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

Jui-Jung Liao; Wei-Chun Lee; Kuo-Nan Huang; Yung-Fu Huang

doi:10.3934/jimo.2017012

Article Contents

2017, Volume 13, Issue 4: 1661-1683. Doi: 10.3934/jimo.2017012

This issue Previous Article Optimal pension decision under heterogeneous health statuses and bequest motives Next Article Continuity of approximate solution maps to vector equilibrium problems

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

1.
Department of Business Administration, Chihlee University of Technology, Banqiao District, New Taipei City, 22050, Taiwan
2.
Department of Business Administration, Lunghwa University of Science and Technology, Guishan District, Taoyuan City, 33306, Taiwan
3.
Department of Industrial Engineering & Management, St. John's University, Tamsui District, New Taipei City, 25135, Taiwan
4.
Department of Marketing and Logistics Management, Chaoyang University of Technology, Taichung, 41349, Taiwan

* Corresponding author: liaojj@mail.chihlee.edu.tw
* Corresponding author: liaojj@mail.chihlee.edu.tw

Received: October 31, 2013

Revised: October 2016

Published: November 2017

Abstract Full Text(HTML) Figure(5) / Table(5) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 90B05.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. K. Bhunia, C. K. Jaggi, A. Sharma and R. Sharma, A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging, Applied Mathematics and Computation, 232 (2014), 1125-1137. doi: 10.1016/j.amc.2014.01.115.
[2]	L. E. Cárdenas-Barrón, Optimal manufacturing batch size with rework in a single-stage production system-A simple derivation, Computers and Industrial Engineering, 55 (2008), 758-765.
[3]	L. E. Cárdenas-Barrón, K. J. Chung and G. Trevino-Garza, Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris, International Journal of Production Economics, 155 (2014), 1-7.
[4]	C. T. Chang, J. T. Teng and M. S. Chern, Optimal manufacturer's replenishment policies for deteriorating items in a supply chain with up-stream and down-stream trade credits, International Journal of Production Economics, 127 (2010), 197-202. doi: 10.1016/j.ijpe.2010.05.014.
[5]	S. C. Chen, C. T. Chang and J. T. Teng, A comprehensive note on "Lot-sizing decisions for deteriorating items with two warehouses under an order-size-dependent trade credit", International Transactions in Operational Research, 21 (2014), 855-868. doi: 10.1111/itor.12045.
[6]	S. C. Chen and J. T. Teng, Retailer's optimal ordering policy for deteriorating items with maximum lifetime under supplier's trade credit financing, Applied Mathematical Modelling, 38 (2014), 4049-4061. doi: 10.1016/j.apm.2013.11.056.
[7]	M. S. Chern, L. Y. Chan, J. T. Teng and S. K. Goyal, Nash equilibrium solution in a vendor-buyer supply chain model with permissible delay in payments, Computers and Industrial Engineering, 70 (2014), 116-123. doi: 10.1016/j.cie.2014.01.013.
[8]	K. J. Chung and L. E. Cárdenas-Barrón, The simplified solution procedure for deteriorating items under stock-dependent demand and two-level trade credit in the supply chain management, Applied Mathematical Modelling, 37 (2013), 4653-4660. doi: 10.1016/j.apm.2012.10.018.
[9]	K. J. Chung and J. J. Liao, Lot sizing decisions under trade credit depending on the ordering quantity, Computers and Operation Research, 31 (2004), 909-928. doi: 10.1016/S0305-0548(03)00043-1.
[10]	K. J. Chung and J. J. Liao, The optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach, European Journal of Operational Research, 196 (2009), 563-568. doi: 10.1016/j.ejor.2008.04.018.
[11]	K. J. Chung, S. D. Lin and H. M. Srivastava, The inventory models under conditional trade credit in a supply chain system, Applied Mathematical Modelling, 37 (2013), 10036-10052. doi: 10.1016/j.apm.2013.05.044.
[12]	K. J. Chung and P. S. Ting, The inventory model under supplier's partial trade credit policy in a supply chain system, Journal of Industrial and Management Optimization, 11 (2015), 1175-1183. doi: 10.3934/jimo.2015.11.1175.
