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May  2020, 16(3): 1273-1296. doi: 10.3934/jimo.2019002

## Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering

 1 Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran 2 Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul, 03722, South Korea 3 Islamic Azad University, Tehran South branch, School of Industrial Engineering, Tehran, Iran

* Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146

Received  May 2017 Revised  April 2018 Published  March 2019

This study develops a single-machine manufacturing system for multi-product with defective items and delayed payment policy. Contradictory to the literature limited production capacity and partial backlogging are considered for more realistic result. The objective of this research is to obtain the optimal cycle length, optimal production quantity, and optimal backorder quantity of each product such that the expected total cost is minimum. The model is solved analytically. Three efficient lemmas are developed to obtain the global optimum solution of the model. An improved algorithm is designed to obtain the numerical solution of the model. An illustrative numerical example and sensitivity analysis are provided to show the practical usage of proposed method.

Citation: Ata Allah Taleizadeh, Biswajit Sarkar, Mohammad Hasani. Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1273-1296. doi: 10.3934/jimo.2019002
##### References:
 [1] L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, A simple method to compute economic order quantities: Some observations, Applied Mathematical Modelling, 34 (2010), 1684-1688. doi: 10.1016/j.apm.2009.08.024.  Google Scholar [2] L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, B. Sarkar and G. Treviño-Garza, Easy and improved algorithms to joint determination of the replenishment lot size and number of shipments for an EPQ model with rework, Mathematical & Computational Applications, 18 (2013), 132-138. Google Scholar [3] S. W. Chiu, Production lot size problem with failure in repair and backlogging derived without derivatives, European Journal of Operational Research, 188 (2008), 610-615.  doi: 10.1016/j.ejor.2007.04.049.  Google Scholar [4] S. W. Chiu, S. L. Wang and Y. S. P. Chiu, Determining the optimal run time for EPQ model with scrap, rework, and stochastic break downs, European Journal of Operational Research, 180 (2007), 664-676.   Google Scholar [5] K. K. Damghani and A. Shahrokh, Solving a new multi-period multi-objective multi-product aggregate production planning problem using fuzzy goal programming, Industrial Engineering & Management Systems, 13 (2014), 369-382.   Google Scholar [6] S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal of Production Research, 15 (1977), 107-111.  doi: 10.1080/00207547708943107.  Google Scholar [7] S. K. Goyal, A joint economic lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241.  doi: 10.1111/j.1540-5915.1988.tb00264.x.  Google Scholar [8] S. K. Goyal and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Note on economic production quantity model for items with imperfect quality a practical approach, International Journal of Production Economics, 77 (2002), 85-87. doi: 10.1016/S0925-5273(01)00203-1.  Google Scholar [9] Y. F. Huang, Optimal retailer replenishment decisions in EPQ model under two levels of trade credit policy, European Journal of Operational Research, 176 (2007), 1577-1591.   Google Scholar [10] C. W. Kang, M. Ullah, B. Sarkar, H. Iftikhar and A. Rehman, Impact of random defective rate on lot size focusing work-in-process inventory in manufacturing system, International Journal of Production Research, 55 (2017), 1748-1766.  doi: 10.1080/00207543.2016.1235295.  Google Scholar [11] S. J. Kim and B. Sarkar, Supply chain model with stochastic lead time, trade-credit financing, and transportation discounts, Mathematical Problems in Engineering, (2017), Article ID 6465912, 1-14. doi: 10.1155/2017/6465912.  