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July  2019, 15(3): 1289-1315. doi: 10.3934/jimo.2018096

Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers

Prasenjit Pramanik 1,, , Sarama Malik Das 2, and  Manas Kumar Maiti 3,

1. 

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, Paschim Medinipur, West Bengal 721102, India
2. 

Department of Mathematics, Calcutta Institute of Technology, Uluberia, Howrah, West Bengal 711316, India
3. 

Department of Mathematics, Mahishadal Raj College, Mahishadal, Purba Medinipur, West Bengal 721628, India

* Corresponding author

Received  August 2017 Revised  February 2018 Published  July 2019 Early access  July 2018

In the recently published paper [Gour Chandra Mahata and Sujit Kumar De, Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit-risk customers, International Journal of Management Science and Engineering Management, 2017, DOI:10.1080/17509653.2015.1109482], a supplier-retailer supply chain model of a deteriorating item with maximum lifetime and partial trade credit to credit risk customers is studied. In their study, unfortunately the amount of the payable bank interest due to the deteriorated units is omitted in the retailer's profit function for making the marketing decision. Some other unrealistic studies are also found in the numerical section of the paper. In this study those non-trivial flaws are identified and technically corrected. After correction, the theoretical existence of the optimal solutions of different scenarios are established and the solutions are derived using a soft computing technique.

Keywords: Supply chain, EOQ model, deterioration, partial trade credit, maximum lifetime.
Mathematics Subject Classification: Primary: 90B05; Secondary 91B60.
Citation: Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096
References:
[1]

S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662.  doi: 10.2307/2584538.  Google Scholar

[2]

K. Annaduari and R. Uthayakumar, Analysis of partial trade credit financing in a supply chain by EOQ-based model for decaying items with shortage, The International Journal of Advanced Manufacturing Technology, 61 (2012), 1139-1159.  doi: 10.1007/s00170-011-3765-9.  Google Scholar

[3]

K. Annaduari and R. Uthayakumar, Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit, Optimization Letters, 7 (2013), 1227-1249.  doi: 10.1007/s11590-012-0499-z.  Google Scholar

[4]

M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European Journal of Operational Research, 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004.  Google Scholar

[5]

C. T. Chang, L. Y. Ouyang and J. T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, 27 (2003), 983-996.  doi: 10.1016/S0307-904X(03)00131-8.  Google Scholar

[6]

K. J. Chung and T. S. Huang, The optimal retailer's ordering policies for deteriorating items with limited storage capacity under trade credit financing, International Journal of Production Economics, 106 (2007), 127-145.  doi: 10.1016/j.ijpe.2006.05.008.  Google Scholar

[7]

R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transitions, 5 (1973), 323-326.  doi: 10.1080/05695557308974918.  Google Scholar

[8]

U. Dave and L. K. Patel, (T, $S_{i}$) policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, 32 (1981), 137-142.  doi: 10.1057/jors.1981.27.  Google Scholar

[9]

A. P. Engelbrecht, Fundamentals of Computational Swarm Intelligence, John Wiley and Sons Ltd., 2005. Google Scholar

[10]

P. M. Ghare and G. H. Schrader, A model for exponentially decaying inventory system, Journal of Industrial Engineering, 21 (1963), 449-460.   Google Scholar

[11]

A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, The Journal of the Operational Research Society, 42 (1991), 1105-1110.  doi: 10.2307/2582957.  Google Scholar

[12]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payment, The Journal of the Operational Research Society, 36 (1985), 335-338.  doi: 10.2307/2582421.  Google Scholar

[13]

P. Guchhait, M. K. Maiti and M. Maiti, Inventory model of a deteriorating item with price and credit linked fuzzy demand : A fuzzy differential equation approach, OPSEARCH, 51 (2014), 321-353.  doi: 10.1007/s12597-013-0153-2.  Google Scholar

[14]

P. Guchhait, M. K. Maiti and M. Maiti, Two storage inventory model of a deteriorating item with variable demand under partial credit period, Applied Soft Computing, 13 (2013), 428-448.  doi: 10.1016/j.asoc.2012.07.028.  Google Scholar

