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# Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit

• In today's competitive markets, the supplier let the buyer to pay the purchasing cost after receiving the items, this strategy motivates the retailer to buy more items from the supplier and gains some benefit from the money which they did not pay at the time of receiving of the items. However, the retailer will be unable to pay off the debt obligations to the supplier in the future, so this study extends Yen et al. (2012) to consider the above situation and assumes the retailer can either pay off all accounts at the end of the delay period or delay incurring interest charges on the unpaid and overdue balance due to the difference between interest earned and interest charged. We will discuss the explorations of the function behaviors of the objection function to demonstrate the retailer's optimal replenishment cycle time not only exists but also is unique. Finally, numerical examples are given to illustrate the theorems and gained managerial insights.

Mathematics Subject Classification: 90B05.

 Citation:

• Figure 1.  The interest charged when $W^* < T$

Figure 2.  The total accumulation of interest earned when $0<T\le N$

Figure 3.  The total accumulation of interest earned when $N<T\le M$

Figure 4.  The total accumulation of interest earned when $M<T\le W^{\ast }$

Figure 5.  The total accumulation of interest earned when W*T

Table 1.  The ordering policy by using Theorem 1

 $D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{1}$ $\Delta_{2}$ $\Delta_{3}$ $\Delta_{4}$ $\Delta_{5}$ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$ $A1$ 100 0.010 1 2 1.0 0.955 0.000115 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0141$ 98.6175 $A2$ 100 0.010 1 2 1.0 0.955 0.115000 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $<$0 $<$0 $<$0 $>$0 $<$0 $T_{1}^{*}=0.0141$ 98.6175 $A3$ 100 0.010 5 20 0.9 0.100 0.130000 0.15 0.01700000 0.020000 1.65 0.0165 0.0800 $<$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.9999$ 1521.51 $B1$ 100 0.002 10 15 0.9 0.100 0.130000 0.15 0.01700000 0.030000 1.65 0.0165 0.0451 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.0169$ 502.332 $B2$ 100 0.002 5 15 0.9 0.100 0.130000 0.15 0.01700000 0.030000 1.65 0.0165 0.0901 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{2}^{*}=0.0169$ 1002.30 $B3$ 100 0.002 4 15 0.9 0.100 0.130000 0.15 0.01700000 0.030000 1.65 0.0165 0.1126 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3110$ 1102.80 $C1$ 100 0.010 10 15 0.9 0.100 0.130000 0.15 0.01700000 0.03000 1.65 0.0165 0.0451 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{3}^{*}=0.0190$ 501.880 $C2$ 100 0.010 5 10 0.9 0.100 0.130000 0.15 0.01700000 0.020000 1.65 0.0165 0.0400 $>$0 $>$0 $<$0 $>$0 $<$0 $T_{4}^{*}=0.0300$ 499.948 $C3$ 100 0.010 5 16 0.9 0.100 0.130000 0.15 0.01700000 0.020000 1.65 0.0165 0.0640 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.0871$ 1101.50 $D1$ 100 0.014 1 2 1.0 0.955 0.000115 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.0175$ 98.3643 $D2$ 100 0.090 1 2 1.0 0.955 0.000115 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.0420$ 95.8097

Table 2.  The optimal ordering policy by using Theorem 2

 $D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{6}$ $\Delta_{7}$ $\Delta_{3}$ $\Delta_{4}$ $\Delta_{5}$ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$ $A1$ 35 2 1 3 3.5 3.00 0.12 0.15 0.200 0.300 10 0.2857 0.9090 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1952$ 50.7661 $A2$ 500 3 10 20.8 1.12 1.118 0.20 0.21 0.123 0.300 100 0.2 0.6397 $<$0 $<$0 $<$0 $>$0 $<$0 $T_{1}^{*}=0.1036$ 5716.50 $A3$ 300 3 10 24 1.12 1.118 0.20 0.21 0.300 0.400 100 0.3333 0.9768 $<$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=1.0000$ 4330.40 $B1$ 35 1 5 8 3.00 1.00 0.12 0.15 0.200 0.300 10 0.2857 0.4848 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.2208$ 99.9341 $B2$ 35 2 5 8 3.00 1.00 0.12 0.15 0.200 0.300 10 0.2857 0.4848 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{6}^{*}=0.0169$ 95.9332 $B3$ 37 1.4 10 20.8 1.12 1.11 0.20 0.21 0.120 0.179 6 0.1622 0.3760 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3796$ 399.6874 $C1$ 250 10 5 10 3.00 1.00 0.12 0.15 0.200 0.300 50 0.2000 0.6060 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{3}^{*}=0.2230$ 1206.30 $C2$ 250 30 5 10 3.00 1.00 0.12 0.15 0.200 0.300 50 0.2000 0.6060 $>$0 $>$0 $<$0 $>$0 $<$0 $T_{4}^{*}=0.3510$ 1132.50 $C3$ 37 1 9 18 1.12 1.10 0.20 0.21 0.160 0.179 6 0.1622 0.3593 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3629$ 330.0850 $D1$ 250 100 5 10 3.00 1.00 0.12 0.15 0.200 0.300 50 0.2000 0.6060 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.5570$ 782.8170 $D2$ 250 160 5 10 3.00 1.00 0.12 0.15 0.200 0.300 50 0.2000 0.6060 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.6838$ 881.6120

