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A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands

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  • In this paper, an $ (s, S) $ continuous inventory model with perishable items and retrial demands is proposed. In addition, replenishment lead times that are independent and identically distributed according to phase-type distribution are implemented. The proposed system is modeled as a three-dimensional Markov process using a level-dependent quasi-birth-death (QBD) process. The ergodicity of the modeled Markov system is demonstrated and the best method for efficiently approximating the steady-state distribution at the inventory level is determined. This paper also provides performance measure formulas based on the steady-state distribution of the proposed approximation method. Furthermore, in order to minimize the system cost, the optimum values of $ s $ and $ S $ are determined numerically and sensitivity analysis is performed on the main parameters.

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

    Citation:

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  • Figure 1.  Inventory Model

    Figure 2.  Contour Plot of TCR

    Figure 3.  The effect of $ \lambda $

    Figure 4.  The effect of $ \mu $

    Table 1.  Total Cost Rate(TCR)

    $S \diagdown s$ 1 2 3 4 5 6 7 8 9 10
    15 367.40 363.25 361.55 362.46 366.06 372.45 381.92 394.31 409.75 429.18
    16 366.82 362.49 360.49 360.94 363.82 369.16 377.05 387.83 400.29 415.71
    17 366.83 362.40 360.21 360.34 362.72 367.30 374.09 383.24 394.32 407.17
    18 367.32 362.86 360.56 360.46 362.49 366.53 372.52 380.51 390.68 401.73
    19 368.23 363.77 361.42 361.18 362.95 366.59 372.00 379.14 388.10 398.40
    20 369.49 365.07 362.70 362.38 363.97 367.32 372.29 378.80 386.87 396.60
    21 371.06 366.69 364.35 363.98 365.45 368.58 373.22 379.25 386.65 395.44
    22 372.88 368.60 366.29 365.91 367.30 370.27 374.65 380.32 387.20 395.29
    23 374.93 370.74 368.49 368.12 369.45 372.31 376.50 381.89 388.37 395.91
    24 377.18 373.09 370.91 370.56 371.87 374.65 378.69 383.85 390.02 397.14
    25 379.60 375.62 373.52 373.21 374.51 377.23 381.16 386.14 392.06 398.84
    26 382.17 378.31 376.29 376.03 377.34 380.02 383.86 388.71 394.42 400.93
    27 384.88 381.13 379.20 379.00 380.33 382.99 386.76 391.50 397.05 403.34
    28 387.71 384.07 382.24 382.10 383.46 386.10 389.83 394.48 399.91 406.01
    29 390.64 387.12 385.38 385.31 386.70 389.35 393.05 397.63 402.94 408.90
    30 393.66 390.27 388.62 388.62 390.05 392.71 396.39 400.91 406.14 411.98
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