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doi: 10.3934/jimo.2020138

Two-echelon trade credit with default risk in an EOQ model for deteriorating items under dynamic demand

1. 

Department of Commerce, Srikrishna College, P.O. - Bagula, Dist. - Nadia, PIN - 741502, West Bengal, India

2. 

Department of Mathematics, Sidho-Kanho-Birsha University, Ranchi Road, P.O.- Purulia Sainik School, Purulia 723104, West Bengal, India

* Corresponding author: Gour Chandra Mahata

Received  September 2019 Revised  March 2020 Published  September 2020

In today's competitive markets, offering delay payments has become a commonly adopted method. In this paper, we examine an optimal dynamic decision-making problem for a retailer selling a single deteriorating product, the demand rate of which varies simultaneously with on-hand inventory level and the length of credit period that is offered to the customers. In addition, the risk of default increases with the credit period length. In this study, not only the supplier would offer fixed credit period to the retailer, but retailer also adopt the trade credit policy to his customer in order to promote the market competition. The retailer can accumulate revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. A generalized model is presented to determine the optimal trade credit and replenishment strategies that maximize the retailer's total profit after the default risk occurs over a planning period. For the objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. Some properties of the optimal solutions are shown to find the optimal ordering policies of the considered problem. At the end of this paper, some numerical examples and the results of a sensitivity and elasticity analysis are used to illustrate the features of the proposed model; we then offer our concluding remarks.

Citation: Puspita Mahata, Gour Chandra Mahata. Two-echelon trade credit with default risk in an EOQ model for deteriorating items under dynamic demand. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020138
References:
[1]

H. J. Chang and C. Y. Dye, An inventory model for deteriorating items with partial backlogging and permissible delay in payments, International Journal of Systems Science, 32 (2001), 345-352.  doi: 10.1080/002077201300029700.  Google Scholar

[2]

C. T. ChangJ. T. Teng and S. K. Goyal, Optimal replenishment polices for non-instantaneous deteriorating items with stock-dependent demand, International Journal of Production Economics, 123 (2010), 62-68.   Google Scholar

[3]

P. ChuK. H. Chang and S. P Lan, Economic order quantity of deteriorating items under permissible delay in payment, Computers and Operations Research, 25 (1998), 817-824.  doi: 10.1016/S0305-0548(98)00006-9.  Google Scholar

[4]

K. J. Chung, The complete proof on the optimal ordering policy under cash discount and trade credit, International Journal of Systems Science, 41 (2010), 467-475.  doi: 10.1080/00207720903045866.  Google Scholar

[5]

K. J. Chung and Y. F. Huang, The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics, 84 (2003), 307-318.  doi: 10.1016/S0925-5273(02)00465-6.  Google Scholar

[6]

K. J. ChungS. D. Lin and H. M. Srivastava, The complete and concrete solution procedures for integrated vendor-buyer cooperative inventory models with trade credit financing in supply chain management, Applied Mathematics and Information Sciences, 10 (2016), 155-171.  doi: 10.18576/amis/100115.  Google Scholar

[7]

K. J. Chung and J. J. Liao, The simplified solution algorithm for an integrated supplier-buyer-inventory model with two-part trade credit in a supply chain system, European Journal of Operational Research, 213 (2011), 156-165.  doi: 10.1016/j.ejor.2011.03.018.  Google Scholar

[8]

K. J. ChungJ. J. LiaoS. D. LinS. T. Chuang and H. M. Srivastava, Manufacturer's optimal pricing and lot-sizing policies under trade-credit financing, Mathematical Methods in the Applied Sciences, 43 (2020), 1-18.  doi: 10.1002/mma.6104.  Google Scholar

[9]

C. Y. Dye and C. T. Yang, Sustainable trade credit and replenishment decisions with credit-linked demand under carbon emission constraints, European Journal of Operational Research, 244 (2015), 187-200.  doi: 10.1016/j.ejor.2015.01.026.  Google Scholar