Table 3.  The optimal ordering policy by using Theorem 3

 $D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{9}$ $\Delta_{10}$ $\Delta_{5}$ $\Delta_{8}$ $\Delta_{6}$ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$ $A1$ 50 0.2 5 7 3 1 0.12 0.15 0.10 0.15 10 0.2000 0.2111 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0894$ 144.3954 $A2$ 50 0.3 5 7 3 1 0.12 0.15 0.10 0.15 10 0.2000 0.2111 $<$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1053$ 97.6279 $B1$ 50 0.4 3 6 3 1 0.12 0.15 0.13 0.15 10 0.2000 0.3007 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1265$ 142.3945 $B2$ 50 0.7 7 7 3 1 0.12 0.15 0.10 0.15 10 0.2000 0.2111 $>$0 $<$0 $<$0 $<$0 $>$0 $T_{7}^{*}=0.1590$ 93.3635 $B3$ 50 0.85 5 7 3 1 0.12 0.15 0.10 0.15 10 0.2000 0.2111 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{7}^{*}=0.1680$ 92.4482 $C1$ 49 1 5 7 1.5 1.2 0.12 0.15 0.19 0.20 10 0.2041 0.2803 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1844$ 87.5672 $C2$ 50 1.2 5 7 3 1 0.12 0.15 0.10 0.19 10 0.2000 0.2682 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1751$ 91.8729 $C3$ 50 1.5 5 7 3 1 0.20 0.15 0.10 0.15 10 0.2000 0.2111 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{4}^{*}=0.2100$ 89.2023 $D1$ 50 0.01 10 20 0.25 0.2 0.12 0.15 0.05 0.08 5 0.1000 0.1605 $>$0 $>$0 $>$0 $<$0 $<$0 $T_{1}^{*}=0.0447$ 503.1528 $D2$ 50 0.1 10 20 0.25 0.2 0.12 0.15 0.05 0.08 5 0.1000 0.1605 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.1901$ 502.4800 $D3$ 50 2 5 7 3 1 0.12 0.15 0.10 0.15 10 0.2000 0.2111 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.2237$ 86.8978

Table 4.  The optimal ordering policy by using Theorem 4

 $D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{9}$ $\Delta_{11}$ $\Delta_{12}$ $\Delta_{8}$ $\Delta_{6}$ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$ $A1$ 30 0.020 3 4 0.12 0.1 0.12 0.15 0.13 0.18 10 0.33 0.2412 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1140$ 30.4546 $A2$ 30 0.030 3 4 0.12 0.1 0.12 0.15 0.13 0.18 10 0.33 0.2412 $<$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1320$ 30.2946 $B1$ 30 0.010 3 4 0.12 0.1 0.12 0.15 0.15 0.18 10 0.33 0.2408 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0816$ 30.1871 $B2$ 30 0.040 3 4 0.12 0.1 0.12 0.15 0.15 0.18 10 0.33 0.2408 $>$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1524$ 29.9407 $B3$ 30 0.125 3 4 0.12 0.1 0.12 0.15 0.15 0.18 10 0.33 0.2408 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{8}^{*}=0.2408$ 29.5262 $C1$ 50 0.010 10 20 0.25 0.2 0.12 0.15 0.12 0.15 80 1.6 0.3010 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{8}^{*}=0.3740$ 504.7746 $C2$ 50 0.500 10 20 0.25 0.2 0.12 0.15 0.12 0.15 80 1.6 0.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4420$ 503.5738 $C3$ 50 1.000 10 20 0.25 0.2 0.12 0.15 0.12 0.15 80 1.6 0.3010 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{8}^{*}=0.5030$ 502.5159 $D1$ 50 0.050 10 20 0.25 0.2 0.12 0.15 0.12 0.15 18 0.36 0.3010 $>$0 $>$0 $>$0 $<$0 $<$0 $T_{5}^{*}=0.3774$ 504.6673 $D2$ 50 0.500 10 20 0.25 0.2 0.12 0.15 0.12 0.15 18 0.36 0.3010 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.4330$ 503.5567 $D3$ 50 0.800 10 20 0.25 0.2 0.12 0.15 0.12 0.15 18 0.36 0.3010 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.4663$ 502.8894

Table 5.  Sensitivity analysis with respect to parameters $A$, $C$ and $W$

 $D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{1}$ $\Delta_{2}$ $\Delta_{3}$ $\Delta_{4}$ $\Delta_{5}$ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$ 100 0.010 1 2 1.00 0.955 0.000115 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0141$ 98.6175 100 0.014 1 2 1.00 0.955 0.000115 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.0175$ 98.3643 100 0.090 1 2 1.00 0.955 0.000115 0.15 0.01650098 0.016501 1.65 0.0165 0.0330 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.0420$ 95.8097 100 0.002 4 15 0.90 0.100 0.130000 0.15 0.01700000 0.030000 1.65 0.0165 0.1126 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3110$ 1102.800 100 0.002 5 15 0.90 0.100 0.130000 0.15 0.01700000 0.030000 1.65 0.0165 0.0901 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{2}^{*}=0.0169$ 1002.300 100 0.002 10 15 0.90 0.100 0.130000 0.15 0.01700000 0.030000 1.65 0.0165 0.0451 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.0169$ 502.3318 50 0.500 10 20 0.25 0.200 0.120000 0.15 0.12000000 0.15000 18 0.36 0.3010 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.4330$ 503.5567 50 0.500 10 20 0.25 0.200 0.120000 0.15 0.12000000 0.15000 50 1.00 0.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4390$ 503.5631 50 0.500 10 20 0.25 0.200 0.120000 0.15 0.12000000 0.15000 80 1.60 0.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4420$ 503.5738

Figures(5)

Tables(5)