[15]

M. A. Hariga, Optimal EOQ models for deteriorating items with time-varying demand, The Journal of the Operation Research Society, 47 (1996), 1228-1246.  doi: 10.2307/3010036.  Google Scholar

[16]

C. K. Huang, An integrated inventory model under conditions of order processing cost reduction and permissible delay in payments, Applied Mathematical Modelling, 34 (2010), 1352-1359.  doi: 10.1016/j.apm.2009.08.015.  Google Scholar

[17]

D. Huang, L. Q. Ouyang and H. Zhou, Note on: Managing multi-echelon multi-item channels with trade allowances under credit period, International Journal Production Economics, 138 (2012), 117-124.  doi: 10.1016/j.ijpe.2012.03.008.  Google Scholar

[18]

Y. F. Huang, Economic order quantity under conditionally permissible delay in payments, European Journal of Operational Research, 176 (2007), 911-924.  doi: 10.1016/j.ejor.2005.08.017.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

G. C. Mahata and S. K. De, Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit-risk customers, International Journal of Management Science and Engineering Management, 12 (2017), 21-32.  doi: 10.1080/17509653.2015.1109482.  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]

M. K. Maiti, A fuzzy genetic algorithm with varying population size to solve an inventory model with credit-linked promotional demand in an imprecise planning horizon, European Journal of Operational Research, 213 (2011), 96-106.  doi: 10.1016/j.ejor.2011.02.014.  Google Scholar

[25]

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.  Google Scholar

[26]

L. Y. Ouyang, J. T. Teng, S. K. Goyal and C. T. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments to order quantity, European Journal of Operational Research, 194 (2009), 418-431.  doi: 10.1016/j.ejor.2007.12.018.  Google Scholar

[27]

G. C. Philip, A generalized EOQ model for items with Weibull distribution deterioration, AIIE Transactions, 6 (1974), 159-162.  doi: 10.1080/05695557408974948.  Google Scholar

[28]

P. Pramanik, M. K. Maiti and M. Maiti, A supply chain with variable demand under three level trade credit policy, Computers & Industrial Engineering, 106 (2017), 205-221.  doi: 10.1016/j.cie.2017.02.007.  Google Scholar

[29]

P. Pramanik, M. K. Maiti and M. Maiti, An appropriate business strategy for a sale item, OPSEARCH, 55 (2018), 85-106.  doi: 10.1007/s12597-017-0310-0.  Google Scholar

[30]

P. Pramanik, M. K. Maiti and M. Maiti, Three level partial trade credit with promotional cost sharing, Applied Soft Computing, 58 (2017), 553-575.  doi: 10.1016/j.asoc.2017.04.013.  Google Scholar

[31]

B. Sarkar and S. Sarkar, Variable deterioration and demand-An inventory model, Economic Modelling, 31 (2013), 548-556.  doi: 10.1016/j.econmod.2012.11.045.  Google Scholar

[32]

D. Seifert, R. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: opportunity for research in operations, European Journal of Operational Research, 231 (2013), 245-256.  doi: 10.1016/j.ejor.2013.03.016.  Google Scholar

[33]

B. K. Sett, S. Sarkar, B. Sarkar and W. Y. Yun, Optimal replenishment policy with variable deterioration for fixed-lifetime products, Scientia Iranica, 23 (2016), 2318-2329.  doi: 10.24200/sci.2016.3959.  Google Scholar

[34]

T. Singh and H. Pattanayak, An EOQ model for deteriorating items with linear demand, variable deterioration and partial backlogging, Journal of Service Science and Management, 6 (2013), 186-190.  doi: 10.4236/jssm.2013.62019.  Google Scholar

[35]

S. Tayal, S. R. Singh and R. Sharma, Multi Item Inventory Model for Deteriorating Items with Expiration Date and Allowable Shortages, Indian Journal of Science and Technology, 7 (2014), 463-471.   Google Scholar

[36]

J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad customers, International Journal of Production Economics, 119 (2009), 415-423.  doi: 10.1016/j.ijpe.2009.04.004.  Google Scholar

