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

• 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:

• 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 15 367.4 363.25 361.55 362.46 366.06 372.45 381.92 394.31 409.75 429.18 16 366.82 362.49 360.49 360.94 363.82 369.16 377.05 387.83 400.29 415.71 17 366.83 362.4 360.21 360.34 362.72 367.3 374.09 383.24 394.32 407.17 18 367.32 362.86 360.56 360.46 362.49 366.53 372.52 380.51 390.68 401.73 19 368.23 363.77 361.42 361.18 362.95 366.59 372 379.14 388.1 398.4 20 369.49 365.07 362.7 362.38 363.97 367.32 372.29 378.8 386.87 396.6 21 371.06 366.69 364.35 363.98 365.45 368.58 373.22 379.25 386.65 395.44 22 372.88 368.6 366.29 365.91 367.3 370.27 374.65 380.32 387.2 395.29 23 374.93 370.74 368.49 368.12 369.45 372.31 376.5 381.89 388.37 395.91 24 377.18 373.09 370.91 370.56 371.87 374.65 378.69 383.85 390.02 397.14 25 379.6 375.62 373.52 373.21 374.51 377.23 381.16 386.14 392.06 398.84 26 382.17 378.31 376.29 376.03 377.34 380.02 383.86 388.71 394.42 400.93 27 384.88 381.13 379.2 379 380.33 382.99 386.76 391.5 397.05 403.34 28 387.71 384.07 382.24 382.1 383.46 386.1 389.83 394.48 399.91 406.01 29 390.64 387.12 385.38 385.31 386.7 389.35 393.05 397.63 402.94 408.9 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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