[10]

B. C. Giri and T. Maiti, Supply chain model with price-and trade credit-sensitive demand under two-level permissible delay in payments, International Journal of Systems Science, 44 (2013), 937-948.  doi: 10.1080/00207721.2011.649367.  Google Scholar

[11]

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

[12]

C. H. Ho, The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit, Computers and Industrial Engineering, 60 (2011), 117-126.  doi: 10.1016/j.cie.2010.10.009.  Google Scholar

[13]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the Operational research society, 54 (2003) 1011–1015. doi: 10.1057/palgrave.jors.2601588.  Google Scholar

[14]

Y. F. Huang, Optimal retailer's replenishment decisions in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 176 (2007), 1577-1591.  doi: 10.1016/j.ejor.2005.10.035.  Google Scholar

[15]

A. M. M. JamalB. R. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the Operational Research Society, 48 (1997), 826-833.   Google Scholar

[16]

C. K JaggiS. K. Goyal and S. K. Goel, Retailer's optimal replenishment decisions with credit-linked demand under permissible delay in payments, European Journal of Operational Research, 190 (2008), 130-135.  doi: 10.1016/j.ejor.2007.05.042.  Google Scholar

[17]

C. K. JaggiP. K. KapurS. K. Goyal and S. K. Goel, Optimal replenishment and credit policy in EOQ model under two-levels of trade credit policy when demand is influenced by credit period, International Journal of System Assurance Engineering and Management, 3 (2012), 352-359.  doi: 10.1007/s13198-012-0106-9.  Google Scholar

[18]

R. I. Levin, C. P. McLaughlin, R. P. Lamone and J. F. Kattas, Productions/Operations Management: Contemporary Policy for Managing Operating Systems, McGraw Hill, New York, 1972. Google Scholar

[19]

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

[20]

K. R. Lou and W. C. Wang, Optimal trade credit and order quantity when trade credit impacts on both demand rate and default risk, Journal of the Operational Research Society, 11 (2012), 1551-1556.  doi: 10.1057/jors.2012.134.  Google Scholar

[21]

G. C. Mahata and A. Goswami, An EOQ model for deteriorating items under trade credit financing in the fuzzy sense, Production Planning and Control, 18 (2007), 681-692.  doi: 10.1080/09537280701619117.  Google Scholar

[22]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert System with Applications, 39 (2012), 3537-3550.  doi: 10.1016/j.eswa.2011.09.044.  Google Scholar

[23]

P. Mahata and G. C. Mahata, Economic production quantity model with trade credit financing and price-discount offer for non-decreasing time varying demand pattern, International Journal of Procurement Management, 7 (2014), 563-581.  doi: 10.1504/IJPM.2014.064619.  Google Scholar

[24]

G. C. Mahata, Partial trade credit policy of retailer in economic order quantity models for deteriorating items with expiration dates and price sensitive demand, Journal of Mathematical Modelling and Algorithms in Operations Research, 14 (2015), 363-392.  doi: 10.1007/s10852-014-9269-5.  Google Scholar

[25]

G. C. Mahata, Retailer's optimal credit period and cycle time in a supply chain for deteriorating items with up-stream and down-stream trade credits, Journal of Industrial Engineering International, 11 (2015), 353-366.  doi: 10.1007/s40092-015-0106-x.  Google Scholar

[26]

P. MahataG. C. Mahata and S. K. De, Optimal replenishment and credit policy in supply chain inventory model under two levels of trade credit with time- and credit-sensitive demand involving default risk, Journal of Industrial Engineering International, 14 (2018), 31-42.  doi: 10.1007/s40092-017-0208-8.  Google Scholar

[27]

P. MahataG. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, International Journal of Systems Science: Operations & Logistics, 7 (2020), 1-17.  doi: 10.1080/23302674.2018.1473526.  Google Scholar

