# American Institute of Mathematical Sciences

October  2016, 12(4): 1333-1347. doi: 10.3934/jimo.2016.12.1333

## An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest

 1 Department of Marketing and Supply Chain Management, Overseas Chinese University, Taichung 40721, Taiwan 2 Department of Business Administration, National Chung Cheng University, Chia-Yi 621, Taiwan 3 Department of Industrial Engineering and Management, Overseas Chinese University, Taichung 40721, Taiwan

Received  October 2014 Revised  October 2015 Published  January 2016

In this paper we develop a new inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest. Three highlights are included in this new model: (1) interest is earned by depositing the sales revenue from the perfect and imperfect items into an interest-bearing account (2) the screening rate may not be given but is a decision variable (3) the supplier offers quantity discounts to trigger the retailer into ordering greater lot sizes. This scenario has not been discussed in previous EOQ models with imperfect quality. Our model could determine two decision variables, order quantity and screening rate, to maximize retailer profit. The expected total profit function is derived with two special cases explored to validate the proposed model. An algorithm is developed to help the manager determine the optimal order quantity and screening rate. A numerical example is given to illustrate the proposed model and algorithm. Sensitivity analyses are carried out to investigate the model parameters effects on the optimal solution. Managerial insights are also included.
Citation: Tien-Yu Lin, Ming-Te Chen, Kuo-Lung Hou. An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1333-1347. doi: 10.3934/jimo.2016.12.1333
##### References:
 [1] W. C. Benton and S. Park, A classification of literature on determining the lot size under quantity discounts, Eur. J. Oper. Res., 92 (1996), 219-238. doi: 10.1016/0377-2217(95)00315-0. [2] T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-depend demand under quantity and freight discounts, Int. J. Prod. Econ., 48 (1997), 141-155. [3] L. E. Cárdenas-Barrón, Observation on: Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 67 (2000), 201. [4] L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Comput. Ind. Eng., 57 (2009), 1105-1113. [5] W.-M. Chan, R. N. Ibrahim and P. B. Lochert, A new EPQ model: Integrating lower pricing rework and reject situations, Prod. Plan. Control, 14 (2003), 588-595. doi: 10.1080/09537280310001626179. [6] H. -C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality, Comput. Oper. Res., 31 (2004), 2079-2092. doi: 10.1016/S0305-0548(03)00166-7. [7] H.-C. Chang and C.-H. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering, Omega, 38 (2010), 233-237. doi: 10.1016/j.omega.2009.09.006. [8] A. Eroglu and A. Ozdemir, An economic order quantity model with defective items and shortages, Int. J. Prod. Econ., 106 (2007), 544-549. doi: 10.1016/j.ijpe.2006.06.015. [9] S. K. Goyal and L. E. Cárdenas-Barrón, Note on: Economic production quantity model for items with imperfect quality-a practical approach, Int. J. Prod. Econ., 77 (2002), 85-87. doi: 10.1016/S0925-5273(01)00203-1. [10] J. T. Hsu and L. F. Hsu, An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns, Int. J. Prod. Econ., 143 (2013), 162-170. [11] W. K. Hsu and H. F. Yu, An EOQ model with imperfective quality items under an announced price increase, J. Chinese Ins. Ind. Eng., 28 (2011), 34-44. doi: 10.1080/10170669.2010.532347. [12] C.