Google Scholar [12] J. Li, S. Wang and T. C. E. Cheng, Analysis of postponement strategy by EPQ based models with planned backorders, Omega, 36 (2008), 777-788.  doi: 10.1016/j.omega.2006.03.002.  Google Scholar [13] 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 [14] L. Y Ouyang, C. T. Yang, Y. L. Chan and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, A comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Applied Mathematics and Computation, 224 (2013), 268-277. doi: 10.1016/j.amc.2013.08.062.  Google Scholar [15] L. Y. Ouyang, N. C. Yeh and K. S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of Operations Research Socity, 47 (1996), 829-832.   Google Scholar [16] L. Y. Ouyang, B. R. Chuang and Y. J. Lin, Impact of backorder discounts on periodic review inventory model, International Journal of Information & Managment Science, 14 (2003), 1-13.   Google Scholar [17] C. H. Pan and Y. C. Hsiao, Inventory models with back-order discounts and variable lead time, Internatinal Journal of System Science, 32 (2010), 925-929.  doi: 10.1080/00207720010004449.  Google Scholar [18] S. H. A. Pasandide and S.T. A. Niaki, A genetic algorithm approach to optimize a multi products EPQ model with discrete delivery orders and constrained space, Applied Mathematics and Computation, 195 (2008), 506-514.  doi: 10.1016/j.amc.2007.05.007.  Google Scholar [19] W. D. Pentico, M. J. Drake and C. Toews, The deterministic EPQ with partial backordering: A new approach, Omega, 37 (2009), 624-636.   Google Scholar [20] M. K. Salameh and M. Y. Jaber, Economic order quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64.   Google Scholar [21] S. S. Sana, S. K. Goyal and K. S. Chaudhuri, An imperfect production process in a volume flexible inventory model, International Journal of Production Economics, 105 (2007), 548-559.   Google Scholar [22] B. Sarkar, An inventory model with reliability in an imperfect production process, Applied Mathematics and Computations, 218 (2012a), 4881-4891.  doi: 10.1016/j.amc.2011.10.053.  Google Scholar [23] B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55 (2012b), 367-377.  doi: 10.1016/j.mcm.2011.08.009.  Google Scholar [24] B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products, Mathematical Problem in Engineering, 2016 (2016), Article ID 6318737, 1-14. doi: 10.1155/2016/6318737.  Google Scholar [25] B. Sarkar, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, M. Sarkar and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435. doi: 10.1016/j.jmsy.2014.02.001.  Google Scholar [26] B. Sarkar, H. Gupta, K. S. Chaudhuri and S. K. Goyal, An integrated inventory model with variable lead time, defective units and delay in payments, Applied Mathematics and Computations, 237 (2014), 650-658.  doi: 10.1016/j.amc.2014.03.061.  Google Scholar [27] B. Sarkar, A. Majumder, M. Sarkar, B. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Management Optimization, 13 (2017), 1085-1104.  doi: 10.3934/jimo.2016063.  Google Scholar [28] B. Sarkar, B. Mondal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.  Google Scholar [29] B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, Applied Mathematics and Computations, 217 (2011), 6159-6167.  doi: 10.1016/j.amc.2010.12.098.  Google Scholar [30] B. Sarkar, S. S. Sana and K. S. Chaudhuri, Optimal reliability, production lotsize and safety stock: An economic manufacturing quantity model, International Journal of Management Science & Engineering Management, 5 (2010), 192-202.   Google Scholar [31] B. Sarkar, S. S. Sana and K. S. Chaudhuri, A stock-dependent inventory model in an imperfect production process, International Journal of Procurement Management, 3 (2010), 361-378.  doi: 10.1504/IJPM.2010.035467.  Google Scholar [32] B. Sarkar, S. S. Sana and K. S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money - An EMQ model, Expert Systems with Applications, 38 (2011), 13543-13548.   