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May  2021, 17(3): 1101-1118. doi: 10.3934/jimo.2020013

## Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages

 Economics and Management College, Zhaoqing University, Zhaoqing City 526061, Guangdong Province, China

* Corresponding author: Tien-Yu Lin

Received  December 2018 Revised  August 2019 Published  May 2021 Early access  January 2020

This paper explores the retailer's optimal lot sizing and quantity backordering for a deteriorating production system with a two-state Markov production process in which quantity discounts are provided by the supplier. The products are sold with the policy of free reasonable repair warranty employing the fraction of nonconforming items in a lot size. Unlike the traditional economic production quantity (EPQ) model with warranty policy based on the elapsed time of the system in the control state follows an exponential distribution, this paper not only constructs an alternative mathematical model for EPQ model based on the fraction of nonconforming items in a lot size for an imperfect production system but also extends the topics of optimal quantity and shortage to a wider scope of academic research and further finds that some results are different from the traditional EPQ models. We seek to minimize the expected total relevant cost through optimal lot sizing and quantity backordering. We also demonstrate that the optimal lot size is bounded in a finite interval. An efficient algorithm is developed to determine the optimal solution. Moreover, a numerical example is given and sensitivity analysis is conducted to highlight management insights.

Citation: Tien-Yu Lin. Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1101-1118. doi: 10.3934/jimo.2020013
##### References:
 [1] A. Y. Alqahtani, S. M. Gupta and K. Nakashima, Warranty and maintenance analysis of sensor embedded products using internet of things in industry 4.0, Int. J. Prod. Econ., 208 (2019), 483-499.  doi: 10.1016/j.ijpe.2018.12.022. [2] Y. Barron and D. Hermel, Shortage decision policies for a fluid production model with MAP arrivals, Int. J. Prod. Res., 55 (2017), 3946-3969.  doi: 10.1080/00207543.2016.1218083. [3] W. R. Blischke and D. N. P. Murthy, Product warranty management Ⅲ: A review and mathematical models, Eur. Oper. Res., 62 (1992), 1-34. [4] S. Chand, Lot sizes and setup frequency with learning in setups and process quality, Eur. J. Oper. Res., 42 (1989), 190-202.  doi: 10.1016/0377-2217(89)90321-4. [5] C.-K. Chen and C.-C. Lo, Optimal production run length for products sold with warranty in an imperfect production system with allowable shortages, Math. Comput. Model., 44 (2006), 319-331.  doi: 10.1016/j.mcm.2006.01.019. [6] Y.-H. Chien, Z. G. Zhang and X. L. Yin, On optimal preventive-maintenance policy for generalized Polya process repairable products under free-repair warranty, Eur. J. Oper. Res., 279 (2019), 68-78.  doi: 10.1016/j.ejor.2019.03.042. [7] K.-J. Chung and K.-L. Hou, An optimal production run time with imperfect production processes and allowable shortages, Comput. Oper. Res., 30 (2003), 483-490.  doi: 10.1016/S0305-0548(01)00091-0. [8] A. Eroglu and G. 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] P. A. Hayek and M. K. Salameh, Production lot sizing with the reworking of imperfect quality items produced, Prod. Plan. Control, 12 (2001), 584-590.  doi: 10.1080/095372801750397707. [10] K.-L. Hou, Optimal production run length for deteriorating production system with a two-state continuous-time Markovian processes under allowable shortages, J. Oper. Res. Soc., 56 (2005), 346-350.  doi: 10.1057/palgrave.jors.2601792. [11] K.-L. Hou, L.-C. Lin and T.-Y. Lin, Optimal lot sizing with maintenance actions and imperfect production processes, Int. J. Syst. Sci., 46 (2015), 2749-2755.  