[37]

J. T. Teng, H. J. Chang, C. Y. Dye and C. H. Hung, An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging, Operation Research Letters, 30 (2002), 387-393.  doi: 10.1016/S0167-6377(02)00150-5.  Google Scholar

[38]

J. T. Teng, M. S. Chern, H. L. Yang and Y. J. Wang, Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand, Operation Research Letters, 24 (1999), 65-72.  doi: 10.1016/S0167-6377(98)00042-X.  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]

Y. C. Tsao, Managing multi-echelon multi-item channels with trade allowances under credit period, International Journal Production Economics, 127 (2010), 226-237.  doi: 10.1016/j.ijpe.2009.08.010.  Google Scholar

show all references

References:
[1]

S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662.  doi: 10.2307/2584538.  Google Scholar

[2]

K. Annaduari and R. Uthayakumar, Analysis of partial trade credit financing in a supply chain by EOQ-based model for decaying items with shortage, The International Journal of Advanced Manufacturing Technology, 61 (2012), 1139-1159.  doi: 10.1007/s00170-011-3765-9.  Google Scholar

[3]

K. Annaduari and R. Uthayakumar, Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit, Optimization Letters, 7 (2013), 1227-1249.  doi: 10.1007/s11590-012-0499-z.  Google Scholar

[4]

M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European Journal of Operational Research, 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004.  Google Scholar

[5]

C. T. Chang, L. Y. Ouyang and J. T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, 27 (2003), 983-996.  doi: 10.1016/S0307-904X(03)00131-8.  Google Scholar

[6]

K. J. Chung and T. S. Huang, The optimal retailer's ordering policies for deteriorating items with limited storage capacity under trade credit financing, International Journal of Production Economics, 106 (2007), 127-145.  doi: 10.1016/j.ijpe.2006.05.008.  Google Scholar

[7]

R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transitions, 5 (1973), 323-326.  doi: 10.1080/05695557308974918.  Google Scholar

[8]

U. Dave and L. K. Patel, (T, $S_{i}$) policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, 32 (1981), 137-142.  doi: 10.1057/jors.1981.27.  Google Scholar

[9]

A. P. Engelbrecht, Fundamentals of Computational Swarm Intelligence, John Wiley and Sons Ltd., 2005. Google Scholar

[10]

P. M. Ghare and G. H. Schrader, A model for exponentially decaying inventory system, Journal of Industrial Engineering, 21 (1963), 449-460.   Google Scholar

[11]

A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, The Journal of the Operational Research Society, 42 (1991), 1105-1110.  doi: 10.2307/2582957.  Google Scholar

[12]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payment, The Journal of the Operational Research Society, 36 (1985), 335-338.  doi: 10.2307/2582421.  Google Scholar

[13]

P. Guchhait, M. K. Maiti and M. Maiti, Inventory model of a deteriorating item with price and credit linked fuzzy demand : A fuzzy differential equation approach, OPSEARCH, 51 (2014), 321-353.  doi: 10.1007/s12597-013-0153-2.  Google Scholar

[14]

P. Guchhait, M. K. Maiti and M. Maiti, Two storage inventory model of a deteriorating item with variable demand under partial credit period, Applied Soft Computing, 13 (2013), 428-448.  doi: 10.1016/j.asoc.2012.07.028.  Google Scholar

[15]

M. A. Hariga, Optimal EOQ models for deteriorating items with time-varying demand, The Journal of the Operation Research Society, 47 (1996), 1228-1246.  doi: 10.2307/3010036.  Google Scholar

[16]

C. K. Huang, An integrated inventory model under conditions of order processing cost reduction and permissible delay in payments, Applied Mathematical Modelling, 34 (2010), 1352-1359.  doi: 10.1016/j.apm.2009.08.015.  Google Scholar

[17]

D. Huang, L. Q. Ouyang and H. Zhou, Note on: Managing multi-echelon multi-item channels with trade allowances under credit period, International Journal Production Economics, 138 (2012), 117-124.  doi: 10.1016/j.ijpe.2012.03.008.  Google Scholar