Google Scholar [33] B. Sarkar and S. Saren, Partial trade-credit policy of retailer with exponentially deteriorating items, International Journal of Applied and Computational Mathematics, 1 (2015), 343-368.  doi: 10.1007/s40819-014-0019-1.  Google Scholar [34] B. Sarkar and S. Sarkar, An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand, Economic Modelling, 30 (2013), 924-932.   Google Scholar [35] H. Scarf, A Min-Max Solution of an Inventory Problem, In: Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Redwood City, CA, (1958), 201-209. Google Scholar [36] E. W. Taft, The most economical production lot, The Iron Age, 101 (1918), 1410-1412.   Google Scholar [37] A. H. Tai, Economic production quantity models for deteriorating/imperfect products and service with rework, Computers & Industrial Engineering, 66 (2013), 879-888.  doi: 10.1016/j.cie.2013.09.007.  Google Scholar [38] A. A. Taleizadeh, An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments, Applied Mathematical Modeling, 38 (2014), 5357-5366.  doi: 10.1016/j.apm.2014.02.014.  Google Scholar [39] A. A. Taleizadeh, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, J. Biabani and R. R. Nikousokhan, Multi products single machine EPQ model with immediate rework process, International Journal of Industrial Engineering Computations, 3 (2012), 93-102. doi: 10.5267/j.ijiec.2011.09.001.  Google Scholar [40] A. A. Taleizadeh, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n and B. Mohammadi, Multi product single machine epq model with backordering, scraped products, rework and interruption in manufacturing process, International Journal of Production Economic, 150 (2014), 9-27. doi: 10.1016/j.ijpe.2013.11.023.  Google Scholar [41] A. A. Taleizadeh, S. G. H. Jalali-Naini, H. M. Wee and T. C. Kuo, An Imperfect, Multi Product Production System with Rework, Scientia Iranica, 20 (2013), 811-823.   Google Scholar [42] A. A. Taleizadeh, H. Moghadasi, S. T. A. Niaki and A. K. Eftekhari, An EOQ-Joint Replenishment Policy to Supply Expensive Imported Raw Materials with Payment in Advance, Journal of Applied Science, 8 (2009), 4263-4273.   Google Scholar [43] A. A. Taleizadeh, A. A. Najafi and S. T. A. Niaki, Economic production quantity model with scraped items and limited production capacity, Scientia Iranica, 17 (2010), 58-69.   Google Scholar [44] A. A. Taleizadeh, S. T. Niaki and M. B. Aryanezhad, Multi-product multi-constraint inventory control systems with stochastic replenishment and discount under fuzzy purchasing price and holding costs, American Journal of Applied Science, 8 (2008), 1228-1234.   Google Scholar [45] A. A. Taleizadeh and D. W. Pentico, An Economic Order Quantity Model with Partial Backordering and All-units Discount, International Journal of Production Economic, 155 (2014), 172-184.  doi: 10.1016/j.ijpe.2014.01.012.  Google Scholar [46] A. A. Taleizadeh, H. M. Wee and S. Gh. R. Jalali-Naini, Economic production quantity model with repair failure and limited capacity, Applied Mathematical Modeling, 37 (2013), 2765-2774.  doi: 10.1016/j.apm.2012.06.006.  Google Scholar [47] A. A. Taleizadeh, H. M. Wee and F. Jolai, Revisiting fuzzy rough economic order quantity model for deteriorating items considering quantity discount and prepayment, Mathematical and Computer Modeling, 57 (2013), 1466-1479.  doi: 10.1016/j.mcm.2012.12.008.  Google Scholar [48] A. A. Taleizadeh, H. M. Wee and S. J. Sadjadi, Multi-product production quantity model with repair failure and partial backordering, Computers & Industrial Engineering, 59 (2010), 45-54.  doi: 10.1016/j.cie.2010.02.015.  Google Scholar [49] J. T. Teng, C. T. Chang and S. K. Goyal, Optimal pricing and ordering policy under permissible delay in payments, International Journal of Production Economics, 97 (2005), 121-129.  doi: 10.1016/j.ijpe.2004.04.010.  