[28]

J. MinY. W. ZhouG. Q. Liu and S. D. Wang, An EPQ model for deteriorating items with inventory-level-dependent demand and permissible delay in payments, International Journal of Systems Science, 43 (2012), 1039-1053.  doi: 10.1080/00207721.2012.659685.  Google Scholar

[29]

H. QiuL. LiangY. G. Yu and S. F. Du, EOQ model under three levels of order-size-dependent delay in payments, Journal of Systems & Management, 16 (2007), 669-672.   Google Scholar

[30]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55, (2012), 367–377. doi: 10.1016/j.mcm.2011.08.009.  Google Scholar

[31]

N. H. Shah and L. E. Cardenas-Barron, Retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount, Applied Mathematics and Computation, 259 (2015), 569-578.  doi: 10.1016/j.amc.2015.03.010.  Google Scholar

[32]

X. Shi and S. Zhang, An incentive-compatible solution for trade credit term incorporating default risk, European Journal of Operational Research, 206 (2010), 178-196.  doi: 10.1016/j.ejor.2010.02.003.  Google Scholar

[33]

N. H. Soni, Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment, International Journal of Production Economics, 146 (2013), 259-268.  doi: 10.1016/j.ijpe.2013.07.006.  Google Scholar

[34]

C. H. SuL. Y. OuyangC. H. Ho and C. T. Chang, Retailer's inventory policy and supplier's delivery policy under two-level trade credit strategy, Asia-Pacific Journal of Operational Research, 24 (2007), 613-630.  doi: 10.1142/S0217595907001413.  Google Scholar

[35]

J. T. Teng, On the economic order quantity under conditions 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. TengI. P KrommydaK. Skouri and K. R. Lou, A comprehensive extension of optimal ordering policy for stock-dependent demand under progressive payment scheme, European Journal of Operational Research, 215 (2011), 97-104.  doi: 10.1016/j.ejor.2011.05.056.  Google Scholar

[38]

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

[39]

J. T. Teng and S. K. Goyal, Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits, Journal of the Operational Research Society, 58 (2007), 1252-1255.  doi: 10.1057/palgrave.jors.2602404.  Google Scholar

[40]

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

[41]

J. T. TengJ. Min and Q. H. 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

[42]

J. T. TengK. R. Lou and L. Wang, Optimal trade credit and lot size policies in economic production quantity models with learning curve production costs, International Journal of Production Economics, 155 (2014), 318-323.  doi: 10.1016/j.ijpe.2013.10.012.  Google Scholar

[43]

A. Thangam and R. Uthayakumar, Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period, Computers and Industrial Engineering, 57 (2009), 773-786.  doi: 10.1016/j.cie.2009.02.005.  Google Scholar

[44]

J. WuL. Y. OuyangL. E. Cardenas-Barron and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.  Google Scholar

[45]

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

[46]

H. L. YangJ. T. Teng and M. S. Chern, An inventory model under inflation for deteriorating items with stock-dependent consumption rate and partial backlogging shortages, International Journal of Production Economics, 123 (2010), 8-19.   Google Scholar

[47]

S. A. Yang and J. R. Birge, How inventory is (should be) financed: Trade credit in supply chains with demand uncertainty and costs of financial distress, SSRN, (2013), 1734682, 37 pp. doi: 10.2139/ssrn.1734682.  Google Scholar

[48]

Q. ZhangM. DongJ. Luo and A. Segerstedt, Supply chain coordination with trade credit and quantity discount incorporating default risk, International Journal of Production Economics, 153 (2014), 352-360.  doi: 10.1016/j.ijpe.2014.03.019.  Google Scholar

show all references

References:
[1]

H. J. Chang and C. Y. Dye, An inventory model for deteriorating items with partial backlogging and permissible delay in payments, International Journal of Systems Science, 32 (2001), 345-352.  doi: 10.1080/002077201300029700.  Google Scholar

