 157 (2004), 357-371.  doi: 10.1016/S0377-2217(03)00222-4.
    [14] N. H. ShahU. Chaudhari and L. E. Cárdenas-Barrón, Integrating credit and replenishment policies for deteriorating items under quadratic demand in a three echelon supply chain, International Journal of Systems Science: Operations and Logistics, 7 (2020), 34-45. 
    [15] N. H. Shah and M. K. Naik, Inventory policies for deteriorating items with time-price backlog dependent demand, International Journal of Systems Science: Operations and Logistics, 7 (2020), 76-89.  doi: 10.1080/23302674.2018.1506062.
    [16] J.-T. TengL. E. Cárdenas-BarrónH.-J. ChangJ. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Appl. Math. Model., 40 (2016), 8605-8616.  doi: 10.1016/j.apm.2016.05.022.
    [17] S. TiwariL. E. Cárdenas-BarrónM. Goh and A. A. Shaikh, Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, International Journal of Production Economics, 200 (2018), 16-36.  doi: 10.1016/j.ijpe.2018.03.006.
    [18] S. TiwariL. E. Cárdenas-BarrónA. Khanna and C. K. Jaggi, Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169.  doi: 10.1016/j.ijpe.2016.03.016.
    [19] Q. WangJ. WuN. Zhao and Q. Zhu, Inventory control and supply chain management: A green growth perspective, Resources, Conservation and Recycling, 145 (2019), 78-85.  doi: 10.1016/j.resconrec.2019.02.024.
    [20] J. WuF. B. Al-KhateebJ.-T. Teng and L. E. Cárdenas-Barrón, Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis, International Journal of Production Economics, 171 (2016), 105-115.  doi: 10.1016/j.ijpe.2015.10.020.
    [21] J. WuL.-Y. OuyangL. E. Cárdenas-Barrón and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.
    [22] J. ZhangG. LiuQ. Zhang and Z. Bai, Coordinating a supply chain for deteriorating items with a revenue sharing and cooperative investment contract, Omega, 56 (2015), 37-49.  doi: 10.1016/j.omega.2015.03.004.
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