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October  2014, 10(4): 1261-1277. doi: 10.3934/jimo.2014.10.1261

Optimal pricing policy for deteriorating items with preservation technology investment

 1 Institute of Systems Engineering, Tianjin University, Tianjin 300072, China, China, China

Received  February 2013 Revised  September 2013 Published  February 2014

This paper considers the problem of simultaneously determining the price and inventory control strategies for deteriorating items. It is assumed that the rate of deterioration can be reduced by means of effective preservation technology investment and the demand rate is a function of selling price. The goal of this study is to maximize the total profit per unit time by simultaneously determining the optimal selling price, length of replenishment cycle and preservation technology investment. First, for a given preservation technology investment, we prove that the optimal selling price and the optimal length of replenishment cycle exist and are unique. Next, it is shown that the total profit per unit time is a concave function of the preservation technology investment. Then, an effective algorithm is designed to find the optimal joint policy. Finally, numerical examples to illustrate the solution procedure and some managerial implications are provided.
Citation: Jianxiong Zhang, Zhenyu Bai, Wansheng Tang. Optimal pricing policy for deteriorating items with preservation technology investment. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1261-1277. doi: 10.3934/jimo.2014.10.1261
References:
 [1] P. L. Abad, Optimal price and order size for a reseller under partial backordering,, Computers $&$ Operations Research, 28 (2001), 53. doi: 10.1016/S0305-0548(99)00086-6. Google Scholar [2] A. A. Alamri and Z. T. Balkhi, The effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand and deterioration rates,, International Journal of Production Economics, 107 (2007), 125. doi: 10.1016/j.ijpe.2006.08.004. Google Scholar [3] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001,, European Journal of Operational Research, 221 (2012), 275. doi: 10.1016/j.ejor.2012.03.004. Google Scholar [4] Z. T. Balkhi, On the global optimal solution to an integrated inventory system with general time varying demand, production and deterioration rates,, European Journal of Operational Research, 114 (1999), 29. doi: 10.1016/S0377-2217(98)00155-6. Google Scholar [5] H. J. Chang and C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging,, Journal of the Operational Research Society, 50 (1999), 1176. doi: 10.2307/3010088. 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. doi: 10.1016/j.ijpe.2006.05.008. Google Scholar [7] P. S. Deng, R. H. J. Lin and P. Chu, A note on the inventory models for deteriorating items with ramp type demand rate,, European Journal of Operational Research, 178 (2007), 112. doi: 10.1016/j.ejor.2006.01.028. Google Scholar [8] C. Y. Dye and T. P. Hsieh, An optimal replenishment policy for deteriorating items with effective investment in preservation technology,, European Journal of Operational Research, 218 (2012), 106. doi: 10.1016/j.ejor.2011.10.016. Google Scholar [9] B. C. Giri, A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with time varying demand and costs,, Journal of the Operational Research Society, 47 (1996), 1398. doi: 10.2307/3010205. Google Scholar [10] A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand,, Journal of the Operational Research Society, 42 (1991), 1105. doi: 10.2307/2582957. Google Scholar [11] O. K. Gupta, N. H. Shah and A. R. Patel, An integrated deteriorating inventory model with permissible delay in payments and price-sensitive stock-dependent demand,, International Journal of Operational Research, 11 (2011), 425. doi: 10.1504/IJOR.2011.041801. Google Scholar [12] T. P. Hsieh and C. Y. Dye, A production inventory model incorporating the effect of preservation technology investment when demand is fluctuating with time,, Journal of Computational and Applied Mathematics, 239 (2013), 25. doi: 