[18]

Y. F. Huang, Economic order quantity under conditionally permissible delay in payments, European Journal of Operational Research, 176 (2007), 911-924.  doi: 10.1016/j.ejor.2005.08.017.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

G. C. Mahata and S. K. De, Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit-risk customers, International Journal of Management Science and Engineering Management, 12 (2017), 21-32.  doi: 10.1080/17509653.2015.1109482.  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]

M. K. Maiti, A fuzzy genetic algorithm with varying population size to solve an inventory model with credit-linked promotional demand in an imprecise planning horizon, European Journal of Operational Research, 213 (2011), 96-106.  doi: 10.1016/j.ejor.2011.02.014.  Google Scholar

[25]

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.  Google Scholar

[26]

L. Y. Ouyang, J. T. Teng, S. K. Goyal and C. T. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments to order quantity, European Journal of Operational Research, 194 (2009), 418-431.  doi: 10.1016/j.ejor.2007.12.018.  Google Scholar

[27]

G. C. Philip, A generalized EOQ model for items with Weibull distribution deterioration, AIIE Transactions, 6 (1974), 159-162.  doi: 10.1080/05695557408974948.  Google Scholar

[28]

P. Pramanik, M. K. Maiti and M. Maiti, A supply chain with variable demand under three level trade credit policy, Computers & Industrial Engineering, 106 (2017), 205-221.  doi: 10.1016/j.cie.2017.02.007.  Google Scholar

[29]

P. Pramanik, M. K. Maiti and M. Maiti, An appropriate business strategy for a sale item, OPSEARCH, 55 (2018), 85-106.  doi: 10.1007/s12597-017-0310-0.  Google Scholar

[30]

P. Pramanik, M. K. Maiti and M. Maiti, Three level partial trade credit with promotional cost sharing, Applied Soft Computing, 58 (2017), 553-575.  doi: 10.1016/j.asoc.2017.04.013.  Google Scholar

[31]

B. Sarkar and S. Sarkar, Variable deterioration and demand-An inventory model, Economic Modelling, 31 (2013), 548-556.  doi: 10.1016/j.econmod.2012.11.045.  Google Scholar

[32]

D. Seifert, R. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: opportunity for research in operations, European Journal of Operational Research, 231 (2013), 245-256.  doi: 10.1016/j.ejor.2013.03.016.  Google Scholar

[33]

B. K. Sett, S. Sarkar, B. Sarkar and W. Y. Yun, Optimal replenishment policy with variable deterioration for fixed-lifetime products, Scientia Iranica, 23 (2016), 2318-2329.  doi: 10.24200/sci.2016.3959.  Google Scholar

[34]

T. Singh and H. Pattanayak, An EOQ model for deteriorating items with linear demand, variable deterioration and partial backlogging, Journal of Service Science and Management, 6 (2013), 186-190.  doi: 10.4236/jssm.2013.62019.  Google Scholar

[35]

S. Tayal, S. R. Singh and R. Sharma, Multi Item Inventory Model for Deteriorating Items with Expiration Date and Allowable Shortages, Indian Journal of Science and Technology, 7 (2014), 463-471.   Google Scholar

[36]

J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad customers, International Journal of Production Economics, 119 (2009), 415-423.  doi: 10.1016/j.ijpe.2009.04.004.  Google Scholar

[37]

J. T. Teng, H. J. Chang, C. Y. Dye and C. H. Hung, An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging, Operation Research Letters, 30 (2002), 387-393.  doi: 10.1016/S0167-6377(02)00150-5.  Google Scholar

[38]

J. T. Teng, M. S. Chern, H. L. Yang and Y. J. Wang, Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand, Operation Research Letters, 24 (1999), 65-72.  doi: 10.1016/S0167-6377(98)00042-X.  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]

Y. C. Tsao, Managing multi-echelon multi-item channels with trade allowances under credit period, International Journal Production Economics, 127 (2010), 226-237.  doi: 10.1016/j.ijpe.2009.08.010.  Google Scholar