Google Scholar [50] 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 [51] H. Wee, G. Widyadanab, A. Taleizadeh and J. Biabanid, Multi products single machine economic production quantity model with multiple batch size, International Journal of Industrial Engineering Computations, 2 (2011), 213-224.   Google Scholar [52] G. A. Widyadana L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n and H. M. Wee, Economic order quantity model for deteriorating items with planned backorder level, Mathematical and Computer Modelling, 54 (2011), 1569-1575. doi: 10.1016/j.mcm.2011.04.028.  Google Scholar [53] W. Wisittipanich and P. Hengmeechai, A multi-objective differential evolution for just-in-time door assignment and truck scheduling in multi-door cross docking problems, Industrial Engineering & Management Systems, 14 (2015), 299-311.  doi: 10.7232/iems.2015.14.3.299.  Google Scholar [54] S. H. Yoo, D. Kim and M. S. 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show all references

##### References:
 [1] L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, A simple method to compute economic order quantities: Some observations, Applied Mathematical Modelling, 34 (2010), 1684-1688. doi: 10.1016/j.apm.2009.08.024.  Google Scholar [2] L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, B. Sarkar and G. Treviño-Garza, Easy and improved algorithms to joint determination of the replenishment lot size and number of shipments for an EPQ model with rework, Mathematical & Computational Applications, 18 (2013), 132-138. Google Scholar [3] S. W. Chiu, Production lot size problem with failure in repair and backlogging derived without derivatives, European Journal of Operational Research, 188 (2008), 610-615.  doi: 10.1016/j.ejor.2007.04.049.  Google Scholar [4] S. W. Chiu, S. L. Wang and Y. S. P. Chiu, Determining the optimal run time for EPQ model with scrap, rework, and stochastic break downs, European Journal of Operational Research, 180 (2007), 664-676.   Google Scholar [5] K. K. Damghani and A. Shahrokh, Solving a new multi-period multi-objective multi-product aggregate production planning problem using fuzzy goal programming, Industrial Engineering & Management Systems, 13 (2014), 369-382.   Google Scholar [6] S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal of Production Research, 15 (1977), 107-111.  doi: 10.1080/00207547708943107.  Google Scholar [7] S. K. Goyal, A joint economic lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241.  doi: 10.1111/j.1540-5915.1988.tb00264.x.  Google Scholar [8] S. K. Goyal and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Note on economic production quantity model for items with imperfect quality a practical approach, International Journal of Production Economics, 77 (2002), 85-87. doi: 10.1016/S0925-5273(01)00203-1.  Google Scholar [9] Y. F. Huang, Optimal retailer replenishment decisions in EPQ model under two levels of trade credit policy, European Journal of Operational Research, 176 (2007), 1577-1591.   Google Scholar [10] C. W. Kang, M. Ullah, B. Sarkar, H. Iftikhar and A. Rehman, Impact of random defective rate on lot size focusing work-in-process inventory in manufacturing system, International Journal of Production Research, 55 (2017), 1748-1766.  doi: 10.1080/00207543.2016.1235295.  Google Scholar [11] S. J. Kim and B. Sarkar, Supply chain model with stochastic lead time, trade-credit financing, and transportation discounts, Mathematical Problems in Engineering, (2017), Article ID 6465912, 1-14. doi: 10.1155/2017/6465912.  Google Scholar [12] J. Li, S. Wang and T. C. E. Cheng, Analysis of postponement strategy by EPQ based models with planned backorders, Omega, 36 (2008), 777-788.  doi: 10.1016/j.omega.2006.03.002.  Google Scholar [13] 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 [14] L. Y Ouyang, C. T. Yang, Y. L. Chan and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, A comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Applied Mathematics and Computation, 224 (2013), 268-277. doi: 10.1016/j.amc.2013.08.062.  