doi: 10.1080/00207721.2013.879229. [12] B. Huang and A. Wu, Reduce shortage with self-reservation policy for a manufacturer paying both fixed and variable stockout expenditure, Eur. J. Oper. Res., 262 (2017), 944-953.  doi: 10.1016/j.ejor.2017.03.063. [13] M. Y. Jaber, M. Bonney and I. Moualek, An economic order quantity model for an imperfect production process with entropy cost, Int. J. Prod. Econ., 118 (2009), 26-33.  doi: 10.1016/j.ijpe.2008.08.007. [14] M. Y. Jaber, S. Zanoni and L. E. Zavanella, Economic order quantity models for imperfect items with buy and repair options, Int. J. Prod. Econ., 155 (2014), 126-131.  doi: 10.1016/j.ijpe.2013.10.014. [15] 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. [16] R. S. Kumar and A. Goswami, EPQ model with learning consideration, imperfect production and partial backlogging in fuzzy random environment, Int. J. Syst. Sci., 46 (2015), 1486-1497. [17] H. Lee, J. H. Cha and M. Finkelstein, On information-based warranty policy for repairable products from heterogeneous population, Eur. J. Oper. Res., 253 (2016), 204-215.  doi: 10.1016/j.ejor.2016.02.020. [18] J. S. Lee and K. S. Park, Joint determination of production cycle and inspection intervals in a deteriorating production system, J. Oper. Res. Soc., 42 (1991), 775-783. [19] T.-Y. Lin, Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality, Comput. Ind. Eng., 66 (2013), 53-62.  doi: 10.1016/j.cie.2013.06.012. [20] B. Liu, J. Wu and M. Xie, Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty, Eur. J. Oper. Res., 243 (2015), 874-882.  doi: 10.1016/j.ejor.2015.01.030. [21] M. Luo and S. M. Wu, A comprehensive analysis of warranty claims and optimal policies, Eur. Oper. Res., 276 (2019), 144-159.  doi: 10.1016/j.ejor.2018.12.034. [22] B. Maddah, L. Moussawi and M. Y. Jaber, Lot sizing with a Markov production process and imperfect items scrapped, Int. J. Prod. Econ., 124 (2010), 340-347.  doi: 10.1016/j.ijpe.2009.11.029. [23] V. Makis, Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections, Nav. Res. Log., 45 (1998), 165-186.  doi: 10.1002/(SICI)1520-6750(199803)45:2<165::AID-NAV3>3.0.CO;2-6. [24] L. Moussawi-Haidar, M. Salameh and W. Nasr, Production lot sizing with quality screening and rework, Appl. Math. Model., 40 (2016), 3242-3256.  doi: 10.1016/j.apm.2015.09.095. [25] D. N. P. Murthy and W. R. Blischke, Product warranty managemen-Ⅱ: An integrated framework for study, Eur. J. Oper. Res., 62 (1992), 261-281.  doi: 10.1016/0377-2217(92)90117-R. [26] D. N. P. Murthy and W. R. Blischke, Product warranty management-Ⅲ: A review of mathematical models, Eur. J. Oper. Res., 63 (1992), 1-34.  doi: 10.1016/0377-2217(92)90052-B. [27] D. N. P. Murthy and I. Djamaludin, New product warranty: A literature review, Int. J. Prod. Econ., 79 (2002), 231-260.  doi: 10.1016/S0925-5273(02)00153-6. [28] L.-Y. Ouyang and C.-T. Chang, Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging, Int. J. Prod. Econ., 144 (2013), 610-617.  doi: 10.1016/j.ijpe.2013.04.027. [29] B. Pal, S. S. Sana and K. Chaudhuri, Three-layer Supply Chain- a Production- inventory model for reworkable items, Appl. Math. Comput., 219 (2012), 530-543.  doi: 10.1016/j.amc.2012.06.038. [30] B. Pal, S. S. Sana and K. Chaudhuri, Maximizing profits for an EPQ model with unreliable machine and rework of random defective items, Int. J. Syst. Sci., 44 (2013), 582-594.  doi: 10.1080/00207721.2011.617896. [31] B. Pal, S. S. Sana and K. Chaudhuri, A mathematical model on EPQ for stochastic demand in an imperfect production system, J. Manuf. Sys, 32 (2013), 260-270.  doi: 10.1016/j.jmsy.2012.11.009. [32] E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Oper. Res., 34 (1986), 137-144.  doi: 10.1287/opre.34.1.137. [33] M. J. Rosenblatt and H. L. Lee, Economic production cycle with imperfect production processes, IIE Trans., 18 (1986), 48-55.  