# American Institute of Mathematical Sciences

January  2012, 8(1): 263-269. doi: 10.3934/jimo.2012.8.263

## Using the algebraic approach to determine the replenishment optimal policy with defective products, backlog and delay of payments in the supply chain management

 1 School of Economics and Management, Southwest Jiaotong University, Chengdu, Sichuan 610031 2 College of Business, Chung Yuan Christian University, Chung Li 32023, Taiwan

Received  October 2009 Revised  October 2011 Published  November 2011

Li et al. [Journal of Industrial and Management Optimization 5 (2009) 867-880] develop a model to determine an optimal replenishment policy with defective items and shortage backlogging under conditions of permissible delay in payments. They used the optimization technique based on differential calculus to determine an optimal replenishment policy. The main purpose of this paper is to solve the inventory problem in Li et al. (2009) by the algebraic method to simplify the solution procedures.
Citation: Jun Li, Hairong Feng, Kun-Jen Chung. Using the algebraic approach to determine the replenishment optimal policy with defective products, backlog and delay of payments in the supply chain management. Journal of Industrial and Management Optimization, 2012, 8 (1) : 263-269. doi: 10.3934/jimo.2012.8.263
##### References:
 [1] L. E. Cárdenas-Barrón, Observation on: Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 67 (2000), 201-201. doi: 10.1016/S0925-5273(00)00059-1. [2] L. E. Cárdenas-Barrón, 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. [3] B. E. Castello and A. J. Goldman, EOQ rides again!, Perspectives in Operations Research, 36 (2006), 307-331. doi: 10.1007/978-0-387-39934-8_18. [4] W. M. Chan, R. N. Ibrahim and P. B Lochert, A new EPQ model: Integrating lower pricing rework and reject situations, Production Planning and Control, 14 (2003), 588-595. doi: 10.1080/09537280310001626179. [5] H.-C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality items, Computers and Operations Research, 31 (2004), 2079-2092. doi: 10.1016/S0305-0548(03)00166-7. [6] H. C. Chang and C. H. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backlogging, Omega, 38 (2010), 233-237. doi: 10.1016/j.omega.2009.09.006. [7] S. K. J. Chang, J. P. C. Chuang and H. J. Chen, Short comments on technical note-the EOQ and EPQ models with shortages derived without derivatives, International Journal of Production Economics, 97 (2005), 241-243. doi: 10.1016/j.ijpe.2004.07.002. [8] L. H. Chen and F. S. Kang., Coordination between vendor and buyer considering trade credit and items with imperfect quality, International Journal of Production Economics, 123 (2010), 52-61. [9] K.-J. Chung, C. C. Her and S. D. Lin, A two-warehouse inventory model with imperfect quality production processes, Computers and Industrial Engineering, 56 (2009), 193-197. doi: 10.1016/j.cie.2008.05.005. [10] K.-J. Chung and C.-K. Huang, An ordering policy with allowable shortage and permissible delay in payments, Applied Mathematical Modelling, 33 (2009), 2518-2525. doi: 10.1016/j.apm.2008.07.016. [11] K.-J. Chung and Y. F. Huang, Retailer's optimal cycle times in the EOQ model with imperfect quality and a permissible credit period, Quality and Quantity, 40 (2006), 59-77. doi: 10.1007/s11135-005-5356-z. [12] A. Eroglu and G. Ozdemir, An economic order quantity model with defective items and shortages, International Journal of Production Economics, 106 (2007), 544-549. doi: 10.1016/j.ijpe.2006.06.015. [13] S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 36 (1985), 335-338. [14] S. K. Goyal and L. E. Cárdenas-Barrón, 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. [15] R. W. Grubbstrom and A. Erdem, EOQ with backlogging derived without derivatives, International Journal of Production Economics, 59 (1999), 529-530. doi: 10.1016/S0925-5273(98)00015-2. [16] M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of extensions of a modified EOQ model for imperfect quality items, International Journal of Production Economics, 132 (2011), 1-12. doi: 10.1016/j.ijpe.2011.03.009. [17] V. B. Kreng and S. J. Tan, Optimal replenishment decision in an EPQ model with defective items under supply chain trade credit policy, Expert Systems with Applications, 38 (2011), 9888-9899. doi: 10.1016/j.eswa.2011.02.040. [18] J. Li, H. Feng and M. Wang, A replenishment policy with defective products, backlog and delay of payments, Journal of Industrial and Management Optimization, 5 (2009), 867-880. doi: 10.3934/jimo.2009.5.867. [19] B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quality: Revisited, International Journal of Production Economics, 112 (2008), 808-815. doi: 10.1016/j.ijpe.2007.07.003. [20] Y. H. Oh and H. Hwang, Deterministic inventory model for recycling system, Journal of Intelligent Manufacturing, 17 (2006), 423-428. doi: 10.1007/s10845-005-0015-8. [21] R. Ronald, G. K. Yang and P. Chu, Technical note: The EOQ and EPQ models with shortages derived without derivatives, International Journal of Production Economics, 92 (2004), 197-200. doi: 10.1016/j.ijpe.2003.10.013. [22] M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. [23] J.-T. Teng, A simple method to compute economic order quantities, European Journal of Operational Research, 198 (2009), 351-353. doi: 10.1016/j.ejor.2008.05.019. [24] D. Varberg, E. J. Purcell and S. E. Steven, "Calculus," 9th edition, Pearson Education, Inc., Upper Saddle River, NJ, 07458, 2007. [25] H. M. Wee, J. Yu and M. C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega, 35 (2007), 7-11. doi: 10.1016/j.omega.2005.01.019.