Google Scholar [15] L. Y. Ouyang, N. C. Yeh and K. S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of Operations Research Socity, 47 (1996), 829-832.   Google Scholar [16] L. Y. Ouyang, B. R. Chuang and Y. J. Lin, Impact of backorder discounts on periodic review inventory model, International Journal of Information & Managment Science, 14 (2003), 1-13.   Google Scholar [17] C. H. Pan and Y. C. Hsiao, Inventory models with back-order discounts and variable lead time, Internatinal Journal of System Science, 32 (2010), 925-929.  doi: 10.1080/00207720010004449.  Google Scholar [18] S. H. A. Pasandide and S.T. A. Niaki, A genetic algorithm approach to optimize a multi products EPQ model with discrete delivery orders and constrained space, Applied Mathematics and Computation, 195 (2008), 506-514.  doi: 10.1016/j.amc.2007.05.007.  Google Scholar [19] W. D. Pentico, M. J. Drake and C. Toews, The deterministic EPQ with partial backordering: A new approach, Omega, 37 (2009), 624-636.   Google Scholar [20] M. K. Salameh and M. Y. Jaber, Economic order quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64.   Google Scholar [21] S. S. Sana, S. K. Goyal and K. S. Chaudhuri, An imperfect production process in a volume flexible inventory model, International Journal of Production Economics, 105 (2007), 548-559.   Google Scholar [22] B. Sarkar, An inventory model with reliability in an imperfect production process, Applied Mathematics and Computations, 218 (2012a), 4881-4891.  doi: 10.1016/j.amc.2011.10.053.  Google Scholar [23] B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55 (2012b), 367-377.  doi: 10.1016/j.mcm.2011.08.009.  Google Scholar [24] B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products, Mathematical Problem in Engineering, 2016 (2016), Article ID 6318737, 1-14. doi: 10.1155/2016/6318737.  Google Scholar [25] B. Sarkar, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, M. Sarkar and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435. doi: 10.1016/j.jmsy.2014.02.001.  Google Scholar [26] B. Sarkar, H. Gupta, K. S. Chaudhuri and S. K. Goyal, An integrated inventory model with variable lead time, defective units and delay in payments, Applied Mathematics and Computations, 237 (2014), 650-658.  doi: 10.1016/j.amc.2014.03.061.  Google Scholar [27] B. Sarkar, A. Majumder, M. Sarkar, B. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Management Optimization, 13 (2017), 1085-1104.  doi: 10.3934/jimo.2016063.  Google Scholar [28] B. Sarkar, B. Mondal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.  Google Scholar [29] B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, Applied Mathematics and Computations, 217 (2011), 6159-6167.  doi: 10.1016/j.amc.2010.12.098.  Google Scholar [30] B. Sarkar, S. S. Sana and K. S. Chaudhuri, Optimal reliability, production lotsize and safety stock: An economic manufacturing quantity model, International Journal of Management Science & Engineering Management, 5 (2010), 192-202.   Google Scholar [31] B. Sarkar, S. S. Sana and K. S. Chaudhuri, A stock-dependent inventory model in an imperfect production process, International Journal of Procurement Management, 3 (2010), 361-378.  doi: 10.1504/IJPM.2010.035467.  Google Scholar [32] B. Sarkar, S. S. Sana and K. S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money - An EMQ model, Expert Systems with Applications, 38 (2011), 13543-13548.   Google Scholar [33] B. Sarkar and S. Saren, Partial trade-credit policy of retailer with exponentially deteriorating items, International Journal of Applied and Computational Mathematics, 1 (2015), 343-368.  doi: 10.1007/s40819-014-0019-1.  Google Scholar [34] B. Sarkar and S. Sarkar, An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand, Economic Modelling, 30 (2013), 924-932.   Google Scholar [35] H. Scarf, A Min-Max Solution of an Inventory Problem, In: Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Redwood City, CA, (1958), 201-209. Google Scholar [36] E. W. Taft, The most economical production lot, The Iron Age, 101 (1918), 1410-1412.   