[2]

C. T. ChangJ. T. Teng and S. K. Goyal, Optimal replenishment polices for non-instantaneous deteriorating items with stock-dependent demand, International Journal of Production Economics, 123 (2010), 62-68.   Google Scholar

[3]

P. ChuK. H. Chang and S. P Lan, Economic order quantity of deteriorating items under permissible delay in payment, Computers and Operations Research, 25 (1998), 817-824.  doi: 10.1016/S0305-0548(98)00006-9.  Google Scholar

[4]

K. J. Chung, The complete proof on the optimal ordering policy under cash discount and trade credit, International Journal of Systems Science, 41 (2010), 467-475.  doi: 10.1080/00207720903045866.  Google Scholar

[5]

K. J. Chung and Y. F. Huang, The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics, 84 (2003), 307-318.  doi: 10.1016/S0925-5273(02)00465-6.  Google Scholar

[6]

K. J. ChungS. D. Lin and H. M. Srivastava, The complete and concrete solution procedures for integrated vendor-buyer cooperative inventory models with trade credit financing in supply chain management, Applied Mathematics and Information Sciences, 10 (2016), 155-171.  doi: 10.18576/amis/100115.  Google Scholar

[7]

K. J. Chung and J. J. Liao, The simplified solution algorithm for an integrated supplier-buyer-inventory model with two-part trade credit in a supply chain system, European Journal of Operational Research, 213 (2011), 156-165.  doi: 10.1016/j.ejor.2011.03.018.  Google Scholar

[8]

K. J. ChungJ. J. LiaoS. D. LinS. T. Chuang and H. M. Srivastava, Manufacturer's optimal pricing and lot-sizing policies under trade-credit financing, Mathematical Methods in the Applied Sciences, 43 (2020), 1-18.  doi: 10.1002/mma.6104.  Google Scholar

[9]

C. Y. Dye and C. T. Yang, Sustainable trade credit and replenishment decisions with credit-linked demand under carbon emission constraints, European Journal of Operational Research, 244 (2015), 187-200.  doi: 10.1016/j.ejor.2015.01.026.  Google Scholar

[10]

B. C. Giri and T. Maiti, Supply chain model with price-and trade credit-sensitive demand under two-level permissible delay in payments, International Journal of Systems Science, 44 (2013), 937-948.  doi: 10.1080/00207721.2011.649367.  Google Scholar

[11]

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

[12]

C. H. Ho, The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit, Computers and Industrial Engineering, 60 (2011), 117-126.  doi: 10.1016/j.cie.2010.10.009.  Google Scholar

[13]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the Operational research society, 54 (2003) 1011–1015. doi: 10.1057/palgrave.jors.2601588.  Google Scholar

[14]

Y. F. Huang, Optimal retailer's replenishment decisions in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 176 (2007), 1577-1591.  doi: 10.1016/j.ejor.2005.10.035.  Google Scholar

[15]

A. M. M. JamalB. R. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the Operational Research Society, 48 (1997), 826-833.   Google Scholar

[16]

C. K JaggiS. K. Goyal and S. K. Goel, Retailer's optimal replenishment decisions with credit-linked demand under permissible delay in payments, European Journal of Operational Research, 190 (2008), 130-135.  doi: 10.1016/j.ejor.2007.05.042.  Google Scholar

[17]

C. K. JaggiP. K. KapurS. K. Goyal and S. K. Goel, Optimal replenishment and credit policy in EOQ model under two-levels of trade credit policy when demand is influenced by credit period, International Journal of System Assurance Engineering and Management, 3 (2012), 352-359.  doi: 10.1007/s13198-012-0106-9.  Google Scholar

[18]

R. I. Levin, C. P. McLaughlin, R. P. Lamone and J. F. Kattas, Productions/Operations Management: Contemporary Policy for Managing Operating Systems, McGraw Hill, New York, 1972. Google Scholar