show all references

##### References:
 [1] L. E. Cárdenas-Barrón, Observation on: Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 67 (2000), 201-201. doi: 10.1016/S0925-5273(00)00059-1. [2] L. E. Cárdenas-Barrón, 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. [3] B. E. Castello and A. J. Goldman, EOQ rides again!, Perspectives in Operations Research, 36 (2006), 307-331. doi: 10.1007/978-0-387-39934-8_18. [4] W. M. Chan, R. N. Ibrahim and P. B Lochert, A new EPQ model: Integrating lower pricing rework and reject situations, Production Planning and Control, 14 (2003), 588-595. doi: 10.1080/09537280310001626179. [5] H.-C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality items, Computers and Operations Research, 31 (2004), 2079-2092. doi: 10.1016/S0305-0548(03)00166-7. [6] H. C. Chang and C. H. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backlogging, Omega, 38 (2010), 233-237. doi: 10.1016/j.omega.2009.09.006. [7] S. K. J. Chang, J. P. C. Chuang and H. J. Chen, Short comments on technical note-the EOQ and EPQ models with shortages derived without derivatives, International Journal of Production Economics, 97 (2005), 241-243. doi: 10.1016/j.ijpe.2004.07.002. [8] L. H. Chen and F. S. Kang., Coordination between vendor and buyer considering trade credit and items with imperfect quality, International Journal of Production Economics, 123 (2010), 52-61. [9] K.-J. Chung, C. C. Her and S. D. Lin, A two-warehouse inventory model with imperfect quality production processes, Computers and Industrial Engineering, 56 (2009), 193-197. doi: 10.1016/j.cie.2008.05.005. [10] K.-J. Chung and C.-K. Huang, An ordering policy with allowable shortage and permissible delay in payments, Applied Mathematical Modelling, 33 (2009), 2518-2525. doi: 10.1016/j.apm.2008.07.016. [11] K.-J. Chung and Y. F. Huang, Retailer's optimal cycle times in the EOQ model with imperfect quality and a permissible credit period, Quality and Quantity, 40 (2006), 59-77. doi: 10.1007/s11135-005-5356-z. [12] A. Eroglu and G. Ozdemir, An economic order quantity model with defective items and shortages, International Journal of Production Economics, 106 (2007), 544-549. doi: 10.1016/j.ijpe.2006.06.015. [13] S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 36 (1985), 335-338. [14] S. K. Goyal and L. E. Cárdenas-Barrón, 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. [15] R. W. Grubbstrom and A. Erdem, EOQ with backlogging derived without derivatives, International Journal of Production Economics, 59 (1999), 529-530. doi: 10.1016/S0925-5273(98)00015-2. [16] M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of extensions of a modified EOQ model for imperfect quality items, International Journal of Production Economics, 132 (2011), 1-12. doi: 10.1016/j.ijpe.2011.03.009. [17] V. B. Kreng and S. J. Tan, Optimal replenishment decision in an EPQ model with defective items under supply chain trade credit policy, Expert Systems with Applications, 38 (2011), 9888-9899. doi: 10.1016/j.eswa.2011.02.040. [18] J. Li, H. Feng and M. Wang, A replenishment policy with defective products, backlog and delay of payments, Journal of Industrial and Management Optimization, 5 (2009), 867-880. doi: 10.3934/jimo.2009.5.867. [19] B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quality: Revisited, International Journal of Production Economics, 112 (2008), 808-815. doi: 10.1016/j.ijpe.2007.07.003. [20] Y. H. Oh and H. Hwang, Deterministic inventory model for recycling system, Journal of Intelligent Manufacturing, 17 (2006), 423-428. doi: 10.1007/s10845-005-0015-8. [21] R. Ronald, G. K. Yang and P. Chu, Technical note: The EOQ and EPQ models with shortages derived without derivatives, International Journal of Production Economics, 92 (2004), 197-200. doi: 10.1016/j.ijpe.2003.10.013. [22] M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. [23] J.-T. Teng, A simple method to compute economic order quantities, European Journal of Operational Research, 198 (2009), 351-353. doi: 10.1016/j.ejor.2008.05.019. [24] D. Varberg, E. J. Purcell and S. E. Steven, "Calculus," 9th edition, Pearson Education, Inc., Upper Saddle River, NJ, 07458, 2007. [25] H. M. Wee, J. Yu and M. C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega, 35 (2007), 7-11. doi: 10.1016/j.omega.2005.01.019.

2020 Impact Factor: 1.801