October  2017, 13(4): 1661-1683. doi: 10.3934/jimo.2017012

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

Received  November 2013 Revised  October 02, 2016 Published  December 2016

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.

Citation: Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang, Yung-Fu Huang. Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012
References:
[1]

A. K. BhuniaC. K. JaggiA. 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.  Google Scholar

[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.   Google Scholar

[3]

L. E. Cárdenas-BarrónK. 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.   Google Scholar

[4]

C. T. ChangJ. 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.  Google Scholar

[5]

S. C. ChenC. 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.  Google Scholar

[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.  Google Scholar

[7]

M. S. ChernL. Y. ChanJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

K. J. ChungS. 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.  Google Scholar

[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.  Google Scholar

[13]

J. FengH. 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.  Google Scholar

[14]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 36 (1985), 335-338.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. J. LiaoK. 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.  Google Scholar

[24]

J. J. LiaoK. 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.  Google Scholar

[25]

J. J. LiaoK. 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.  Google Scholar

[26]

J. J. LiaoK. 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.  Google Scholar

[27]

J. MinY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. SarkarS. 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.  Google Scholar

[31]

K. SkouriI. KonstantarasS. 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.  Google Scholar

[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.   Google Scholar

[33]

A. A. TaleizadehS. 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.  Google Scholar

[34]

A. A. TaleizadehS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

J. T. TengJ. 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.  Google Scholar

[40]

J. T. TengH. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

C. T. TungP. 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.  Google Scholar

[45]

W. C. WangJ. 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.  Google Scholar

[46]

H. M. WeeW. 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.  Google Scholar

[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.  Google Scholar

[48]

J. WuK. SkouriJ. 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.  Google Scholar

[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.  Google Scholar

[50]

G. F. YenK. 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.  Google Scholar

[51]

J. ZhangZ. 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.  Google Scholar

show all references

References:
[1]

A. K. BhuniaC. K. JaggiA. 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.  Google Scholar

[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.   Google Scholar

[3]

L. E. Cárdenas-BarrónK. 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.   Google Scholar

[4]

C. T. ChangJ. 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.  Google Scholar

[5]

S. C. ChenC. 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.  Google Scholar

[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.  Google Scholar

[7]

M. S. ChernL. Y. ChanJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

K. J. ChungS. 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.  Google Scholar

[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.  Google Scholar

[13]

J. FengH. 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.  Google Scholar

[14]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 36 (1985), 335-338.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. J. LiaoK. 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.  Google Scholar

[24]

J. J. LiaoK. 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.  Google Scholar

[25]

J. J. LiaoK. 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.  Google Scholar

[26]

J. J. LiaoK. 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.  Google Scholar

[27]

J. MinY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. SarkarS. 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.  Google Scholar

[31]

K. SkouriI. KonstantarasS. 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.  Google Scholar

[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.   Google Scholar

[33]

A. A. TaleizadehS. 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.  Google Scholar

[34]

A. A. TaleizadehS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

J. T. TengJ. 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.  Google Scholar

[40]

J. T. TengH. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

C. T. TungP. 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.  Google Scholar

[45]

W. C. WangJ. 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.  Google Scholar

[46]

H. M. WeeW. 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.  Google Scholar

[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.  Google Scholar

[48]

J. WuK. SkouriJ. 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.  Google Scholar

[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.  Google Scholar

[50]

G. F. YenK. 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.  Google Scholar

[51]