[13]	J. Feng, H. Li and Y. Zeng, Inventory games with permissible delay in payments, European Journal of Operational Research, 234 (2014), 694-700. doi: 10.1016/j.ejor.2013.11.008.
[14]	S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 36 (1985), 335-338.
[15]	Y. F. Huang and K. H. Hsu, An EOQ model under retailer partial trade credit policy in supply chain, International Journal of Production Economics, 112 (2008), 655-664. doi: 10.1016/j.ijpe.2007.05.014.
[16]	K. N. Huang and J. J. Liao, A simple method to locate the optimal solution for exponentially deteriorating items under trade credit financing, Computers and Mathematics with Applications, 56 (2008), 965-977. doi: 10.1016/j.camwa.2007.08.049.
[17]	M. Y. Jaber and I. H. Osman, Coordinating a two-level supply chain with delay in payments and profit sharing, Computers and Industrial Engineering, 50 (2006), 385-400. doi: 10.1016/j.cie.2005.08.004.
[18]	V. B. Kreng and S. J. Tan, The optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Expert Systems with Applications, 37 (2010), 5514-5522. doi: 10.1016/j.eswa.2009.12.014.
[19]	Y. Liang and F. Zhou, A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment, Applied Mathematical Modelling, 35 (2011), 2221-2231. doi: 10.1016/j.apm.2010.11.014.
[20]	J. J. Liao, An EOQ model with noninstantaneous receipt exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. doi: 10.1016/j.ijpe.2007.09.006.
[21]	J. J. Liao and K. N. Huang, An inventory model for deteriorating items with two levels of trade credit taking account of time discounting, Acta Applicandae Mathematicae, 110 (2010), 313-326. doi: 10.1007/s10440-008-9411-3.
[22]	J. J. Liao and K. N. Huang, Deterministic inventory model for deteriorating items with trade credit financing and capacity constraints, Computers and Industrial Engineering, 59 (2010), 611-618. doi: 10.1016/j.cie.2010.07.006.
[23]	J. J. Liao, K. N. Huang and K. J. Chung, Lot-sizing decisions for deteriorating items with two warehouses under an order-size-dependent trade credit, International Journal of Production Economics, 137 (2012), 102-115. doi: 10.1016/j.ijpe.2012.01.020.
[24]	J. J. Liao, K. N. Huang and K. J. Chung, Optimal pricing and ordering policy for perishable items with limited storage capacity and partial trade credit, IMA Journal of Management Mathematics, 24 (2013), 45-61. doi: 10.1093/imaman/dps003.
[25]	J. J. Liao, K. N. Huang and K. J. Chung, A deterministic inventory model for deteriorating items with two warehouses and trade credit in a supply chain system, International Journal of Production Economics, 146 (2013), 557-565. doi: 10.1016/j.ijpe.2013.08.001.
[26]	J. J. Liao, K. N. Huang and P. S. Ting, Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy, Applied Mathematics and Computation, 233 (2014), 647-658. doi: 10.1016/j.amc.2014.01.077.
[27]	J. Min, Y. W. Zhou and J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Applied Mathematical Modelling, 34 (2010), 3273-3285. doi: 10.1016/j.apm.2010.02.019.
[28]	A. Musa and B. Sani, Inventory ordering policies of delayed deteriorating items under permissible delay in payments, International Journal of Production Economics, 136 (2012), 75-83. doi: 10.1016/j.ijpe.2011.09.013.
[29]	L. Y. Ouyang and C. T. Chang, Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging, International Journal of Production Economics, 144 (2013), 610-617. doi: 10.1016/j.ijpe.2013.04.027.
[30]	B. Sarkar, S. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 677-702. doi: 10.1007/s10479-014-1745-9.
[31]	K. Skouri, I. Konstantaras, S. Papachristios and J. T. Teng, Supply chain models for deteriorating products with ramp type demand rate under permissible delay in payments, Expert Systems with Applications, 38 (2011), 14861-14869. doi: 10.1016/j.eswa.2011.05.061.
