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Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering

  • * Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146

    * Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146 
Abstract / Introduction Full Text(HTML) Figure(4) / Table(7) Related Papers Cited by
  • This study develops a single-machine manufacturing system for multi-product with defective items and delayed payment policy. Contradictory to the literature limited production capacity and partial backlogging are considered for more realistic result. The objective of this research is to obtain the optimal cycle length, optimal production quantity, and optimal backorder quantity of each product such that the expected total cost is minimum. The model is solved analytically. Three efficient lemmas are developed to obtain the global optimum solution of the model. An improved algorithm is designed to obtain the numerical solution of the model. An illustrative numerical example and sensitivity analysis are provided to show the practical usage of proposed method.

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

    Citation:

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  • Figure 1.  Graphical representation of inventory system

    Figure 2.  Graphical representation of interest earned and interest charged for $ M<t_a $

    Figure 3.  Graphical representation of interest earned and interest charged for $ t_d \leq M < t_a $

    Figure 4.  Graphical representation of interest earned and interest charged for $ M \geq t_d $

    Table 1.  Author(s) contribution Table

    Author(s) EPQ Imperfect Production Multi-product Single machine Delay-in-payment Backorder Repair
    C$ \acute{a} $rdenas-Barr$ \acute{o} $net al. [2] $ \surd $ $ \surd $
    Chiu et al. [4] $ \surd $ $ \surd $ $ \surd $
    Goyal and C$ \acute{a} $rdenas-Barr$ \acute{o} $n [8] $ \surd $ $ \surd $
    Huang [9] $ \surd $
    Taleizadeh [38] $ \surd $ $ \surd $
    Sana et al.[21] $ \surd $ $ \surd $ $ \surd $
    Li et al.[12] $ \surd $ $ \surd $
    Taleizadh et al. [48] $ \surd $ $ \surd $ $ \surd $ $ \surd $
    Sarkar et al. [25] $ \surd $ $ \surd $ $ \surd $ $ \surd $
    Kang et al. [10] $ \surd $ $ \surd $ $ \surd $
    Ouyang et al. [16] $ \surd $
    This Model $ \surd $ $ \surd $ $ \surd $ $ \surd $ $ \surd $ $ \surd $
     | Show Table
    DownLoad: CSV

    Table 2.  The values of the parameters

    $ P $ $ P_i $ $ P_i^1 $ $ \lambda_i $ $ K_i $ $ C_i $ $ C_R^i $ $ b_i $ $ h_i $
    1 10000 600 2000 750 2 0.5 0.25 0.2
    2 10500 650 1500 700 1.5 0.6 0.5 0.15
    3 11000 750 1000 650 1 0.7 0.75 0.1
     | Show Table
    DownLoad: CSV

    Table 3.  The parametric values

    $P$ $M_j$ $I_c$ $I_e$ $S_i$ $v_i$ $SE_i$ $X_i$
    1 0.04 0.09 0.05 4 1.5 0.003 0.05
    2 0.04 0.09 0.05 3 1 0.004 0.075
    3 0.04 0.09 0.05 2 0.5 0.005 0.1
     | Show Table
    DownLoad: CSV

    Table 4.  Optimal solutions table

    $P$ $T_{Min}$ $T$ $T^*=T_1$ $Q_i$ $B_i$ $Z$
    1 5158 2331.9
    2 0.128 2.579 2.579 3868.5 1061 9046.93
    3 2579 375.7
     | Show Table
    DownLoad: CSV

    Table 5.  Optimal solutions for different values of $ M $

    $ M $ $ T_{Min} $ $ T $ $ T^* $ $ Q_i $ $ B_i $ $ Z^*=Z_1 $
    5159 2332
    0.03 0.128 2.579 2.579 3869 1061 9051.666
    2579 376
    5158 2331.9
    0.04 0.128 2.579 2.579 3868.5 1061 9046.93
    2579 375.7
    5157 2331.5
    0.05 0.128 2.578 2.578 3868 1061 9042.125
    2578.6 375.5
    51555 2330
    0.06 0.128 2.577 2.577 3866 1060 9037.253
    2577 375
     | Show Table
    DownLoad: CSV

    Table 6.  Optimal solutions for different values of $ I_c $

    $ I_c $ $ T_{Min} $ $ T $ $ T^* $ $ Q_i $ $ B_i $ $ Z^*=Z_1 $
    5354.9 2323.3
    0.07 0.128 2.67 2.67 4016.2 1037.7 8990.8
    2677.5 367.2
    5158 2331.9
    0.09 0.128 2.579 2.579 3868.5 1061 9046.93
    2579 375.7
    4987 2336
    0.11 0.128 2.49 2.49 3740 1082 9098.927
    2493 383
    4838 2339
    0.13 0.128 2.41 2.41 3628 1101 9147.354
    2149 392
     | Show Table
    DownLoad: CSV

    Table 7.  Optimal solutions for different values of $ I_e $

    $ I_e $ $ T_{Min} $ $ T $ $ T^* $ $ Q_i $ $ B_i $ $ Z^*=Z_1 $
    5159 2332
    0.03 0.128 2.579 2.579 3869 1061 9047.469
    2579 375
    5158 2331.9
    0.05 0.128 2.579 2.579 3868.5 1061 9046.93
    2579 375.7
    5156 2330
    0.05 0.128 2.57 2.57 3867 1059 9046.39
    2578.6 374
    5154 2329
    0.06 0.128 2.56 2.56 3865 1058 9045.851
    2577 372
     | Show Table
    DownLoad: CSV
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