J. ZhangZ. 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.  Google Scholar

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$1000.010121.00.9550.0001150.150.016500980.0165011.650.01650.0330 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0141$98.6175
$A2$1000.010121.00.9550.1150000.150.016500980.0165011.650.01650.0330 $<$0 $<$0 $<$0 $>$0 $<$0 $T_{1}^{*}=0.0141$98.6175
$A3$1000.0105200.90.1000.1300000.150.017000000.0200001.650.01650.0800 $<$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.9999$1521.51
$B1$1000.00210150.90.1000.1300000.150.017000000.0300001.650.01650.0451 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.0169$502.332
$B2$1000.0025150.90.1000.1300000.150.017000000.0300001.650.01650.0901 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{2}^{*}=0.0169$1002.30
$B3$1000.0024150.90.1000.1300000.150.017000000.0300001.650.01650.1126 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3110$1102.80
$C1$1000.01010150.90.1000.1300000.150.017000000.030001.650.01650.0451 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{3}^{*}=0.0190$501.880
$C2$1000.0105100.90.1000.1300000.150.017000000.0200001.650.01650.0400 $>$0 $>$0 $<$0 $>$0 $<$0 $T_{4}^{*}=0.0300$499.948
$C3$1000.0105160.90.1000.1300000.150.017000000.0200001.650.01650.0640 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.0871$1101.50
$D1$1000.014121.00.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.0175$98.3643
$D2$1000.090121.00.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.0420$95.8097
$D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{1} $ $\Delta_{2} $ $\Delta_{3} $ $\Delta_{4} $ $\Delta_{5} $ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$
$A1$1000.010121.00.9550.0001150.150.016500980.0165011.650.01650.0330 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0141$98.6175
$A2$1000.010121.00.9550.1150000.150.016500980.0165011.650.01650.0330 $<$0 $<$0 $<$0 $>$0 $<$0 $T_{1}^{*}=0.0141$98.6175
$A3$1000.0105200.90.1000.1300000.150.017000000.0200001.650.01650.0800 $<$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.9999$1521.51
$B1$1000.00210150.90.1000.1300000.150.017000000.0300001.650.01650.0451 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.0169$502.332
$B2$1000.0025150.90.1000.1300000.150.017000000.0300001.650.01650.0901 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{2}^{*}=0.0169$1002.30
$B3$1000.0024150.90.1000.1300000.150.017000000.0300001.650.01650.1126 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3110$1102.80
$C1$1000.01010150.90.1000.1300000.150.017000000.030001.650.01650.0451 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{3}^{*}=0.0190$501.880
$C2$1000.0105100.90.1000.1300000.150.017000000.0200001.650.01650.0400 $>$0 $>$0 $<$0 $>$0 $<$0 $T_{4}^{*}=0.0300$499.948
$C3$1000.0105160.90.1000.1300000.150.017000000.0200001.650.01650.0640 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.0871$1101.50
$D1$1000.014121.00.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.0175$98.3643
$D2$1000.090121.00.9550.0001150.150.016500980.0165011.650.01650.0330 $>$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$352133.53.000.120.150.2000.300100.28570.9090 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1952$50.7661
$A2$50031020.81.121.1180.200.210.1230.3001000.20.6397 $<$0 $<$0 $<$0 $>$0 $<$0 $T_{1}^{*}=0.1036$5716.50
$A3$300310241.121.1180.200.210.3000.4001000.33330.9768 $<$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=1.0000$4330.40
$B1$351583.001.000.120.150.2000.300100.28570.4848 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.2208$99.9341
$B2$352583.001.000.120.150.2000.300100.28570.4848 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{6}^{*}=0.0169$95.9332
$B3$371.41020.81.121.110.200.210.1200.17960.16220.3760 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3796$399.6874
$C1$250105103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{3}^{*}=0.2230$1206.30
$C2$250305103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $<$0 $>$0 $<$0 $T_{4}^{*}=0.3510$1132.50
$C3$3719181.121.100.200.210.1600.17960.16220.3593 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3629$330.0850
$D1$2501005103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.5570$782.8170
$D2$2501605103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.6838$881.6120
$D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{6} $ $\Delta_{7} $ $\Delta_{3} $ $\Delta_{4} $ $\Delta_{5} $ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$
$A1$352133.53.000.120.150.2000.300100.28570.9090 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1952$50.7661
$A2$50031020.81.121.1180.200.210.1230.3001000.20.6397 $<$0 $<$0 $<$0 $>$0 $<$0 $T_{1}^{*}=0.1036$5716.50
$A3$300310241.121.1180.200.210.3000.4001000.33330.9768 $<$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=1.0000$4330.40