doi: 10.1080/07408178608975329. [34] S. S. Sana, An economic production lot size model in an imperfect production system, Eur. J. Oper. Res., 201 (2010), 158-170. [35] L. A. San-José, J. Sicilia and J. García-Laguna, Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost, Omega, 54 (2015), 147-157. [36] B. Sarkar, An inventory model with reliability in an imperfect production process, Appl. Math. Comput., 218 (2012), 4881-4891.  doi: 10.1016/j.amc.2011.10.053. [37] B. Sarkar, L. E. Cárdenas-Barrón, 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, J. Manuf. Syst., 33 (2014), 423-435.  doi: 10.1016/j.jmsy.2014.02.001. [38] B. Sarkar, S. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Ann. Oper. Res., 229 (2015), 677-702.  doi: 10.1007/s10479-014-1745-9. [39] E. W. Taft, The most economical production lot, The Iron Age, 101 (1918), 1410-1412. [40] A. H. Tai, Economic production quantity models for deteriorating/imperfect products and service with rework, Comput. Ind. Eng., 66 (2013), 879-888.  doi: 10.1016/j.cie.2013.09.007. [41] A. A. Taleizadeh, L. E. Cárdenas-Barrón and B. Mohammadi, A deterministic multi product single machine EPQ model with backordering, scraped products, rework and interruption in manufacturing process, Int. J. Prod. Econ., 150 (2014), 9-27.  doi: 10.1016/j.ijpe.2013.11.023. [42] A. A. Taleizadeh, S. S. Kalantari and L. E. Cárdenas-Barrón, Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments, Top, 24 (2016), 143-155.  doi: 10.1007/s11750-015-0377-9. [43] A. A. Taleizadeh, H. R. Zarei and B. R. Sarker, An optimal control of inventory under probabilistic replenishment intervals and known price increase, Eur. Oper. Res., 257 (2017), 777-791.  doi: 10.1016/j.ejor.2016.07.041. [44] C. S. Tapiero, P. H. Ritchken and A. Reisman, Reliability, pricing and quality control, Eur. J. Oper. Res., 31 (1987), 37-45.  doi: 10.1016/0377-2217(87)90134-2. [45] B. Van Beek and C. Van Putten, OR contributions to flexibility improvement in production/inventory systems, Eur. J. Oper. Res., 31 (1987), 52-60. [46] M. van der Heijden and B. P. Iskandar, Last time buy decisions for products sold under warranty, Eur. J. Oper. Res., 224 (2013), 302-312.  doi: 10.1016/j.ejor.2012.07.041. [47] C.-H. Wang, The impact of free-repair warranty policy on EMQ model for imperfect production systems, Comput. Oper. Res., 31 (2004), 2021-2035.  doi: 10.1016/S0305-0548(03)00161-8. [48] C.-H. Wang and S.-H. Sheu, Optimal lot sizing for products sold under free-repair warranty, Eur. J. Oper. Res., 149 (2003), 131-141.  doi: 10.1016/S0377-2217(02)00429-0. [49] C. M. Wright and A. Mehrez, An overview of representative research of the relationships between quality and inventory, Omega, 26 (1998), 29-47.  doi: 10.1016/S0305-0483(97)00042-X. [50] C. A. Yano and H. L. Lee, Lot sizing with random yields: A review, Oper. Res., 43 (1995), 311-334.  doi: 10.1287/opre.43.2.311. [51] R. H. Yeh, M. Y. Chen and C. Y. Lin, Optimal periodic replacement policy for repairable products under free-repair warranty, Eur. Oper. Res., 176 (2007), 1678-1686.  doi: 10.1016/j.ejor.2005.10.047. [52] R. H. Yeh, W. T. Ho and S. T. Tseng, Optimal production run length for products sold with warranty, Eur. J. Oper. Res., 120 (2000), 575-582.  doi: 10.1016/S0377-2217(99)00004-1. [53] S. H. Yoo, D. S. Kim and M. S. Park, Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return, Int. J. Prod. Econ., 140 (2012), 922-933.  doi: 10.1016/j.ijpe.2012.07.014. [54] X. Zhang and Y. Gerchak, Joint lot sizing and inspection policy in an EOQ model with random yield, IIE Trans., 22 (1990), 41-47.  doi: 10.1080/07408179008964156. [55] Y.-W. Zhou, J. Y. Chen, Y. Z. Wu and W. H. Zhou, EPQ models for items with imperfect quality and one-time-only discount, Appl. Math. Model., 39 (2015), 1000-1018.  doi: 10.1016/j.apm.2014.07.017.