[19]

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

[20]

K. R. Lou and W. C. Wang, Optimal trade credit and order quantity when trade credit impacts on both demand rate and default risk, Journal of the Operational Research Society, 11 (2012), 1551-1556.  doi: 10.1057/jors.2012.134.  Google Scholar

[21]

G. C. Mahata and A. Goswami, An EOQ model for deteriorating items under trade credit financing in the fuzzy sense, Production Planning and Control, 18 (2007), 681-692.  doi: 10.1080/09537280701619117.  Google Scholar

[22]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert System with Applications, 39 (2012), 3537-3550.  doi: 10.1016/j.eswa.2011.09.044.  Google Scholar

[23]

P. Mahata and G. C. Mahata, Economic production quantity model with trade credit financing and price-discount offer for non-decreasing time varying demand pattern, International Journal of Procurement Management, 7 (2014), 563-581.  doi: 10.1504/IJPM.2014.064619.  Google Scholar

[24]

G. C. Mahata, Partial trade credit policy of retailer in economic order quantity models for deteriorating items with expiration dates and price sensitive demand, Journal of Mathematical Modelling and Algorithms in Operations Research, 14 (2015), 363-392.  doi: 10.1007/s10852-014-9269-5.  Google Scholar

[25]

G. C. Mahata, Retailer's optimal credit period and cycle time in a supply chain for deteriorating items with up-stream and down-stream trade credits, Journal of Industrial Engineering International, 11 (2015), 353-366.  doi: 10.1007/s40092-015-0106-x.  Google Scholar

[26]

P. MahataG. C. Mahata and S. K. De, Optimal replenishment and credit policy in supply chain inventory model under two levels of trade credit with time- and credit-sensitive demand involving default risk, Journal of Industrial Engineering International, 14 (2018), 31-42.  doi: 10.1007/s40092-017-0208-8.  Google Scholar

[27]

P. MahataG. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, International Journal of Systems Science: Operations & Logistics, 7 (2020), 1-17.  doi: 10.1080/23302674.2018.1473526.  Google Scholar

[28]

J. MinY. W. ZhouG. Q. Liu and S. D. Wang, An EPQ model for deteriorating items with inventory-level-dependent demand and permissible delay in payments, International Journal of Systems Science, 43 (2012), 1039-1053.  doi: 10.1080/00207721.2012.659685.  Google Scholar

[29]

H. QiuL. LiangY. G. Yu and S. F. Du, EOQ model under three levels of order-size-dependent delay in payments, Journal of Systems & Management, 16 (2007), 669-672.   Google Scholar

[30]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55, (2012), 367–377. doi: 10.1016/j.mcm.2011.08.009.  Google Scholar

[31]

N. H. Shah and L. E. Cardenas-Barron, Retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount, Applied Mathematics and Computation, 259 (2015), 569-578.  doi: 10.1016/j.amc.2015.03.010.  Google Scholar

[32]

X. Shi and S. Zhang, An incentive-compatible solution for trade credit term incorporating default risk, European Journal of Operational Research, 206 (2010), 178-196.  doi: 10.1016/j.ejor.2010.02.003.  Google Scholar

[33]

N. H. Soni, Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment, International Journal of Production Economics, 146 (2013), 259-268.  doi: 10.1016/j.ijpe.2013.07.006.  Google Scholar

[34]

C. H. SuL. Y. OuyangC. H. Ho and C. T. Chang, Retailer's inventory policy and supplier's delivery policy under two-level trade credit strategy, Asia-Pacific Journal of Operational Research, 24 (2007), 613-630.  doi: 10.1142/S0217595907001413.  Google Scholar

[35]

J. T. Teng, On the economic order quantity under conditions 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. TengI. P KrommydaK. Skouri and K. R. Lou, A comprehensive extension of optimal ordering policy for stock-dependent demand under progressive payment scheme, European Journal of Operational Research, 215 (2011), 97-104.  doi: 10.1016/j.ejor.2011.05.056.  Google Scholar