$B1$351583.001.000.120.150.2000.300100.28570.4848 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.2208$99.9341
$B2$352583.001.000.120.150.2000.300100.28570.4848 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{6}^{*}=0.0169$95.9332
$B3$371.41020.81.121.110.200.210.1200.17960.16220.3760 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3796$399.6874
$C1$250105103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{3}^{*}=0.2230$1206.30
$C2$250305103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $<$0 $>$0 $<$0 $T_{4}^{*}=0.3510$1132.50
$C3$3719181.121.100.200.210.1600.17960.16220.3593 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3629$330.0850
$D1$2501005103.001.000.120.150.2000.300500.20000.6060 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.5570$782.8170
$D2$2501605103.001.000.120.150.2000.300500.20000.6060 $>$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$500.257310.120.150.100.15100.20000.2111 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0894$144.3954
$A2$500.357310.120.150.100.15100.20000.2111 $<$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1053$97.6279
$B1$500.436310.120.150.130.15100.20000.3007 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1265$142.3945
$B2$500.777310.120.150.100.15100.20000.2111 $>$0 $<$0 $<$0 $<$0 $>$0 $T_{7}^{*}=0.1590$93.3635
$B3$500.8557310.120.150.100.15100.20000.2111 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{7}^{*}=0.1680$92.4482
$C1$491571.51.20.120.150.190.20100.20410.2803 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1844$87.5672
$C2$501.257310.120.150.100.19100.20000.2682 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1751$91.8729
$C3$501.557310.200.150.100.15100.20000.2111 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{4}^{*}=0.2100$89.2023
$D1$500.0110200.250.20.120.150.050.0850.10000.1605 $>$0 $>$0 $>$0 $<$0 $<$0 $T_{1}^{*}=0.0447$503.1528
$D2$500.110200.250.20.120.150.050.0850.10000.1605 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.1901$502.4800
$D3$50257310.120.150.100.15100.20000.2111 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.2237$86.8978
$D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{9} $ $\Delta_{10} $ $\Delta_{5} $ $\Delta_{8} $ $\Delta_{6} $ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$
$A1$500.257310.120.150.100.15100.20000.2111 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0894$144.3954
$A2$500.357310.120.150.100.15100.20000.2111 $<$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1053$97.6279
$B1$500.436310.120.150.130.15100.20000.3007 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1265$142.3945
$B2$500.777310.120.150.100.15100.20000.2111 $>$0 $<$0 $<$0 $<$0 $>$0 $T_{7}^{*}=0.1590$93.3635
$B3$500.8557310.120.150.100.15100.20000.2111 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{7}^{*}=0.1680$92.4482
$C1$491571.51.20.120.150.190.20100.20410.2803 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1844$87.5672
$C2$501.257310.120.150.100.19100.20000.2682 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1751$91.8729
$C3$501.557310.200.150.100.15100.20000.2111 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{4}^{*}=0.2100$89.2023
$D1$500.0110200.250.20.120.150.050.0850.10000.1605 $>$0 $>$0 $>$0 $<$0 $<$0 $T_{1}^{*}=0.0447$503.1528
$D2$500.110200.250.20.120.150.050.0850.10000.1605 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.1901$502.4800
$D3$50257310.120.150.100.15100.20000.2111 $>$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$300.020340.120.10.120.150.130.18100.330.2412 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1140$30.4546
$A2$300.030340.120.10.120.150.130.18100.330.2412 $<$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1320$30.2946
$B1$300.010340.120.10.120.150.150.18100.330.2408 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0816$30.1871
$B2$300.040340.120.10.120.150.150.18100.330.2408 $>$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1524$29.9407
$B3$300.125340.120.10.120.150.150.18100.330.2408 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{8}^{*}=0.2408$29.5262
$C1$500.01010200.250.20.120.150.120.15801.60.3010 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{8}^{*}=0.3740$504.7746
$C2$500.50010200.250.20.120.150.120.15801.60.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4420$503.5738
$C3$501.00010200.250.20.120.150.120.15801.60.3010 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{8}^{*}=0.5030$502.5159
$D1$500.05010200.250.20.120.150.120.15180.360.3010 $>$0 $>$0 $>$0 $<$0 $<$0 $T_{5}^{*}=0.3774$504.6673
$D2$500.50010200.250.20.120.150.120.15180.360.3010 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.4330$503.5567
$D3$500.80010200.250.20.120.150.120.15180.360.3010 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.4663$502.8894