-K. Huang, An optimal policy for a single-vendor single buyer integrated production-inventory problem with process unreliability consideration, Int. J. Prod. Econ., 91 (2004), 91-98. doi: 10.1016/S0925-5273(03)00220-2. [13] M. Y. Jaber, S. K. Goyal and M. Imran, Economic production quantity model for items with imperfect quality subject to learning effects, Int. J. Prod. Econ., 115 (2008), 143-150. doi: 10.1016/j.ijpe.2008.05.007. [14] M. Y. Jaber, S. Zanoni and L. E. Zavanella, An entropic economic order quantity (EnEOQ) for items with imperfect quality, Appl. Math. Modelling, 37 (2013), 3982-3992. doi: 10.1016/j.apm.2012.07.046. [15] X. Ji and Z. Shao, Model and algorithm for bilevel newsboy problem with fuzzy demands and discounts, Applied Math. Comput., 172 (2006), 163-174. doi: 10.1016/j.amc.2005.01.139. [16] M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items, Int. J. Prod. Econ., 132 (2011), 1-12. doi: 10.1016/j.ijpe.2011.03.009. [17] M. Khan, M. Jaber and M. Wahab, Economic order quantity model for items with imperfect quality with learning in inspection, Int. J. Prod. Econ., 124 (2010), 87-96. doi: 10.1016/j.ijpe.2009.10.011. [18] T. -Y. Lin and K. -L. Hou, An imperfect quality economic order quantity with advanced receiving, TOP, 23 (2015), 535-551. doi: 10.1007/s11750-014-0352-x. [19] T. -Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Applied Math. Model., 34 (2010), 3158-3165. doi: 10.1016/j.apm.2010.02.004. [20] B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quantity: Revisited, Int. J. Prod. Econ., 112 (2008), 808-815. [21] B. Maddah, M. K. Salameh and C. H. Karame, Lot sizing with random yield and different qualities, Appl. Math. Modelling, 33 (2010), 1997-2009. doi: 10.1016/j.apm.2008.05.009. [22] B. Maddah, M. K. Salameh and L. Moussawi-Haidar, Order overlapping: A practical approach for preventing shortages during screening, Comput. Ind. Eng., 58 (2010), 691-695. doi: 10.1016/j.cie.2010.01.014. [23] G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables, Comput. Ind. Eng., 64 (2013), 190-199. doi: 10.1016/j.cie.2012.09.003. [24] A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs, Int. J. Prod. Econ., 113 (2008), 754-765. doi: 10.1016/j.ijpe.2007.10.010. [25] J. P. Monahan, A quantity discount pricing model to increase vendor profits, Manage. Sci., 30 (1984), 720-726. doi: 10.1287/mnsc.30.6.720. [26] J. Paknejad, F. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times, Int. J. Prod. Res., 33 (2005), 2767-2777. doi: 10.1155/JAMDS.2005.177. [27] S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ., 100 (2006), 148-154. doi: 10.1016/j.ijpe.2004.11.004. [28] H. Qin, M. Luo, X. Gao and A. Lim, The freight allocation problem with all-units quantity-based discount: A heuristic algorithm, Omega, 40 (2012), 415-423. doi: 10.1016/j.omega.2011.05.005. [29] S. M. Ross, Introduction to Probability Models, Academic Press, New York, 1993. [30] M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. [31] W. Shih, Optimal inventory policies when stockouts result from defective products, Int. J. Prod. Res., 18 (1980), 677-685. doi: 10.1080/00207548008919699. [32] E. A. Silver, Establishing the reorder quantity when amount received is uncertain, INFOR, 14 (1976), 32-39. [33] J. F. Tsai, An optimization approach for supply chain management models with quantity discount policy, Eur. J. Oper. Res., 177 (2007), 982-994. doi: 10.1016/j.ejor.2006.01.034. [34] J. Vörös, Economic order and production quantity models without constraint on the percentage of defective items, Cent. Eur. J. Oper. Res., 21 (2013), 867-885. doi: 10.1007/s10100-012-0277-0. [35] M. I. M. Wahab and M. Y. Jaber, Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note, Comput. Ind. Eng., 58 (2010), 186-190. doi: 10.1016/j.cie.2009.07.007. [36] 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. [37] H. F. Yu, W. K. Hsu and W. J. Chang, EOQ model where a portion of the defectives can be used as perfect quality, Int. J. Syst. Sci., 43 (2012), 1689-1698. doi: 10.1080/00207721.2010.549593.