Google Scholar [37] A. H. Tai, Economic production quantity models for deteriorating/imperfect products and service with rework, Computers & Industrial Engineering, 66 (2013), 879-888.  doi: 10.1016/j.cie.2013.09.007.  Google Scholar [38] A. A. Taleizadeh, An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments, Applied Mathematical Modeling, 38 (2014), 5357-5366.  doi: 10.1016/j.apm.2014.02.014.  Google Scholar [39] A. A. Taleizadeh, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, J. Biabani and R. R. Nikousokhan, Multi products single machine EPQ model with immediate rework process, International Journal of Industrial Engineering Computations, 3 (2012), 93-102. doi: 10.5267/j.ijiec.2011.09.001.  Google Scholar [40] A. A. Taleizadeh, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n and B. Mohammadi, Multi product single machine epq model with backordering, scraped products, rework and interruption in manufacturing process, International Journal of Production Economic, 150 (2014), 9-27. doi: 10.1016/j.ijpe.2013.11.023.  Google Scholar [41] A. A. Taleizadeh, S. G. H. Jalali-Naini, H. M. Wee and T. C. Kuo, An Imperfect, Multi Product Production System with Rework, Scientia Iranica, 20 (2013), 811-823.   Google Scholar [42] A. A. Taleizadeh, H. Moghadasi, S. T. A. Niaki and A. K. Eftekhari, An EOQ-Joint Replenishment Policy to Supply Expensive Imported Raw Materials with Payment in Advance, Journal of Applied Science, 8 (2009), 4263-4273.   Google Scholar [43] A. A. Taleizadeh, A. A. Najafi and S. T. A. Niaki, Economic production quantity model with scraped items and limited production capacity, Scientia Iranica, 17 (2010), 58-69.   Google Scholar [44] A. A. Taleizadeh, S. T. Niaki and M. B. Aryanezhad, Multi-product multi-constraint inventory control systems with stochastic replenishment and discount under fuzzy purchasing price and holding costs, American Journal of Applied Science, 8 (2008), 1228-1234.   Google Scholar [45] A. A. Taleizadeh and D. W. Pentico, An Economic Order Quantity Model with Partial Backordering and All-units Discount, International Journal of Production Economic, 155 (2014), 172-184.  doi: 10.1016/j.ijpe.2014.01.012.  Google Scholar [46] A. A. Taleizadeh, H. M. Wee and S. Gh. R. Jalali-Naini, Economic production quantity model with repair failure and limited capacity, Applied Mathematical Modeling, 37 (2013), 2765-2774.  doi: 10.1016/j.apm.2012.06.006.  Google Scholar [47] A. A. Taleizadeh, H. M. Wee and F. Jolai, Revisiting fuzzy rough economic order quantity model for deteriorating items considering quantity discount and prepayment, Mathematical and Computer Modeling, 57 (2013), 1466-1479.  doi: 10.1016/j.mcm.2012.12.008.  Google Scholar [48] A. A. Taleizadeh, H. M. Wee and S. J. Sadjadi, Multi-product production quantity model with repair failure and partial backordering, Computers & Industrial Engineering, 59 (2010), 45-54.  doi: 10.1016/j.cie.2010.02.015.  Google Scholar [49] J. T. Teng, C. T. Chang and S. K. Goyal, Optimal pricing and ordering policy under permissible delay in payments, International Journal of Production Economics, 97 (2005), 121-129.  doi: 10.1016/j.ijpe.2004.04.010.  Google Scholar [50] 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 [51] H. Wee, G. Widyadanab, A. 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Graphical representation of inventory system
Graphical representation of interest earned and interest charged for $M<t_a$
Graphical representation of interest earned and interest charged for $t_d \leq M < t_a$
Graphical representation of interest earned and interest charged for $M \geq t_d$
Author(s) contribution Table
