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Optimal ordering policy and preservation technology for deteriorating items with maximum lifetime under a resilient hybrid payment decision

• *Corresponding author: Jui-Jung Liao
• This study demonstrates an inventory system with items changing value over time under various realistic environments. It is assumed that the time-varying deterioration rate depending on the maximum lifetime of items and the items exceeding the maximum lifetime are regarded as scarp and no longer serviceable. As a result, the retailer will invest in preservation technology to reduce the reckless deterioration. On the other hand, the retailer receives an upstream advance-cash-credit payment plan from the supplier while offering a downstream cash-credit payment plan to customers to stimulate sales. As above description, we incorporate the relevant phenomena into the proposed inventory model, then the primary objective is to determine the replenishment cycle time and the preservation technology which maximizes the retailer's total profit. Consequently, the contributions of this study have three parts as follows: (1) Addressing the economic (total profit) and technology (preservation technology) impacts simultaneously; (2) The propositions and theorems are derived along with a solution procedure; (3) It is proved that the optimal solution not only exists but also is unique under some conditions. Next, an algorithm is developed which simplifies the search for the sustainable optimal ordering strategies. Numerical examples and a sensitivity analysis are elaborated to validate the mathematical formulation. Findings are summarized and managerial implications are also discussed.

Mathematics Subject Classification: Primary: 90B50; Secondary: 93A30.

 Citation:

• Figure 1.  Graphical presentation of the hybrid payment scheme

Table 1.  A brief comparison of related papers

 References Advance Cash Credit Deterioration Expiration PT Chen & Teng (2014) $\checkmark$ $\checkmark$ $\checkmark$ Chang et al. (2019) $\checkmark$ $\checkmark$ $\checkmark$ Chaudhari et al. (2020) $\checkmark$ $\checkmark$ Dye & Hsieh (2012) $\checkmark$ Iqbal & Sarkar (2020) $\checkmark$ $\checkmark$ $\checkmark$ Li et al. (2019a) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ Mishra et al. (2018) $\checkmark$ $\checkmark$ Mishra et al. (2020) $\checkmark$ $\checkmark$ Mahata et al. (2020) $\checkmark$ $\checkmark$ Mohanty et al. (2018) $\checkmark$ $\checkmark$ $\checkmark$ Singh et al. (2016) $\checkmark$ $\checkmark$ Shah et al. (2017) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ Wu et al. (2016a) $\checkmark$ $\checkmark$ $\checkmark$ Wu et al. (2018) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ Yang et al. (2015) $\checkmark$ $\checkmark$ Teng et al. (2016) $\checkmark$ $\checkmark$ $\checkmark$ Mahata & De (2017) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ Tiwari et al. (2018) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ Shi et al. (2020) $\checkmark$ $\checkmark$ $\checkmark$ Sepehri (2021) $\checkmark$ $\checkmark$ $\checkmark$ This paper $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$

Table 2.  For $N<M$

 Theorem 1 $A$ $N$ $O_1$ $O_2$ $\bigtriangleup_1\left( M-N\left| \right.\xi\right)$ $\bigtriangleup_1\left( M\left| \right.\xi\right)$ $T^*$ $TP\left( T^*\left| \right.\xi\right)$ $(1\text{-}1)$ 15 0.12 28.9522 29.9728 29.2741 12.7772 $T^*_1$=0.1744 407.9934 $(1\text{-}2)$ 8.0 0.12 14.9522 15.9728 15.2741 -1.2228 $T^*_2$=0.1275 454.3647 $(1\text{-}3)$ 1.0 0.10 0.9038 1.9244 -0.0198 -15.1796 $T^*_3$=0.0452 539.0858 $(2\text{-}1)$ 0.5 0.12 -0.0478 0.9728 0.2741 -16.2228 $T^*_2$=0.0320 548.2944 $(2\text{-}2)$ 0.1 0.12 -0.8478 0.1728 -0.5259 -17.0228 $T^*_3$=0.0143 565.5832 $(3\text{-}1)$ 0.01 0.12 -1.0278 -0.0072 -0.7059 -17.2028 $T^*_3$=0.0045 575.1418

