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An integrated inventory model with variable transportation cost, two-stage inspection, and defective items

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  • The paper deals with an integrated inventory model with make-to-order policy from buyer to vendor. A variable transportation cost is used as a power function of the delivery quantity for either tapering or considering proportional rate data to maintain single-setup multi-delivery (SSMD) policy with reduced transportation cost. A two-stage inspection process is introduced by the vendor to ensure the perfect quality of product even though the first inspection process indicates a constant defective rate of imperfect production is present during the long-run production system and all defective items are reworked with some fixed cost. The aim is to minimize the total cost of the integrated inventory model by using classical optimization technique. Two numerical examples, sensitivity analysis, and graphical representations are given to illustrate the model.

    Mathematics Subject Classification: Primary: 90B05, 90B50; Secondary: 90B30.

    Citation:

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  • Figure 1.  Integrated vendor-buyer production-inventory model.

    Figure 2.  Graphical illustration of the unit time cost with respect to joint decision(n, q) of Example 1.

    Figure 3.  The curve of total cost function TRC when $a$ varies as on Example 1.

    Figure 4.  Graphical illustration of the unit time cost with respect to joint decision(n, q) of Example 2.

    Figure 5.  The curve of total cost function TRC when $a$ varies as on Example 2.

    Table 1.  A summary of production-delivery lot sizing models.

    Author(s) MTO/MTS strategy Production quantity Delivery quantity Delivery costInspection policy
    C$\acute{a}$rdenas$-$Barr$\acute{o}$n and Sana [4,5] MTO Variable $-$ $-$ $-$
    Taleizadeh et al. [42] MTO Variable $-$ $-$ $-$
    Benerjee [1] and Goyal [11] MTO Variable Costant $-$ $-$
    Goyal [10] MTO Costant $-$ $-$ $-$
    Sarker and Parija [28,29] MTO Variable Costant $-$ $-$
    Sarkar and Majumder[24] MTO Costant Variable Considered$-$
    Sarkar et al.[32] MTO Costant Variable $-$ $-$
    Golhar and Sarker [9] MTO Variable Costant $-$ $-$
    Lu [17] MTS Variable Costant $-$ $-$
    Hill [14,15] MTS Variable Variable $-$ $-$
    Chan and Kingsman [6] MTS Variable Variable $-$ $-$
    Ben$-$Daya et al. [2] MTS Variable Variable $-$ $-$
    Glock [8] MTS Variable Variable $-$ $-$
    Ertogral et al. [7] MTS Variable Variable Considered$-$
    Wee and Chung[47] MTO Costant Costant $-$Included
    Lee and Fu [16] MTO Variable Variable Considered $-$
    This study MTO Variable Variable Considered Included
    "$-$" indicates non-availability of the research contribution, $"MTO"$ denotes make$-$to$-$order, and $"MTS"$ denotes make$-$to$-$stock.
     | Show Table
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    Table 2.  Cost sensitiveness based on the parameter $a$ from Example 1

    $a$ $TRC$
    $0$ 15722.9
    $0.1$ 15816.6
    $0.2$ 15935.4
    $0.3$ 16085.9
    $0.4$16276.8
    $0.5$ 16518.7
    $0.6$ 16825.3
    $0.7$ 17214.0
    $0.8$ 17706.6
    $0.9$ 18331.1
    $1$ 19122.8
     | Show Table
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    Table 3.  Cost sensitiveness based on the parameter $a$ from Example 2

    $a$ $TRC$
    $0$ 12126.6
    $0.1$ 12193.3
    $0.2$ 12275.0
    $0.3$ 12375.0
    $0.4$ 12497.4
    $0.5$ 12647.3
    $0.6$ 12830.7
    $0.7$ 13055.3
    $0.8$ 13330.3
    $0.9$ 13666.9
    $1$ 14078.9
     | Show Table
    DownLoad: CSV

    Table 4.  Sensitivity analysis of key parameters

    Parameters Changes(in %) $TRC$
    -50% -12.45
    -25% -5.04
    $A_1$ +25% 6.23
    +50% 12.45
    -50% -2.31
    -25% -1.15
    $A_2$ +25% 1.15
    +50% 2.31
    -50% -12.70
    -25% -6.35
    $h_1$ +25% 6.35
    +50% 12.70
    -50% -8.32
    -25% -4.16
    $h_2$ +25% 4.16
    +50% 8.32
     | Show Table
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