 Author(s) EPQ Imperfect Production Multi-product Single machine Delay-in-payment Backorder Repair C$\acute{a}$rdenas-Barr$\acute{o}$net al. [2] $\surd$ $\surd$ Chiu et al. [4] $\surd$ $\surd$ $\surd$ Goyal and C$\acute{a}$rdenas-Barr$\acute{o}$n [8] $\surd$ $\surd$ Huang [9] $\surd$ Taleizadeh [38] $\surd$ $\surd$ Sana et al.[21] $\surd$ $\surd$ $\surd$ Li et al.[12] $\surd$ $\surd$ Taleizadh et al. [48] $\surd$ $\surd$ $\surd$ $\surd$ Sarkar et al. [25] $\surd$ $\surd$ $\surd$ $\surd$ Kang et al. [10] $\surd$ $\surd$ $\surd$ Ouyang et al. [16] $\surd$ This Model $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
 Author(s) EPQ Imperfect Production Multi-product Single machine Delay-in-payment Backorder Repair C$\acute{a}$rdenas-Barr$\acute{o}$net al. [2] $\surd$ $\surd$ Chiu et al. [4] $\surd$ $\surd$ $\surd$ Goyal and C$\acute{a}$rdenas-Barr$\acute{o}$n [8] $\surd$ $\surd$ Huang [9] $\surd$ Taleizadeh [38] $\surd$ $\surd$ Sana et al.[21] $\surd$ $\surd$ $\surd$ Li et al.[12] $\surd$ $\surd$ Taleizadh et al. [48] $\surd$ $\surd$ $\surd$ $\surd$ Sarkar et al. [25] $\surd$ $\surd$ $\surd$ $\surd$ Kang et al. [10] $\surd$ $\surd$ $\surd$ Ouyang et al. [16] $\surd$ This Model $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
The values of the parameters
 $P$ $P_i$ $P_i^1$ $\lambda_i$ $K_i$ $C_i$ $C_R^i$ $b_i$ $h_i$ 1 10000 600 2000 750 2 0.5 0.25 0.2 2 10500 650 1500 700 1.5 0.6 0.5 0.15 3 11000 750 1000 650 1 0.7 0.75 0.1
 $P$ $P_i$ $P_i^1$ $\lambda_i$ $K_i$ $C_i$ $C_R^i$ $b_i$ $h_i$ 1 10000 600 2000 750 2 0.5 0.25 0.2 2 10500 650 1500 700 1.5 0.6 0.5 0.15 3 11000 750 1000 650 1 0.7 0.75 0.1
The parametric values
 $P$ $M_j$ $I_c$ $I_e$ $S_i$ $v_i$ $SE_i$ $X_i$ 1 0.04 0.09 0.05 4 1.5 0.003 0.05 2 0.04 0.09 0.05 3 1 0.004 0.075 3 0.04 0.09 0.05 2 0.5 0.005 0.1
 $P$ $M_j$ $I_c$ $I_e$ $S_i$ $v_i$ $SE_i$ $X_i$ 1 0.04 0.09 0.05 4 1.5 0.003 0.05 2 0.04 0.09 0.05 3 1 0.004 0.075 3 0.04 0.09 0.05 2 0.5 0.005 0.1
Optimal solutions table
 $P$ $T_{Min}$ $T$ $T^*=T_1$ $Q_i$ $B_i$ $Z$ 1 5158 2331.9 2 0.128 2.579 2.579 3868.5 1061 9046.93 3 2579 375.7
 $P$ $T_{Min}$ $T$ $T^*=T_1$ $Q_i$ $B_i$ $Z$ 1 5158 2331.9 2 0.128 2.579 2.579 3868.5 1061 9046.93 3 2579 375.7
Optimal solutions for different values of $M$
 $M$ $T_{Min}$ $T$ $T^*$ $Q_i$ $B_i$ $Z^*=Z_1$ 5159 2332 0.03 0.128 2.579 2.579 3869 1061 9051.666 2579 376 5158 2331.9 0.04 0.128 2.579 2.579 3868.5 1061 9046.93 2579 375.7 5157 2331.5 0.05 0.128 2.578 2.578 3868 1061 9042.125 2578.6 375.5 51555 2330 0.06 0.128 2.577 2.577 3866 1060 9037.253 2577 375
 $M$ $T_{Min}$ $T$ $T^*$ $Q_i$ $B_i$ $Z^*=Z_1$ 5159 2332 0.03 0.128 2.579 2.579 3869 1061 9051.666 2579 376 5158 2331.9 0.04 0.128 2.579 2.579 3868.5 1061 9046.93 2579 375.7 5157 2331.5 0.05 0.128 2.578 2.578 3868 1061 9042.125 2578.6 375.5 51555 2330 0.06 0.128 2.577 2.577 3866 1060 9037.253 2577 375
Optimal solutions for different values of $I_c$
 $I_c$ $T_{Min}$ $T$ $T^*$ $Q_i$ $B_i$ $Z^*=Z_1$ 5354.9 2323.3 0.07 0.128 2.67 2.67 4016.2 1037.7 8990.8 2677.5 367.2 5158 2331.9 0.09 0.128 2.579 2.579 3868.5 1061 9046.93 2579 375.7 4987 2336 0.11 0.128 2.49 2.49 3740 1082 9098.927 2493 383 4838 2339 0.13 0.128 2.41 2.41 3628 1101 9147.354 2149 392
 $I_c$ $T_{Min}$ $T$ $T^*$ $Q_i$ $B_i$ $Z^*=Z_1$ 5354.9 2323.3 0.07 0.128 2.67 2.67 4016.2 1037.7 8990.8 2677.5 367.2 5158 2331.9 0.09 0.128 2.579 2.579 3868.5 1061 9046.93 2579 375.7 4987 2336 0.11 0.128 2.49 2.49 3740 1082 9098.927 2493 383 4838 2339 0.13 0.128 2.41 2.41 3628 1101 9147.354 2149 392
Optimal solutions for different values of $I_e$
 $I_e$ $T_{Min}$ $T$ $T^*$ $Q_i$ $B_i$ $Z^*=Z_1$ 5159 2332 0.03 0.128 2.579 2.579 3869 1061 9047.469 2579 375 5158 2331.9 0.05 0.128 2.579 2.579 3868.5 1061 9046.93 2579 375.7 5156 2330 0.05 0.128 2.57 2.57 3867 1059 9046.39 2578.6 374 5154 2329 0.06 0.128 2.56 2.56 3865 1058 9045.851 2577 372
 $I_e$ $T_{Min}$ $T$ $T^*$ $Q_i$ $B_i$ $Z^*=Z_1$ 5159 2332 0.03 0.128 2.579 2.579 3869 1061 9047.469 2579 375 5158 2331.9 0.05 0.128 2.579 2.579 3868.5 1061 9046.93 2579 375.7 5156 2330 0.05 0.128 2.57 2.57 3867 1059 9046.39 2578.6 374 5154 2329 0.06 0.128 2.56 2.56 3865 1058 9045.851 2577 372
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