Figure 1.  Pictorial representation of situation 1.1
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Figure 2.  Interest earn and paid situations for the sold units
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Figure 3.  Interest paid situation due to deteriorated units
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Figure 4.  Pictorial representation of situation 1.2
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Figure 5.  Interest earn and paid situations for the sold units
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Figure 6.  Interest paid situation due to deteriorated units
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Figure 7.  Pictorial representation of situation 1.3
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Figure 8.  Interest earn and paid situations for the sold units
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Figure 9.  Interest paid situation due to deteriorated units
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Figure 10.  Pictorial representation of situation 2.1
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Figure 11.  Interest earn and paid situations for the sold units
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Figure 12.  Interest paid situation due to deteriorated units
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Figure 13.  Pictorial representation of situation 2.2
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Figure 14.  Interest earn and paid situations for the sold units
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Figure 15.  Interest paid situation due to deteriorated units
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Table 1.  Innovation of this paper related to the exiting literature
Article Deteriorating item Level of trade credit Pattern of trade credit Deterioration rate Item(s) has expiration time Interest paid for deteriorated units
[8,11,15,37,38] $\surd$ $\times$ NA Constant $\times$ NA
[35] $\surd$ $\times$ NA Constant $\surd$ NA
[1] $\surd$ Supplier-Retailer Full credit Constant $\times$ $\times$
[2,14,26] $\surd$ Supplier-Retailer Ordered quantity based full/ partial credit Constant $\times$ $\times$
[3,6,13,21,25] $\surd$ Supplier-Retailer Full credit Constant $\times$ $\times$
Retailer-Customers Full credit
[20] $\surd$ Supplier-Retailer Full credit Constant $\times$ $\times$
Retailer-Customers Partial credit
[7,27,34] $\surd$ $\times$ NA Time dependent $\times$ NA
[5] $\surd$ Supplier-Retailer Order quantity based full credit Time dependent $\times$ $\times$
[22] $\surd$ Supplier-Retailer Full credit Time dependent $\surd$ $\times$
Retailer-Customers Partial credit
This Paper $\surd$ Supplier-Retailer Full credit Time dependent $\surd$ $\surd$
Retailer-Customers Partial credit
NA stands for ‘Not Applicable’.
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Article Deteriorating item Level of trade credit Pattern of trade credit Deterioration rate Item(s) has expiration time Interest paid for deteriorated units
[8,11,15,37,38] $\surd$ $\times$ NA Constant $\times$ NA
[35] $\surd$ $\times$ NA Constant $\surd$ NA
[1] $\surd$ Supplier-Retailer Full credit Constant $\times$ $\times$
[2,14,26] $\surd$ Supplier-Retailer Ordered quantity based full/ partial credit Constant $\times$ $\times$
[3,6,13,21,25] $\surd$ Supplier-Retailer Full credit Constant $\times$ $\times$
Retailer-Customers Full credit