[32]	X. Song and X. Cai, On optimal payment time for a retailer under permitted delay of payment by the wholesaler, International Journal of Production Economics, 103 (2006), 246-251.
[33]	A. A. Taleizadeh, S. S. Kalantari and L. E. Cardenas-Barron, Determining optimal price, replenishment lot sizes and number of shipments for an EPQ model with rework and multiple shipments, Journal of Industrial and Management Optimization, 11 (2015), 1059-1071. doi: 10.3934/jimo.2015.11.1059.
[34]	A. A. Taleizadeh, S. S. Kalantari and L. E. Cardenas-Barron, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009.
[35]	J. T. Teng, On the economic order quantity under condition of permissible delay in payments, Journal of the Operational Research Society, 53 (2002), 915-918. doi: 10.1057/palgrave.jors.2601410.
[36]	J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers, International Journal of Production Economics, 119 (2009), 415-423. doi: 10.1016/j.ijpe.2009.04.004.
[37]	J. T. Teng and C. T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363. doi: 10.1016/j.ejor.2008.02.001.
[38]	J. T. Teng and K. R. Lou, Seller's optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits, Journal of Global Optimization, 53 (2012), 417-430. doi: 10.1007/s10898-011-9720-3.
[39]	J. T. Teng, J. Min and Q. Pan, Economic order quantity model with trade credit financing for non-decreasing demand, Omega, 40 (2012), 328-335. doi: 10.1016/j.omega.2011.08.001.
[40]	J. T. Teng, H. L. Yang and M. S. Chern, An inventory model for increasing demand under two levels of trade credit linked to order quantity, Applied Mathematical Modelling, 37 (2013), 7624-7632. doi: 10.1016/j.apm.2013.02.009.
[41]	A. Thangam, Retailer's inventory system in a two-level trade credit financing with selling price discount and partial order cancellations, International Journal of Industrial Engineering Journal, 11 (2015), 159-170.
[42]	A. Thangam and R. Uthayakumar, Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both credit period and selling price, Computers and Industrial Engineering, 57 (2009), 773-786. doi: 10.1016/j.cie.2009.02.005.
[43]	P. S. Ting, The EPQ model with deteriorating items under two levels of trade credit in a supply chain system, Journal of Industrial and Management Optimization, 11 (2015), 479-492. doi: 10.3934/jimo.2015.11.479.
[44]	C. T. Tung, P. Deng and J. Chuang, Note on inventory models with a permissible delay in payments, Yugoslav Journal of Operations Research, 24 (2014), 111-118. doi: 10.2298/YJOR120622015T.
[45]	W. C. Wang, J. T. Teng and K. R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321. doi: 10.1016/j.ejor.2013.06.027.
[46]	H. M. Wee, W. T. Wang and L. E. Cárdenas-Barrón, An alternative analysis and solution procedure for the EPQ model with rework process at a single-stage manufacturing system with planned backorders, Computers and Industrial Engineering, 64 (2013), 748-755. doi: 10.1016/j.cie.2012.11.005.
[47]	J. Wu and Y. L. Chan, Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers, International Journal of Production Economics, 155 (2014), 292-301. doi: 10.1016/j.ijpe.2014.03.023.
[48]	J. Wu, K. Skouri, J. T. Teng and L. Y. Ouyang, A note on 'optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment', International Journal of Production Economics, 155 (2014), 324-329. doi: 10.1016/j.ijpe.2013.12.017.
[49]	H. L. Yang and C. T. Chang, A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation, Applied Mathematical Modelling, 37 (2013), 2717-2726. doi: 10.1016/j.apm.2012.05.008.
[50]	G. F. Yen, K. J. Chung and Z. C. Chen, The optimal retailer's ordering policies with trade credit financing and limited storage capacity in the supply chain system, International Journal of Systems Science, 43 (2012), 2144-2159. doi: 10.1080/00207721.2011.565133.
[51]	J. Zhang, Z. Bai and W. Tang, Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277. doi: 10.3934/jimo.2014.10.1261.