10.1016/j.cam.2012.09.016. Google Scholar [13] P. H. Hsu, H. M. Wee and H. M. Teng, Preservation technology investment for deteriorating inventory,, International Journal of Production Economics, 124 (2010). doi: 10.1016/j.ijpe.2009.11.034. Google Scholar [14] K. C. Hung, An inventory model with generalized type demand, deterioration and backorder rates,, European Journal of Operational Research, 208 (2011), 239. doi: 10.1016/j.ejor.2010.08.026. Google Scholar [15] H. H. Lee, The investment model in preventive maintenance in multi-level production systems,, International Journal of Production Economics, 112 (2008), 816. doi: 10.1016/j.ijpe.2007.07.004. Google Scholar [16] R. Maihami and I. N. Kamalabadi, Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand,, International Journal of Production Economics, 136 (2012), 116. doi: 10.1016/j.ijpe.2011.09.020. Google Scholar [17] B. Mandal and A. K. Pal, Order level inventory system with ramp type demand rate for deteriorating items,, Journal of Interdisciplinary Mathematics, 1 (1998), 49. doi: 10.1080/09720502.1998.10700243. Google Scholar [18] S. Mukhopadhyay, R. N. Mukherjee and K. S. Chaudhuri, Joint pricing and ordering policy for a deteriorating inventory,, Computers $&$ Industrial Engineering, 47 (2004), 339. doi: 10.1016/j.cie.2004.06.007. Google Scholar [19] A. Musa and B. Sani, Inventory ordering policies of delayed deteriorating items under permissible delay in payments,, International Journal of Production Economics, 136 (2012), 75. doi: 10.1016/j.ijpe.2011.09.013. Google Scholar [20] L. Y. Ouyang, C. T. Chang and J. T. Teng, An EOQ model for deteriorating items under trade credits,, Journal of the Operational Research Society, 56 (2005), 719. doi: 10.1057/palgrave.jors.2601881. Google Scholar [21] M. S. Sajadieh, M. R. Akbari Jokar, Optimizing shipment, ordering and pricing policies in a two-stage supply chain with price-sensitive demand,, Transportation Research Part E: Logistics and Transportation Review, 45 (2009), 302. doi: 10.1016/j.tre.2008.12.002. Google Scholar [22] S. S. Sana, Optimal selling price and lotsize with time varying deterioration and partial backlogging,, Applied Mathematics and Computation, 217 (2010), 185. doi: 10.1016/j.amc.2010.05.040. Google Scholar [23] N. H. Shah, H. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates,, Omega, 41 (2013), 421. doi: 10.1016/j.omega.2012.03.002. Google Scholar [24] Y. K. Shah and M. C. Jaiswal, An order-level inventory model for a system with constant rate of deterioration,, Opsearch, 14 (1977), 174. Google Scholar [25] K. Skouri, I. Konstantaras, S. Papachristos and I. Ganas, Inventory models with ramp type demand rate, partial backlogging and weibull deterioration rate,, European Journal of Operational Research, 192 (2009), 79. doi: 10.1016/j.ejor.2007.09.003. Google Scholar [26] J. T. Teng, H. L. Yang and L. Y. Ouyang, On an EOQ model for deteriorating items with time-varying demand and partial backlogging,, Journal of the Operational Research Society, 54 (2003), 432. doi: 10.1057/palgrave.jors.2601490. Google Scholar [27] C. Uckun, F. Karaesmen and S. Savas, Investment in improved inventory accuracy in a decentralized supply chain,, International Journal of Production Economics, 113 (2008), 546. doi: 10.1016/j.ijpe.2007.10.012. Google Scholar [28] H. M. Wee, Economic production lot size model for deteriorating items with partial back-ordering,, Computers $&$ Industrial Engineering, 24 (1993), 449. doi: 10.1016/0360-8352(93)90040-5. Google Scholar [29] K. S. Wu and L. Y. Ouyang, Replenishment policy for deteriorating items with ramp type demand rate,, Proceedings of the National Science Council, 24 (2000), 279. Google Scholar [30] X. L. Xu and X. Q. Cai, Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium,, Journal of Industrial and Management Optimization, 4 (2008), 843. doi: 10.3934/jimo.2008.4.843. Google Scholar [31] C. T. Yang, L. Y. Ouyang, H. F. Yen and K. L. Lee, Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase,, Journal of Industrial and Management Optimization, 9 (2013), 