show all references

##### References:
 [1] A. Y. Alqahtani, S. M. Gupta and K. Nakashima, Warranty and maintenance analysis of sensor embedded products using internet of things in industry 4.0, Int. J. Prod. Econ., 208 (2019), 483-499.  doi: 10.1016/j.ijpe.2018.12.022. [2] Y. Barron and D. Hermel, Shortage decision policies for a fluid production model with MAP arrivals, Int. J. Prod. Res., 55 (2017), 3946-3969.  doi: 10.1080/00207543.2016.1218083. [3] W. R. Blischke and D. N. P. Murthy, Product warranty management Ⅲ: A review and mathematical models, Eur. Oper. Res., 62 (1992), 1-34. [4] S. Chand, Lot sizes and setup frequency with learning in setups and process quality, Eur. J. Oper. Res., 42 (1989), 190-202.  doi: 10.1016/0377-2217(89)90321-4. [5] C.-K. Chen and C.-C. Lo, Optimal production run length for products sold with warranty in an imperfect production system with allowable shortages, Math. Comput. Model., 44 (2006), 319-331.  doi: 10.1016/j.mcm.2006.01.019. [6] Y.-H. Chien, Z. G. Zhang and X. L. Yin, On optimal preventive-maintenance policy for generalized Polya process repairable products under free-repair warranty, Eur. J. Oper. Res., 279 (2019), 68-78.  doi: 10.1016/j.ejor.2019.03.042. [7] K.-J. Chung and K.-L. Hou, An optimal production run time with imperfect production processes and allowable shortages, Comput. Oper. Res., 30 (2003), 483-490.  doi: 10.1016/S0305-0548(01)00091-0. [8] A. Eroglu and G. 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] P. A. Hayek and M. K. Salameh, Production lot sizing with the reworking of imperfect quality items produced, Prod. Plan. Control, 12 (2001), 584-590.  doi: 10.1080/095372801750397707. [10] K.-L. Hou, Optimal production run length for deteriorating production system with a two-state continuous-time Markovian processes under allowable shortages, J. Oper. Res. Soc., 56 (2005), 346-350.  doi: 10.1057/palgrave.jors.2601792. [11] K.-L. Hou, L.-C. Lin and T.-Y. Lin, Optimal lot sizing with maintenance actions and imperfect production processes, Int. J. Syst. Sci., 46 (2015), 2749-2755.  doi: 10.1080/00207721.2013.879229. [12] B. Huang and A. Wu, Reduce shortage with self-reservation policy for a manufacturer paying both fixed and variable stockout expenditure, Eur. J. Oper. Res., 262 (2017), 944-953.  doi: 10.1016/j.ejor.2017.03.063. [13] M. Y. Jaber, M. Bonney and I. Moualek, An economic order quantity model for an imperfect production process with entropy cost, Int. J. Prod. Econ., 118 (2009), 26-33.  doi: 10.1016/j.ijpe.2008.08.007. [14] M. Y. Jaber, S. Zanoni and L. E. Zavanella, Economic order quantity models for imperfect items with buy and repair options, Int. J. Prod. Econ., 155 (2014), 126-131.  doi: 10.1016/j.ijpe.2013.10.014. [15] 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. [16] R. S. Kumar and A. Goswami, EPQ model with learning consideration, imperfect production and partial backlogging in fuzzy random environment, Int. J. Syst. Sci., 46 (2015), 1486-1497. [17] H. Lee, J. H. Cha and M. Finkelstein, On information-based warranty policy for repairable products from heterogeneous population, Eur. J. Oper. Res., 253 (2016), 204-215.  