[38]

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

[39]

J. T. Teng and S. K. Goyal, Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits, Journal of the Operational Research Society, 58 (2007), 1252-1255.  doi: 10.1057/palgrave.jors.2602404.  Google Scholar

[40]

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

[41]

J. T. TengJ. Min and Q. H. 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

[42]

J. T. TengK. R. Lou and L. Wang, Optimal trade credit and lot size policies in economic production quantity models with learning curve production costs, International Journal of Production Economics, 155 (2014), 318-323.  doi: 10.1016/j.ijpe.2013.10.012.  Google Scholar

[43]

A. Thangam and R. Uthayakumar, Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period, Computers and Industrial Engineering, 57 (2009), 773-786.  doi: 10.1016/j.cie.2009.02.005.  Google Scholar

[44]

J. WuL. Y. OuyangL. E. Cardenas-Barron and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.  Google Scholar

[45]

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

[46]

H. L. YangJ. T. Teng and M. S. Chern, An inventory model under inflation for deteriorating items with stock-dependent consumption rate and partial backlogging shortages, International Journal of Production Economics, 123 (2010), 8-19.   Google Scholar

[47]

S. A. Yang and J. R. Birge, How inventory is (should be) financed: Trade credit in supply chains with demand uncertainty and costs of financial distress, SSRN, (2013), 1734682, 37 pp. doi: 10.2139/ssrn.1734682.  Google Scholar

[48]

Q. ZhangM. DongJ. Luo and A. Segerstedt, Supply chain coordination with trade credit and quantity discount incorporating default risk, International Journal of Production Economics, 153 (2014), 352-360.  doi: 10.1016/j.ijpe.2014.03.019.  Google Scholar