$D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{9} $ $\Delta_{11} $ $\Delta_{12} $ $\Delta_{8} $ $\Delta_{6} $ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$
$A1$300.020340.120.10.120.150.130.18100.330.2412 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.1140$30.4546
$A2$300.030340.120.10.120.150.130.18100.330.2412 $<$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1320$30.2946
$B1$300.010340.120.10.120.150.150.18100.330.2408 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0816$30.1871
$B2$300.040340.120.10.120.150.150.18100.330.2408 $>$0 $<$0 $<$0 $<$0 $>$0 $T_{6}^{*}=0.1524$29.9407
$B3$300.125340.120.10.120.150.150.18100.330.2408 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{8}^{*}=0.2408$29.5262
$C1$500.01010200.250.20.120.150.120.15801.60.3010 $>$0 $>$0 $<$0 $<$0 $<$0 $T_{8}^{*}=0.3740$504.7746
$C2$500.50010200.250.20.120.150.120.15801.60.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4420$503.5738
$C3$501.00010200.250.20.120.150.120.15801.60.3010 $>$0 $>$0 $<$0 $>$0 $>$0 $T_{8}^{*}=0.5030$502.5159
$D1$500.05010200.250.20.120.150.120.15180.360.3010 $>$0 $>$0 $>$0 $<$0 $<$0 $T_{5}^{*}=0.3774$504.6673
$D2$500.50010200.250.20.120.150.120.15180.360.3010 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.4330$503.5567
$D3$500.80010200.250.20.120.150.120.15180.360.3010 $>$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} $ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$
1000.010121.000.9550.0001150.150.016500980.0165011.650.01650.0330 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0141$98.6175
1000.014121.000.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.0175$98.3643
1000.090121.000.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.0420$95.8097
1000.0024150.900.1000.1300000.150.017000000.0300001.650.01650.1126 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3110$1102.800
1000.0025150.900.1000.1300000.150.017000000.0300001.650.01650.0901 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{2}^{*}=0.0169$1002.300
1000.00210150.900.1000.1300000.150.017000000.0300001.650.01650.0451 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.0169$502.3318
500.50010200.250.2000.1200000.150.120000000.15000180.360.3010 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.4330$503.5567
500.50010200.250.2000.1200000.150.120000000.15000501.000.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4390$503.5631
500.50010200.250.2000.1200000.150.120000000.15000801.600.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4420$503.5738
$D$ $A$ $C$ $p$ $k$ $h$ $I_{e}$ $I_{p}$ $N$ $M$ $W$ $W/D$ $W^*$ $\Delta_{1} $ $\Delta_{2} $ $\Delta_{3} $ $\Delta_{4} $ $\Delta_{5} $ $T^{\ast }$ $Z(T^{\ast })_{\mathrm{}}$
1000.010121.000.9550.0001150.150.016500980.0165011.650.01650.0330 $<$0 $<$0 $<$0 $<$0 $<$0 $T_{1}^{*}=0.0141$98.6175
1000.014121.000.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $<$0 $T_{4}^{*}=0.0175$98.3643
1000.090121.000.9550.0001150.150.016500980.0165011.650.01650.0330 $>$0 $>$0 $>$0 $>$0 $>$0 $T_{5}^{*}=0.0420$95.8097
1000.0024150.900.1000.1300000.150.017000000.0300001.650.01650.1126 $>$0 $<$0 $<$0 $>$0 $>$0 $T_{5}^{*}=0.3110$1102.800
1000.0025150.900.1000.1300000.150.017000000.0300001.650.01650.0901 $>$0 $<$0 $<$0 $>$0 $<$0 $T_{2}^{*}=0.0169$1002.300
1000.00210150.900.1000.1300000.150.017000000.0300001.650.01650.0451 $>$0 $<$0 $<$0 $<$0 $<$0 $T_{2}^{*}=0.0169$502.3318
500.50010200.250.2000.1200000.150.120000000.15000180.360.3010 $>$0 $>$0 $>$0 $<$0 $>$0 $T_{5}^{*}=0.4330$503.5567
500.50010200.250.2000.1200000.150.120000000.15000501.000.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4390$503.5631
500.50010200.250.2000.1200000.150.120000000.15000801.600.3010 $>$0 $>$0 $<$0 $<$0 $>$0 $T_{8}^{*}=0.4420$503.5738
[1]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049

[2]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[3]

Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[5]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[6]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[7]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[8]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[9]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[10]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[11]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[12]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[13]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[14]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[15]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[16]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[17]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[18]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[19]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[20]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (144)
  • HTML views (513)
  • Cited by (0)

[Back to Top]