doi: 10.1016/j.ejor.2016.02.020. [18] J. S. Lee and K. S. Park, Joint determination of production cycle and inspection intervals in a deteriorating production system, J. Oper. Res. Soc., 42 (1991), 775-783. [19] T.-Y. Lin, Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality, Comput. Ind. Eng., 66 (2013), 53-62.  doi: 10.1016/j.cie.2013.06.012. [20] B. Liu, J. Wu and M. Xie, Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty, Eur. J. Oper. Res., 243 (2015), 874-882.  doi: 10.1016/j.ejor.2015.01.030. [21] M. Luo and S. M. Wu, A comprehensive analysis of warranty claims and optimal policies, Eur. Oper. Res., 276 (2019), 144-159.  doi: 10.1016/j.ejor.2018.12.034. [22] B. Maddah, L. Moussawi and M. Y. Jaber, Lot sizing with a Markov production process and imperfect items scrapped, Int. J. Prod. Econ., 124 (2010), 340-347.  doi: 10.1016/j.ijpe.2009.11.029. [23] V. Makis, Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections, Nav. Res. Log., 45 (1998), 165-186.  doi: 10.1002/(SICI)1520-6750(199803)45:2<165::AID-NAV3>3.0.CO;2-6. [24] L. Moussawi-Haidar, M. Salameh and W. Nasr, Production lot sizing with quality screening and rework, Appl. Math. Model., 40 (2016), 3242-3256.  doi: 10.1016/j.apm.2015.09.095. [25] D. N. P. Murthy and W. R. Blischke, Product warranty managemen-Ⅱ: An integrated framework for study, Eur. J. Oper. Res., 62 (1992), 261-281.  doi: 10.1016/0377-2217(92)90117-R. [26] D. N. P. Murthy and W. R. Blischke, Product warranty management-Ⅲ: A review of mathematical models, Eur. J. Oper. Res., 63 (1992), 1-34.  doi: 10.1016/0377-2217(92)90052-B. [27] D. N. P. Murthy and I. Djamaludin, New product warranty: A literature review, Int. J. Prod. Econ., 79 (2002), 231-260.  doi: 10.1016/S0925-5273(02)00153-6. [28] L.-Y. Ouyang and C.-T. Chang, Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging, Int. J. Prod. Econ., 144 (2013), 610-617.  doi: 10.1016/j.ijpe.2013.04.027. [29] B. Pal, S. S. Sana and K. Chaudhuri, Three-layer Supply Chain- a Production- inventory model for reworkable items, Appl. Math. Comput., 219 (2012), 530-543.  doi: 10.1016/j.amc.2012.06.038. [30] B. Pal, S. S. Sana and K. Chaudhuri, Maximizing profits for an EPQ model with unreliable machine and rework of random defective items, Int. J. Syst. Sci., 44 (2013), 582-594.  doi: 10.1080/00207721.2011.617896. [31] B. Pal, S. S. Sana and K. Chaudhuri, A mathematical model on EPQ for stochastic demand in an imperfect production system, J. Manuf. Sys, 32 (2013), 260-270.  doi: 10.1016/j.jmsy.2012.11.009. [32] E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Oper. Res., 34 (1986), 137-144.  doi: 10.1287/opre.34.1.137. [33] M. J. Rosenblatt and H. L. Lee, Economic production cycle with imperfect production processes, IIE Trans., 18 (1986), 48-55.  doi: 10.1080/07408178608975329. [34] S. S. Sana, An economic production lot size model in an imperfect production system, Eur. J. Oper. Res., 201 (2010), 158-170. [35] L. A. San-José, J. Sicilia and J. García-Laguna, Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost, Omega, 54 (2015), 147-157. [36] B. Sarkar, An inventory model with reliability in an imperfect production process, Appl. Math. Comput., 218 (2012), 4881-4891.  doi: 10.1016/j.amc.2011.10.053. [37] B. Sarkar, L. E. Cárdenas-Barrón, 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, J. Manuf. Syst., 33 (2014), 423-435.  