437. doi: 10.3934/jimo.2013.9.437. Google Scholar [32] M. J. Yao and Y. C. Wang, Theoretical analysis and a search procedure for the joint replenishment problem with deteriorating products,, Journal of Industrial and Management Optimization, 1 (2005), 359. doi: 10.3934/jimo.2005.1.359. Google Scholar [33] P. S. You, Inventory policy for products with price and time-dependent demands,, Journal of the Operational Research Society, 56 (2005), 870. doi: 10.1057/palgrave.jors.2601905. Google Scholar [34] J. C. P. Yu, H. M. Wee and K. J. Wang, Supply chain partnership for Three-Echelon deteriorating inventory model,, Journal of Industrial and Management Optimization, 4 (2008), 827. doi: 10.3934/jimo.2008.4.827. Google Scholar

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References:
 [1] P. L. Abad, Optimal price and order size for a reseller under partial backordering,, Computers $&$ Operations Research, 28 (2001), 53. doi: 10.1016/S0305-0548(99)00086-6. Google Scholar [2] A. A. Alamri and Z. T. Balkhi, The effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand and deterioration rates,, International Journal of Production Economics, 107 (2007), 125. doi: 10.1016/j.ijpe.2006.08.004. Google Scholar [3] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001,, European Journal of Operational Research, 221 (2012), 275. doi: 10.1016/j.ejor.2012.03.004. Google Scholar [4] Z. T. Balkhi, On the global optimal solution to an integrated inventory system with general time varying demand, production and deterioration rates,, European Journal of Operational Research, 114 (1999), 29. doi: 10.1016/S0377-2217(98)00155-6. Google Scholar [5] H. J. Chang and C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging,, Journal of the Operational Research Society, 50 (1999), 1176. doi: 10.2307/3010088. 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. doi: 10.1016/j.ijpe.2006.05.008. Google Scholar [7] P. S. Deng, R. H. J. Lin and P. Chu, A note on the inventory models for deteriorating items with ramp type demand rate,, European Journal of Operational Research, 178 (2007), 112. doi: 10.1016/j.ejor.2006.01.028. Google Scholar [8] C. Y. Dye and T. P. Hsieh, An optimal replenishment policy for deteriorating items with effective investment in preservation technology,, European Journal of Operational Research, 218 (2012), 106. doi: 10.1016/j.ejor.2011.10.016. Google Scholar [9] B. C. Giri, A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with time varying demand and costs,, Journal of the Operational Research Society, 47 (1996), 1398. doi: 10.2307/3010205. Google Scholar [10] A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand,, Journal of the Operational Research Society, 42 (1991), 1105. doi: 10.2307/2582957. Google Scholar [11] O. K. Gupta, N. H. Shah and A. R. Patel, An integrated deteriorating inventory model with permissible delay in payments and price-sensitive stock-dependent demand,, International Journal of Operational Research, 11 (2011), 425. doi: 10.1504/IJOR.2011.041801. Google Scholar [12] T. P. Hsieh and C. Y. Dye, A production inventory model incorporating the effect of preservation technology investment when demand is fluctuating with time,, Journal of Computational and Applied Mathematics, 239 (2013), 25. doi: 10.1016/j.cam.2012.09.016. Google Scholar [13] P. H. Hsu, H. M. Wee and H. M. Teng, Preservation technology investment for deteriorating inventory,, International Journal of Production Economics, 124 (2010). doi: 10.1016/j.ijpe.2009.11.034. Google Scholar [14] K. C. Hung, An inventory model with generalized type demand, deterioration and backorder rates,, European Journal of Operational Research, 208 (2011), 239. doi: 10.1016/j.ejor.2010.08.026. Google Scholar [15] H. H. Lee, The investment model in preventive maintenance in multi-level production systems,, International Journal of Production Economics, 112 (2008), 816. doi: 10.1016/j.ijpe.2007.07.004. Google Scholar [16] R. Maihami and I. N. Kamalabadi, Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand,, International Journal