Figure 1.  Optimal profit graph for Example 1
Figure 2.  Optimal profit graph for Example 2
Figure 3.  Comparative Graphical analysis about the Effect over Demand due to trade credit: Real situation vs. proposed demand in this model
Table 2.  Sensitivity analysis on parameters
Parameter $ N^* $ $ T^* $ $ \Pi^*(N, T) $
10 1.6346 0.0125 $27979.15
$ A $ 14 1.6337 0.0148 $27685.27
18 1.6328 0.0168 $27431.85
22 1.6314 0.0181 $27105.88
100 1.6346 0.0125 $27979.15
$ D_0 $ 150 1.6336 0.0124 $28060.09
200 1.6325 0.0123 $28141.27
250 1.6317 0.0125 $28222.71
30 1.6345 0.0125 $27979.15
$ p $ 35 1.8886 0.0066 $102463.6
40 2.1081 0.0038 314389.6
45 2.3019 0.0023 844938.8
10 1.6345 0.0125 $27979.15
$ v $ 11 1.4765 0.0182 $13830.44
12 1.3304 0.0257 $7283.99
13 1.1924 0.0353 $4063.65
5 1.6345 0.0125 $27979.15
$ h $ 7 1.6337 0.0110 $27770.88
9 1.6330 0.0100 $27584.54
11 1.6323 0.0093 $27414.47
0.5 1.6345 0.0125 $27979.15
$ M $ 0.6 1.6540 0.0119 $30602.06
0.7 1.6737 0.0113 $33493.21
0.8 1.6934 0.0107 $36682.29
0.5 1.6345 0.0125 27979.15
$ k $ 0.6 1.3669 0.0243 $8658.22
0.7 1.1563 0.0406 $3698.78
0.8 0.9745 0.0619 $2007.86
5 1.6345 0.0125 $27979.15
$ b $ 7 1.6358 0.0105 $39454.92
9 1.6367 0.0092 $50971.26
11 1.6373 0.0083 $62515.11
0.2 1.6345 0.0125 $27979.15
$ a $ 0.3 1.6344 0.0126 $27997.37
0.4 1.6343 0.0127 $28015.81
0.5 1.6342 0.0129 $28034.48
5 1.6345 0.0125 $27979.15
$ c $ 6 1.6755 0.0048 $151018.6
7 1.7026 0.0019 $831067.7
8 1.7221 0.0007 $4642183.0
0.1 1.6345 0.0125 $27979.15
$ \theta $ 0.2 1.6340 0.0116 $27855.04
0.3 1.6335 0.0109 $27739.36
0.4 1.6331 0.0103 $27630.62
Parameter $ N^* $ $ T^* $ $ \Pi^*(N, T) $
10 1.6346 0.0125 $27979.15
$ A $ 14 1.6337 0.0148 $27685.27
18 1.6328 0.0168 $27431.85
22 1.6314 0.0181 $27105.88
100 1.6346 0.0125 $27979.15
$ D_0 $ 150 1.6336 0.0124 $28060.09
200 1.6325 0.0123 $28141.27
250 1.6317 0.0125 $28222.71
30 1.6345 0.0125 $27979.15
$ p $ 35 1.8886 0.0066 $102463.6
40 2.1081 0.0038 314389.6
45 2.3019 0.0023 844938.8
10 1.6345 0.0125 $27979.15
$ v $ 11 1.4765 0.0182 $13830.44
12 1.3304 0.0257 $7283.99
13 1.1924 0.0353 $4063.65
5 1.6345 0.0125 $27979.15
$ h $ 7 1.6337 0.0110 $27770.88
9 1.6330 0.0100 $27584.54
11 1.6323 0.0093 $27414.47
0.5 1.6345 0.0125 $27979.15
$ M $ 0.6 1.6540 0.0119 $30602.06
0.7 1.6737 0.0113 $33493.21
0.8 1.6934 0.0107 $36682.29
0.5 1.6345 0.0125 27979.15
$ k $ 0.6 1.3669 0.0243 $8658.22
0.7 1.1563 0.0406 $3698.78
0.8 0.9745 0.0619 $2007.86
5 1.6345 0.0125 $27979.15
$ b $ 7 1.6358 0.0105 $39454.92
9 1.6367 0.0092 $50971.26
11 1.6373 0.0083 $62515.11
0.2 1.6345 0.0125 $27979.15
$ a $ 0.3 1.6344 0.0126 $27997.37
0.4 1.6343 0.0127 $28015.81
0.5 1.6342 0.0129 $28034.48
5 1.6345 0.0125 $27979.15
$ c $ 6 1.6755 0.0048 $151018.6
7 1.7026 0.0019 $831067.7
8 1.7221 0.0007 $4642183.0
0.1 1.6345 0.0125 $27979.15
$ \theta $ 0.2 1.6340 0.0116 $27855.04
0.3 1.6335 0.0109 $27739.36