Table 3.  For $N\geq M$

 Theorem 2 $A$ $O_3$ $\bigtriangleup^*\left( M\left| \right.\xi\right)$ $T^*$ $TP\left( T^*\left| \right.\xi\right)$ (1) 6.00 11.6734 6.1703 $T^*_4$=0.1562 207.9781 (2) 2.00 3.6734 -1.8297 $T^*_3$=0.0901 240.4305 (3) 0.05 -0.2266 -5.7297 $T^*_5$=0.0141 277.6508

Table 4.  Outcome for the example 3

 $\xi_0$ $O_1$ $O_2$ $\bigtriangleup_1\left( M-N\right)$ $\bigtriangleup_1\left( M\right)$ $T^*$ $\xi^*$ $TP\left( T^*,\xi^*\right)$ Iterations 5 18.952184 19.972784 18.9794 2.373136 0.1553 19.373632 433.76756 3 15 18.952184 19.972784 19.2502 2.373136 0.1553 19.373632 433.7675608 3 20 18.952184 19.972784 19.2502 2.37311 0.1553 19.373632 433.7675608 2

Table 5.  Impact of purchase cost ($c$)

 $c$ $O_1$ $O_2$ $\bigtriangleup_1\left( M-N\left| \right.\xi\right)$ $\bigtriangleup_1\left( M\left| \right.\xi\right)$ $T^*$ $\xi^*$ $Q^*$ $TP\left( \xi^*,T^*\right)$ Iterations 1 $>0$ $>0$ 19.36395 5.682702 $T^*_1$=0.1722 14.3749771 51.6611 1670.75725 3 3 $>0$ $>0$ 19.30694 5.682702 $T^*_1$=0.1629 18.0695784 48.8702 1050.90845 2 5 $>0$ $>0$ 19.25022 2.37313 $T^*_1$=0.1553 19.3736332 46.5901 433.767561 3

Table 6.  Impact of ordering cost($A$)

 $A$ $O_1$ $O_2$ $\bigtriangleup_1\left( M-N\left| \right.\xi\right)$ $\bigtriangleup_1\left( M\left| \right.\xi\right)$ $T^*$ $\xi^*$ $Q^*$ $TP\left(\xi^*, T^*\right)$ Iterations 5 $>0$ $>0$ 9.250213 -7.626891 $T^*_2$=0.1098 20.162955 32.9400 470.716489 2 10 $>0$ $>0$ 19.25022 2.37313 $T^*_1$=0.1553 19.373633 46.5901 433.767561 2 15 $>0$ $>0$ 29.25022 12.37316 $T^*_1$=0.1903 18.889421 57.0901 405.314944 3

Table 7.  Impact of holding cost ($h$)

 $h$ $O_1$ $O_2$ $\bigtriangleup_1\left( M-N\left| \right.\xi\right)$ $\bigtriangleup_1\left( M\left| \right.\xi\right)$ $T^*$ $\xi^*$ $Q^*$ $TP\left( \xi^*, T^*\right)$ Iterations 2 $>0$ $>0$ 19.25022 2.37313 $T^*_1$=0.1553 19.373633 46.5901 433.767561 3 4 $>0$ $>0$ 18.71024 -11.12623 $T^*_2$=0.1183 19.821419 35.4900 393.046297 3 6 $>0$ $>0$ 18.17026 -24.62576 $T^*_2$=0.0993 20.089663 29.7900 360.400402 3

Table 8.  Impact of preservation investment efficiency parameter ($r$)

 $r$ $O_1$ $O_2$ $\bigtriangleup_1\left( M-N\left| \right.\xi\right)$ $\bigtriangleup_1\left( M\left| \right.\xi\right)$ $T^*$ $\xi^*$ $Q^*$ $TP\left( \xi^*, T^*\right)$ Iterations 0.25 $>0$ $>0$ 19.25022 2.3732 $T^*_1$=0.1553 35.973433 46.59015 417.16532 3 0.50 $>0$ $>0$ 19.25022 2.3732 $T^*_1$ =0.1553 19.373633 46.59007 433.76767 3 0.75 $>0$ $>0$ 19.25022 2.3732 $T^*_1$=0.1553 13.456513 46.59005 439.68551 3

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Tables(8)