doi: 10.1016/j.jmsy.2014.02.001. [38] B. Sarkar, S. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Ann. Oper. Res., 229 (2015), 677-702.  doi: 10.1007/s10479-014-1745-9. [39] E. W. Taft, The most economical production lot, The Iron Age, 101 (1918), 1410-1412. [40] A. H. Tai, Economic production quantity models for deteriorating/imperfect products and service with rework, Comput. Ind. Eng., 66 (2013), 879-888.  doi: 10.1016/j.cie.2013.09.007. [41] A. A. Taleizadeh, L. E. Cárdenas-Barrón and B. Mohammadi, A deterministic multi product single machine EPQ model with backordering, scraped products, rework and interruption in manufacturing process, Int. J. Prod. Econ., 150 (2014), 9-27.  doi: 10.1016/j.ijpe.2013.11.023. [42] A. A. Taleizadeh, S. S. Kalantari and L. E. Cárdenas-Barrón, Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments, Top, 24 (2016), 143-155.  doi: 10.1007/s11750-015-0377-9. [43] A. A. Taleizadeh, H. R. Zarei and B. R. Sarker, An optimal control of inventory under probabilistic replenishment intervals and known price increase, Eur. Oper. Res., 257 (2017), 777-791.  doi: 10.1016/j.ejor.2016.07.041. [44] C. S. Tapiero, P. H. Ritchken and A. Reisman, Reliability, pricing and quality control, Eur. J. Oper. Res., 31 (1987), 37-45.  doi: 10.1016/0377-2217(87)90134-2. [45] B. Van Beek and C. Van Putten, OR contributions to flexibility improvement in production/inventory systems, Eur. J. Oper. Res., 31 (1987), 52-60. [46] M. van der Heijden and B. P. Iskandar, Last time buy decisions for products sold under warranty, Eur. J. Oper. Res., 224 (2013), 302-312.  doi: 10.1016/j.ejor.2012.07.041. [47] C.-H. Wang, The impact of free-repair warranty policy on EMQ model for imperfect production systems, Comput. Oper. Res., 31 (2004), 2021-2035.  doi: 10.1016/S0305-0548(03)00161-8. [48] C.-H. Wang and S.-H. Sheu, Optimal lot sizing for products sold under free-repair warranty, Eur. J. Oper. Res., 149 (2003), 131-141.  doi: 10.1016/S0377-2217(02)00429-0. [49] C. M. Wright and A. Mehrez, An overview of representative research of the relationships between quality and inventory, Omega, 26 (1998), 29-47.  doi: 10.1016/S0305-0483(97)00042-X. [50] C. A. Yano and H. L. Lee, Lot sizing with random yields: A review, Oper. Res., 43 (1995), 311-334.  doi: 10.1287/opre.43.2.311. [51] R. H. Yeh, M. Y. Chen and C. Y. Lin, Optimal periodic replacement policy for repairable products under free-repair warranty, Eur. Oper. Res., 176 (2007), 1678-1686.  doi: 10.1016/j.ejor.2005.10.047. [52] R. H. Yeh, W. T. Ho and S. T. Tseng, Optimal production run length for products sold with warranty, Eur. J. Oper. Res., 120 (2000), 575-582.  doi: 10.1016/S0377-2217(99)00004-1. [53] S. H. Yoo, D. S. Kim and M. S. Park, Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return, Int. J. Prod. Econ., 140 (2012), 922-933.  doi: 10.1016/j.ijpe.2012.07.014. [54] X. Zhang and Y. Gerchak, Joint lot sizing and inspection policy in an EOQ model with random yield, IIE Trans., 22 (1990), 41-47.  doi: 10.1080/07408179008964156. [55] Y.-W. Zhou, J. Y. Chen, Y. Z. Wu and W. H. Zhou, EPQ models for items with imperfect quality and one-time-only discount, Appl. Math. Model., 39 (2015), 1000-1018.  doi: 10.1016/j.apm.2014.07.017.