0.4 1.6331 0.0103 $27630.62
Table 1.  Demand factors over real case study
Credit periods Customer demand factor
0.1 1.010050167
0.2 1.04452134
0.3 1.019224534
0.4 1.032920774
0.5 1.041841096
0.6 1.069682147
0.7 1.072508181
0.8 1.083287068
0.9 1.129936284
1 1.112734718
1.1 1.16404607
1.2 1.123120852
1.3 1.173786783
1.4 1.085449799
1.5 1.166159193
1.6 1.173510871
1.7 1.177650051
1.8 1.203693163
1.9 1.209249598
Credit periods Customer demand factor
0.1 1.010050167
0.2 1.04452134
0.3 1.019224534
0.4 1.032920774
0.5 1.041841096
0.6 1.069682147
0.7 1.072508181
0.8 1.083287068
0.9 1.129936284
1 1.112734718
1.1 1.16404607
1.2 1.123120852
1.3 1.173786783
1.4 1.085449799
1.5 1.166159193
1.6 1.173510871
1.7 1.177650051
1.8 1.203693163
1.9 1.209249598
Table 3.  Elasticity sensitivities of the parameters
Parameter $ \xi_N $ $ \xi_T $ $ \xi_{\Pi} $
10
$ A $ 14 -0.18% 46.00% -2.63%
18 -0.18% 43.00% -2.45%
22 -0.16% 37.33% -2.30%
100
$ D_0 $ 150 -0.09% -1.60% 0.58%
200 -0.08% -0.80% 0.58%
250 -0.07% 0 0.58%
30
$ p $ 35 93.28% -283.20% 1597.28%
40 86.93% -208.80% 3070.97%
45 81.66% -163.2% 5839.00%
10
$ v $ 11 -96.67% 456.00% -505.69%
12 -93.03% 528.00% -369.83%
13 -90.16% 608.00% -284.92%
5
$ h $ 7 -0.12% -30.00% -1.86%
9 -0.11% -25.00% -1.76%
11 -0.11% -21.33% -1.68%
0.5
$ M $ 0.6 5.97% -24.00% 46.87%
0.7 6.00% -24.00% 49.27%
0.8 6.00% -24.00% 51.84%
0.5
$ k $ 0.6 -81.86% 472.00% -345.27%
0.7 -73.14% 562.00% -216.95%
0.8 -67.29% 658.67% -154.70%
5
$ b $ 7 0.20% -40.00% 102.54%
9 0.17% -33.00% 102.72%
11 0.14% -28.00% 102.86%
0.2
$ a $ 0.3 -0.01% 1.60% 0.13%
0.4 -0.01% 1.60% 0.13%
0.5 0.01% 2.13% 0.13%
5
$ c $ 6 12.54% -308.00% 2198.77%
7 10.42% -212.00% 7175.78%
8 8.93% -197.33% 27485.97%
0.1
$ \theta $ 0.2 -0.03% -6.40% -0.43%
0.3 -0.03% -7.20% -0.44%
0.4 -0.03% -5.87% -0.41%
Parameter $ \xi_N $ $ \xi_T $ $ \xi_{\Pi} $
10
$ A $ 14 -0.18% 46.00% -2.63%
18 -0.18% 43.00% -2.45%
22 -0.16% 37.33% -2.30%
100
$ D_0 $ 150 -0.09% -1.60% 0.58%
200 -0.08% -0.80% 0.58%
250 -0.07% 0 0.58%
30
$ p $ 35 93.28% -283.20% 1597.28%
40 86.93% -208.80% 3070.97%
45 81.66% -163.2% 5839.00%
10
$ v $ 11 -96.67% 456.00% -505.69%
12 -93.03% 528.00% -369.83%
13 -90.16% 608.00% -284.92%
5
$ h $ 7 -0.12% -30.00% -1.86%
9 -0.11% -25.00% -1.76%
11 -0.11% -21.33% -1.68%
0.5
$ M $ 0.6 5.97% -24.00% 46.87%
0.7 6.00% -24.00% 49.27%
0.8 6.00% -24.00% 51.84%
0.5
$ k $ 0.6 -81.86% 472.00% -345.27%
0.7 -73.14% 562.00% -216.95%
0.8 -67.29% 658.67% -154.70%
5
$ b $ 7 0.20% -40.00% 102.54%
9 0.17% -33.00% 102.72%
11 0.14% -28.00% 102.86%
0.2
$ a $ 0.3 -0.01% 1.60% 0.13%
0.4 -0.01% 1.60% 0.13%
0.5 0.01% 2.13% 0.13%
5
$ c $ 6 12.54% -308.00% 2198.77%
7 10.42% -212.00% 7175.78%
8 8.93% -197.33% 27485.97%
0.1
$ \theta $ 0.2 -0.03% -6.40% -0.43%
0.3 -0.03% -7.20% -0.44%
0.4 -0.03% -5.87% -0.41%
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