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##### References:
 [1] W. C. Benton and S. Park, A classification of literature on determining the lot size under quantity discounts, Eur. J. Oper. Res., 92 (1996), 219-238. doi: 10.1016/0377-2217(95)00315-0. [2] T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-depend demand under quantity and freight discounts, Int. J. Prod. Econ., 48 (1997), 141-155. [3] L. E. Cárdenas-Barrón, Observation on: Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 67 (2000), 201. [4] L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Comput. Ind. Eng., 57 (2009), 1105-1113. [5] W.-M. Chan, R. N. Ibrahim and P. B. Lochert, A new EPQ model: Integrating lower pricing rework and reject situations, Prod. Plan. Control, 14 (2003), 588-595. doi: 10.1080/09537280310001626179. [6] H. -C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality, Comput. Oper. Res., 31 (2004), 2079-2092. doi: 10.1016/S0305-0548(03)00166-7. [7] H.-C. Chang and C.-H. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering, Omega, 38 (2010), 233-237. doi: 10.1016/j.omega.2009.09.006. [8] A. Eroglu and A. Ozdemir, An economic order quantity model with defective items and shortages, Int. J. Prod. Econ., 106 (2007), 544-549. doi: 10.1016/j.ijpe.2006.06.015. [9] S. K. Goyal and L. E. Cárdenas-Barrón, Note on: Economic production quantity model for items with imperfect quality-a practical approach, Int. J. Prod. Econ., 77 (2002), 85-87. doi: 10.1016/S0925-5273(01)00203-1. [10] J. T. Hsu and L. F. Hsu, An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns, Int. J. Prod. Econ., 143 (2013), 162-170. [11] W. K. Hsu and H. F. Yu, An EOQ model with imperfective quality items under an announced price increase, J. Chinese Ins. Ind. Eng., 28 (2011), 34-44. doi: 10.1080/10170669.2010.532347. [12] C.-K. Huang, An optimal policy for a single-vendor single buyer integrated production-inventory problem with process unreliability consideration, Int. J. Prod. Econ., 91 (2004), 91-98. doi: 10.1016/S0925-5273(03)00220-2. [13] M. Y. Jaber, S. K. Goyal and M. Imran, Economic production quantity model for items with imperfect quality subject to learning effects, Int. J. Prod. Econ., 115 (2008), 143-150. doi: 10.1016/j.ijpe.2008.05.007. [14] M. Y. Jaber, S. Zanoni and L. E. Zavanella, An entropic economic order quantity (EnEOQ) for items with imperfect quality, Appl. Math. Modelling, 37 (2013), 3982-3992. doi: 10.1016/j.apm.2012.07.046. [15] X. Ji and Z. Shao, Model and algorithm for bilevel newsboy problem with fuzzy demands and discounts, Applied Math. Comput., 172 (2006), 163-174. doi: 10.1016/j.amc.2005.01.139. [16] M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items, Int. J. Prod. Econ., 132 (2011), 1-12. doi: 10.1016/j.ijpe.2011.03.009. [17] M. Khan, M. Jaber and M. Wahab, Economic order quantity model for items with imperfect quality with learning in inspection, Int. J. Prod. Econ., 124 (2010), 87-96. doi: 10.1016/j.ijpe.2009.10.011. [18] T. -Y. Lin and K. -L. Hou, An imperfect quality economic order quantity with advanced receiving, TOP, 23 (2015), 535-551. doi: 10.1007/s11750-014-0352-x. [19] T. -Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Applied Math. Model., 34 (2010), 3158-3165. doi: 10.1016/j.apm.2010.02.004. [20] B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quantity: Revisited, Int. J. Prod. Econ., 112 (2008), 808-815. [21] B. Maddah, M. K. Salameh and C. H. Karame, Lot sizing with random yield and different qualities, Appl. Math. Modelling, 33 (2010), 1997-2009. doi: 10.1016/j.apm.2008.05.009. [22] B. Maddah, M. K. Salameh and L. Moussawi-Haidar, Order overlapping: A practical approach for preventing shortages during screening, Comput. Ind. Eng., 58 (2010), 691-695. doi: 10.1016/j.cie.2010.01.014. [23] G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables, Comput. Ind. Eng., 64 (2013), 190-199. doi: 10.1016/j.cie.2012.09.003. [24] A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs, Int. J. Prod. Econ., 113 (2008), 754-765. doi: 10.1016/j.ijpe.2007.10.010. [25] J. P. Monahan, A quantity discount pricing model to increase vendor profits, Manage. Sci., 30 (1984), 720-726. doi: 10.1287/mnsc.30.6.720. [26] J. Paknejad, F. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times, Int. J. Prod. Res., 33 (2005), 2767-2777. doi: 10.1155/JAMDS.2005.177. [27] S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ., 100 (2006), 148-154. doi: 10.1016/j.ijpe.2004.11.004. [28] H. Qin, M. Luo, X. Gao and A. Lim, The freight allocation problem with all-units quantity-based discount: A heuristic algorithm, Omega, 40 (2012), 415-423. doi: 10.1016/j.omega.2011.05.005. [29] S. M. Ross, Introduction to Probability Models, Academic Press, New York, 1993. [30] M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. [31] W. Shih, Optimal inventory policies when stockouts result from defective products, Int. J. Prod. Res., 18 (1980), 677-685. doi: 10.1080/00207548008919699. [32] E. A. Silver, Establishing the reorder quantity when amount received is uncertain, INFOR, 14 (1976), 32-39. [33] J. F. Tsai, An optimization approach for supply chain management models with quantity discount policy, Eur. J. Oper. Res., 177 (2007), 982-994. doi: 10.1016/j.ejor.2006.01.034. [34] J. Vörös, Economic order and production quantity models without constraint on the percentage of defective items, Cent. Eur. J. Oper. Res., 21 (2013), 867-885. doi: 10.1007/s10100-012-0277-0. [35] M. I. M. Wahab and M. Y. Jaber, Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note, Comput. Ind. Eng., 58 (2010), 186-190. doi: 10.1016/j.cie.2009.07.007. [36] 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. [37] H. F. Yu, W. K. Hsu and W. J. Chang, EOQ model where a portion of the defectives can be used as perfect quality, Int. J. Syst. Sci., 43 (2012), 1689-1698. doi: 10.1080/00207721.2010.549593.

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