The inventory level for imperfect manufacturing system with allowable shortages
The three-dimension graph of the expected total cost
The values of the parameters for numerical example
 Description and parameters Value Unit Production rate ($M$) 10, 000 units/year Demand rate ($D$) 2, 000 units/year Setup cost ($K$) 500 ＄/cycle Holding cost rate for a unit (a fraction of dollar value) ($I$) 0.26 ＄/unit/year Backordering cost ($b$) 6 ＄/unit/year Repair cost/warranty cost ($c_w$) 5 ＄/unit Restoration cost ($R$) 100 ＄/cycle Probability that the system from controlled state shifts to uncontrolled state ($p$) 0.1 N/A Percentage of nonconforming items when the process is controlled state ($\lambda_1$) 0.1 N/A Percentage of nonconforming items when the process is in uncontrolled ($\lambda_2$) 0.75 N/A
 Description and parameters Value Unit Production rate ($M$) 10, 000 units/year Demand rate ($D$) 2, 000 units/year Setup cost ($K$) 500 ＄/cycle Holding cost rate for a unit (a fraction of dollar value) ($I$) 0.26 ＄/unit/year Backordering cost ($b$) 6 ＄/unit/year Repair cost/warranty cost ($c_w$) 5 ＄/unit Restoration cost ($R$) 100 ＄/cycle Probability that the system from controlled state shifts to uncontrolled state ($p$) 0.1 N/A Percentage of nonconforming items when the process is controlled state ($\lambda_1$) 0.1 N/A Percentage of nonconforming items when the process is in uncontrolled ($\lambda_2$) 0.75 N/A
The values of $L^{}, S^{*}$, and $ATC^{*}$ corresponding to 32 combinations of $p, K, c_w, I, R$
 $p$ $K$ $c_w$ $I$ $R$ $L^{*}$ $S^{*}$ $ATC^{*}$ 0.1 500 6 0.2 100 1050 480 84514.92 130 1050 480 84572.06 0.26 100 883.1 448 84652.73 130 905.1 459.2 84719.84 7.8 0.2 100 1050 480 85848.04 130 1050 480 85905.19 0.26 100 879.8 446.4 85984.22 130 901.9 457.6 86051.58 650 6 0.2 100 1050 480 84800.63 130 1062.1 485.5 84857.59 0.26 100 1050 532.7 84958.68 130 1050 532.7 85015.83 7.8 0.2 100 1050 480 86133.76 130 1059 484.1 86190.8 0.26 100 1050 532.7 86291.81 130 1050 532.7 86348.95 0.13 500 6 0.2 100 1050 480 84518.1 130 1050 480 84575.24 0.26 100 884.3 448.7 84656.5 130 906.3 459.8 84723.52 7.8 0.2 100 1050 480 85853.41 130 1050 480 85910.55 0.26 100 881.9 447.4 85990.62 130 903.9 458.6 86057.8 650 6 0.2 100 1050 480 84803.81 130 1050 480 84860.95 0.26 100 1050 532.7 84961.86 130 1050 532.7 85019 7.8 0.2 100 1050 480 86139.12 130 1060.9 485 86196.11 0.26 100 1050 532.7 86297.17 130 1050 532.7 86354.32
 $p$ $K$ $c_w$ $I$ $R$ $L^{*}$ $S^{*}$ $ATC^{*}$ 0.1 500 6 0.2 100 1050 480 84514.92 130 1050 480 84572.06 0.26 100 883.1 448 84652.73 130 905.1 459.2 84719.84 7.8 0.2 100 1050 480 85848.04 130 1050 480 85905.19 0.26 100 879.8 446.4 85984.22 130 901.9 457.6 86051.58 650 6 0.2 100 1050 480 84800.63 130 1062.1 485.5 84857.59 0.26 100 1050 532.7 84958.68 130 1050 532.7 85015.83 7.8 0.2 100 1050 480 86133.76 130 1059 484.1 86190.8 0.26 100 1050 532.7 86291.81 130 1050 532.7 86348.95 0.13 500 6 0.2 100 1050 480 84518.1 130 1050 480 84575.24 0.26 100 884.3 448.7 84656.5 130 906.3 459.8 84723.52 7.8 0.2 100 1050 480 85853.41 130 1050 480 85910.55 0.26 100 881.9 447.4 85990.62 130 903.9 458.6 86057.8 650 6 0.2 100 1050 480 84803.81 130 1050 480 84860.95 0.26 100 1050 532.7 84961.86 130 1050 532.7 85019 7.8 0.2 100 1050 480 86139.12 130 1060.9 485 86196.11 0.26 100 1050 532.7 86297.17 130 1050 532.7 86354.32
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