[20] $\surd$ Supplier-Retailer Full credit Constant $\times$ $\times$
Retailer-Customers Partial credit
[7,27,34] $\surd$ $\times$ NA Time dependent $\times$ NA
[5] $\surd$ Supplier-Retailer Order quantity based full credit Time dependent $\times$ $\times$
[22] $\surd$ Supplier-Retailer Full credit Time dependent $\surd$ $\times$
Retailer-Customers Partial credit
This Paper $\surd$ Supplier-Retailer Full credit Time dependent $\surd$ $\surd$
Retailer-Customers Partial credit
NA stands for ‘Not Applicable’.
Table 2.  Results of the model
Example Appropriate for $T^*$ $Q^*$ $TP^*(T^*)$
5.1 Situation 1.1 $T_{11}=0.1346$ 319.396 $TP_{11}(T_{11})=9824.71$
5.2 Situation 1.2 $T_{12}=0.1404$ 291.159 $TP_{12}(T_{12})=8643.00$
5.3 Situation 1.3 $T_{13}=0.1365$ 296.892 $TP_{13}(T_{13})=8942.34$
5.4 Situation 2.1 $T_{21}=0.1382$ 273.311 $TP_{21}(T_{21})=8087.69$
5.5 Situation 2.2 $T_{22}=0.1374$ 256.299 $TP_{22}(T_{22})=7716.36$
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Example Appropriate for $T^*$ $Q^*$ $TP^*(T^*)$
5.1 Situation 1.1 $T_{11}=0.1346$ 319.396 $TP_{11}(T_{11})=9824.71$
5.2 Situation 1.2 $T_{12}=0.1404$ 291.159 $TP_{12}(T_{12})=8643.00$
5.3 Situation 1.3 $T_{13}=0.1365$ 296.892 $TP_{13}(T_{13})=8942.34$
5.4 Situation 2.1 $T_{21}=0.1382$ 273.311 $TP_{21}(T_{21})=8087.69$
5.5 Situation 2.2 $T_{22}=0.1374$ 256.299 $TP_{22}(T_{22})=7716.36$
Table 3.  Sensitivity Analysis
Parameter $T^*$ $Q^*$ $TP^*(T^*)$
2000 0.1404 291.159 8643.00
D 2500 0.1259 325.220 10991.49
3000 0.1151 355.645 13355.68
0.16 0.1404 291.159 8643.00
M 0.20 0.1395 289.318 8664.18
0.25 0.1392 288.595 8694.01
50 0.1000 205.167 9058.72
A 100 0.1404 291.159 8643.00
200 0.1958 412.145 8048.82
1 0.1404 291.159 8643.00
m 2 0.1569 322.272 8780.27
3 0.1670 341.100 8853.89
0.02 0.1409 292.159 8637.78
$I_{e}$ 0.03 0.1404 291.159 8643.00
0.04 0.1398 289.933 8648.24
0.05 0.1404 291.159 8643.00
$\alpha$ 0.10 0.1402 290.779 8645.21
0.20 0.1403 290.972 8649.619
0.04 0.1396 289.443 8662.98
N 0.08 0.1404 291.159 8643.00
0.12 0.1407 291.855 8626.29
1 0.1894 398.066 16983.74
c 3 0.1595 332.370 12795.03
5 0.1404 291.159 8643.00
1 0.1569 326.853 8795.10
h 2 0.1404 291.159 8643.00
4 0.1189 244.171 8380.12
0.02 0.1410 292.351 8644.47
$I_{c}$ 0.03 0.1404 291.159 8643.00
0.04 0.1400 290.248 8641.556
10 0.1404 291.159 8643.00
s 15 0.1396 289.442 18650.87
20 0.1390 288.053 28658.77
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Parameter $T^*$ $Q^*$ $TP^*(T^*)$
2000 0.1404 291.159 8643.00
D 2500 0.1259 325.220 10991.49
3000 0.1151 355.645 13355.68
0.16 0.1404 291.159 8643.00
M 0.20 0.1395 289.318 8664.18
0.25 0.1392 288.595 8694.01
50 0.1000 205.167 9058.72
A 100 0.1404 291.159 8643.00
200 0.1958 412.145 8048.82
1 0.1404 291.159 8643.00
m 2 0.1569 322.272 8780.27
3 0.1670 341.100 8853.89
0.02 0.1409 292.159 8637.78
$I_{e}$ 0.03 0.1404 291.159 8643.00
0.04 0.1398 289.933 8648.24
0.05 0.1404 291.159 8643.00
$\alpha$ 0.10 0.1402 290.779 8645.21
0.20 0.1403 290.972 8649.619
0.04 0.1396 289.443 8662.98
N 0.08 0.1404 291.159 8643.00
0.12 0.1407 291.855 8626.29
1 0.1894 398.066 16983.74
c 3 0.1595 332.370 12795.03
5 0.1404 291.159 8643.00
1 0.1569 326.853 8795.10
h 2 0.1404 291.159 8643.00
4 0.1189 244.171 8380.12
0.02 0.1410 292.351 8644.47
$I_{c}$ 0.03 0.1404 291.159 8643.00
0.04 0.1400 290.248 8641.556
10 0.1404 291.159 8643.00
s 15 0.1396 289.442 18650.87
20 0.1390 288.053 28658.77
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