of Production Economics, 136 (2012), 116. doi: 10.1016/j.ijpe.2011.09.020. Google Scholar [17] B. Mandal and A. K. Pal, Order level inventory system with ramp type demand rate for deteriorating items,, Journal of Interdisciplinary Mathematics, 1 (1998), 49. doi: 10.1080/09720502.1998.10700243. Google Scholar [18] S. Mukhopadhyay, R. N. Mukherjee and K. S. Chaudhuri, Joint pricing and ordering policy for a deteriorating inventory,, Computers $&$ Industrial Engineering, 47 (2004), 339. doi: 10.1016/j.cie.2004.06.007. Google Scholar [19] A. Musa and B. Sani, Inventory ordering policies of delayed deteriorating items under permissible delay in payments,, International Journal of Production Economics, 136 (2012), 75. doi: 10.1016/j.ijpe.2011.09.013. Google Scholar [20] L. Y. Ouyang, C. T. Chang and J. T. Teng, An EOQ model for deteriorating items under trade credits,, Journal of the Operational Research Society, 56 (2005), 719. doi: 10.1057/palgrave.jors.2601881. Google Scholar [21] M. S. Sajadieh, M. R. Akbari Jokar, Optimizing shipment, ordering and pricing policies in a two-stage supply chain with price-sensitive demand,, Transportation Research Part E: Logistics and Transportation Review, 45 (2009), 302. doi: 10.1016/j.tre.2008.12.002. Google Scholar [22] S. S. Sana, Optimal selling price and lotsize with time varying deterioration and partial backlogging,, Applied Mathematics and Computation, 217 (2010), 185. doi: 10.1016/j.amc.2010.05.040. Google Scholar [23] N. H. Shah, H. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates,, Omega, 41 (2013), 421. doi: 10.1016/j.omega.2012.03.002. Google Scholar [24] Y. K. Shah and M. C. Jaiswal, An order-level inventory model for a system with constant rate of deterioration,, Opsearch, 14 (1977), 174. Google Scholar [25] K. Skouri, I. Konstantaras, S. Papachristos and I. Ganas, Inventory models with ramp type demand rate, partial backlogging and weibull deterioration rate,, European Journal of Operational Research, 192 (2009), 79. doi: 10.1016/j.ejor.2007.09.003. Google Scholar [26] J. T. Teng, H. L. Yang and L. Y. Ouyang, On an EOQ model for deteriorating items with time-varying demand and partial backlogging,, Journal of the Operational Research Society, 54 (2003), 432. doi: 10.1057/palgrave.jors.2601490. Google Scholar [27] C. Uckun, F. Karaesmen and S. Savas, Investment in improved inventory accuracy in a decentralized supply chain,, International Journal of Production Economics, 113 (2008), 546. doi: 10.1016/j.ijpe.2007.10.012. Google Scholar [28] H. M. Wee, Economic production lot size model for deteriorating items with partial back-ordering,, Computers $&$ Industrial Engineering, 24 (1993), 449. doi: 10.1016/0360-8352(93)90040-5. Google Scholar [29] K. S. Wu and L. Y. Ouyang, Replenishment policy for deteriorating items with ramp type demand rate,, Proceedings of the National Science Council, 24 (2000), 279. Google Scholar [30] X. L. Xu and X. Q. Cai, Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium,, Journal of Industrial and Management Optimization, 4 (2008), 843. doi: 10.3934/jimo.2008.4.843. Google Scholar [31] C. T. Yang, L. Y. Ouyang, H. F. Yen and K. L. Lee, Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase,, Journal of Industrial and Management Optimization, 9 (2013), 437. doi: 10.3934/jimo.2013.9.437. Google Scholar [32] M. J. Yao and Y. C. Wang, Theoretical analysis and a search procedure for the joint replenishment problem with deteriorating products,, Journal of Industrial and Management Optimization, 1 (2005), 359. doi: 10.3934/jimo.2005.1.359. Google Scholar [33] P. S. You, Inventory policy for products with price and time-dependent demands,, Journal of the Operational Research Society, 56 (2005), 870. doi: 10.1057/palgrave.jors.2601905. Google Scholar [34] J. C. P. Yu, H. M. Wee and K. J. Wang, Supply chain partnership for Three-Echelon deteriorating inventory model,, Journal of Industrial and Management Optimization, 4 (2008), 827. doi: 10.3934/jimo.2008.4.827. Google Scholar

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