Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns

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July 2015, 11(3): 933-949. doi: 10.3934/jimo.2015.11.933

Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns

Maryam Ghoreishi ^1, , Abolfazl Mirzazadeh ^1, , Gerhard-Wilhelm Weber ^2, and Isa Nakhai-Kamalabadi ^3,

1.	Department of Industrial Engineering, Karazmi University, Mofatteh Avenue, Tehran, Iran, Iran
2.	Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey
3.	Department of Industrial Engineering, Tarbiat Modares University (TMU), Tehran, Iran

Received August 2013 Revised July 2014 Published October 2014

Abstract
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This paper develops an Economic Order Quantity (EOQ) model for non-instantaneous deteriorating items with selling price- and inflation-induced demand under the effect of inflation and customer returns. The customer returns are assumed as a function of demand and price. Shortages are allowed and partially backlogged. The effects of time value of money are studied using the Discounted Cash Flow approach. The main objective is to determine the optimal selling price, the optimal length of time in which there is no inventory shortage, and the optimal replenishment cycle simultaneously such that the present value of total profit is maximized. An efficient algorithm is presented to find the optimal solution of the developed model. Finally, a numerical example is extracted to solve the presented inventory model using the proposed algorithm and the effects of the customer returns, inflation, and non-instantaneous deterioration are also discussed. The paper ends with a conclusion and outlook to future studies.

Keywords: Inventory, customer returns, Inflation- and selling price-dependent demand, partial backlogging, non-instantaneous deteriorating items, pricing..

Mathematics Subject Classification: Primary: 90B05; Secondary: 91B2.

Citation: Maryam Ghoreishi, Abolfazl Mirzazadeh, Gerhard-Wilhelm Weber, Isa Nakhai-Kamalabadi. Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns. Journal of Industrial and Management Optimization, 2015, 11 (3) : 933-949. doi: 10.3934/jimo.2015.11.933

References:

[1]	P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Managem. Sci., 42 (1996), 1093-1104. doi: 10.1287/mnsc.42.8.1093.
[2]	P. L. Abad, Optimal price and order size for a reseller under partial backordering, Comp. and Oper. Res., 28 (2001), 53-65. doi: 10.1016/S0305-0548(99)00086-6.
[3]	E. T. Anderson, K. Hansen, D. Simister and L. K. Wang, How are demand and returns related? Theory and empirical evidence, Working paper, Kellogg School of Management, Northwestern University, February 2006.
[4]	A. K. Bhunia, C. K. Jaggi, A. Sharma and R. Sharma, A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging, Applied Mathematics and Computation, 232 (2014), 1125-1137. doi: 10.1016/j.amc.2014.01.115.
[5]	J. A. Buzacott, Economic order quantity with inflation, Operational Quarterly, 26 (1975), 553-558. doi: 10.2307/3008214.
[6]	C. T. Chang, J. T. Teng and S. K. Goyal, Optimal replenishment policies for non instantaneous deteriorating items with stock-dependent demand. Internat, J. of Prod. Econ, 123 (2010), 62-68.
[7]	H. J. Chang, J. T. Teng, L. Y. Ouyang and C. Y. Dye, Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging, Eur. J. Oper. Res., 168 (2005), 51-64. doi: 10.1016/j.ejor.2004.05.003.
[8]	J. Chen and P. C. Bell, The impact of customer returns on pricing and order decisions, Eur. J. Oper. Res., 195 (2009), 280-295. doi: 10.1016/j.ejor.2008.01.030.
[9]	R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Trans., 5 (1973), 323-326. doi: 10.1080/05695557308974918.
[10]	T. K. Datta and A. K. Pal, Effects of inflation and time value of money on an inventory model with linear time-dependent demand rate and shortages, Eur. J. Oper. Res., 52 (1991), 326-333. doi: 10.1016/0377-2217(91)90167-T.
[11]	C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging, Omega, 35 (2007), 184-189. doi: 10.1016/j.omega.2005.05.002.
[12]	C. Y. Dye, L. Y. Quyang and T. P. Hsieh, Inventory and pricing strategy for deteriorating items with shortages: A discounted cash flow approach, Comput. and Industrial Engineering, 52 (2007), 29-40. doi: 10.1016/j.cie.2006.10.009.
[13]	K. V. Geetha and R. Uthayakumar, Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments, J. of Comp. and Appl. Math., 223 (2010), 2492-2505. doi: 10.1016/j.cam.2009.10.031.
[14]	P. M. Ghare and G. H. Schrader, A model for exponentially decaying inventory system, Internat. J. of Prod. Res., 21 (1963), 449-460.
[15]	A. Gholami-Qadikolaei, A. Mirzazadeh and R. Tavakkoli-Moghaddam, A stochastic multiobjective multiconstraint inventory model under inflationary condition and different inspection scenarios, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 227 (2013), 1057-1074. doi: 10.1177/0954405413481452.
[16]	M. Ghoreishi, A. Arshsadi-Khamseh and A. Mirzazadeh, Joint Optimal Pricing and Inventory Control for Deteriorating Items under Inflation and Customer Returns, Journal of Industrial Engineering, 2013 (2013), Article ID 709083, 7 pages. doi: 10.1155/2013/709083.
[17]	M. Ghoreishi, A. Mirzazadeh and G. W. Weber, Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns, Optimization, 63 (2014), 1785-1804. doi: 10.1080/02331934.2013.853059.
[18]	M. Ghoreishi, A. Mirzazadeh and I. Nakhai-Kamalabadi, Optimal pricing and lot-sizing policies for an economic production quantity model with non-instantaneous deteriorating items, permissible delay in payments, customer returns, and inflation, to appear in Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, (2014), doi: 10.1177/0954405414522215. doi: 10.1177/0954405414522215.
[19]	B. H. Gilding, Inflation and the optimal inventory replenishment schedule within a finite planning horizon, European Journal of Operational Research, 234 (2014), 683-693. doi: 10.1016/j.ejor.2013.11.001.
[20]	S. Goal, Y. P. Gupta and C. R. Bector, Impact of inflation on economic quantity discount schedules to increase vendor profits, Internat. J. of Systems Sci., 22 (1991), 197-207. doi: 10.1080/00207729108910600.
[21]	S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, Eur. J. Oper. Res., 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4.
[22]	A. Guria, B. Das, S. Mondal and M. Maiti, Inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment, Applied Mathematical Modeling, 37 (2013), 240-257. doi: 10.1016/j.apm.2012.02.010.
[23]	R. W. Hall, Price changes and order quantities: Impacts of discount rate and storage costs, IIE Trans., 24 (1992), 104-110. doi: 10.1080/07408179208964207.
[24]	M. A. Hariga, Optimal EOQ models for deteriorating items with time-varying demand, J. Oper. Res. Soc., 47 (1996), 1228-1246. doi: 10.2307/3010036.
[25]	M. A. Hariga and M. Ben-Daya, Optimal time varying lot sizing models under inflationary conditions, Eur. J. Oper. Res., 89 (1996), 313-325. doi: 10.1016/0377-2217(94)00256-8.
[26]	K. J. Heng, J. Labban and R. J. Linn, An order-level lot-size inventory model for deteriorating items with finite replenishment rate, Comp. Ind. Eng., 20 (1991), 187-197.
[27]	J. Hess and G. Mayhew, Modeling merchandise returns in direct marketing, J. of Direct Marketing, 11 (1997), 20-35. doi: 10.1002/(SICI)1522-7138(199721)11:2<20::AID-DIR4>3.3.CO;2-0.
[28]	I. Horowitz, EOQ and inflation uncertainty, International Journal of Prod. Econ., 65 (2000), 217-224. doi: 10.1016/S0925-5273(99)00034-1.
[29]	K. L. Hou and L. C. Lin, Optimal pricing and ordering policies for deteriorating items with multivariate demand under trade credit and inflation, OPSEARCH, 50 (2013), 404-417. doi: 10.1007/s12597-012-0115-0.
[30]	T. P. Hsieh and C. Y. Dye, Pricing and lot-sizing policies for deteriorating items with partial backlogging under inflation, Expert Syst. with Appl., 37 (2010), 7234-7242. doi: 10.1016/j.eswa.2010.04.004.
[31]	C. K. Jaggi, K. K. Aggarwal and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, Int. J. Prod. Econ., 103 (2006), 707-714. doi: 10.1016/j.ijpe.2006.01.004.
[32]	R. Maihami and I. Nakhai Kamalabadi, Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand, Int. J. Prod. Econ., 136 (2012), 116-122. doi: 10.1016/j.ijpe.2011.09.020.
[33]	R. Maihami and I. Nakhai Kamalabadi, Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging, Math. and Comp. Modelling, 55 (2012), 1722-1733. doi: 10.1016/j.mcm.2011.11.017.
[34]	A. Mirzazadeh, M. M. Seyed-Esfehani and S. M. T. Fatemi-Ghomi, An inventory model under uncertain inflationary conditions, finite production rate and inflation-dependent demand rate for deteriorating items with shortages, Internat. J. of Systems Sci., 40 (2009), 21-31. doi: 10.1080/00207720802088264.
[35]	R. B. Misra, A note on optimal inventory management under inflation, Naval Res. Logist. Quart., 26 (1979), 161-165. doi: 10.1002/nav.3800260116.
[36]	I. Moon and S. Lee, The effects of inflation and time value of money on an economic order quantity with a random product life cycle, Eur. J. Oper. Res., 125 (2000), 588-601. doi: 10.1016/S0377-2217(99)00270-2.
[37]	I. Moon, B. C. Giri and B. Ko, Order quantity models for ameliorating/deteriorating items under inflation and time discounting, Eur. J. Oper. Res., 162 (2005), 773-785. doi: 10.1016/j.ejor.2003.09.025.
[38]	A. Musa and B. Sani, Inventory ordering policies of delayed deteriorating items under permissible delay in payments, Internat. J. of Prod. Econ., 136 (2012), 75-83. doi: 10.1016/j.ijpe.2011.09.013.
[39]	L. Y. Ouyang, K. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Comp. and Indust. Eng., 51 (2006), 637-651. doi: 10.1016/j.cie.2006.07.012.
[40]	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), 437-454. doi: 10.3934/jimo.2013.9.437.
[41]	K. S. Park, Inflationary effect on EOQ under trade-credit financing, International Journal on Policy and Information, 10 (1986), 65-69.
[42]	F. Samadi, A. Mirzazadeh and M. M. Pedram, Marketing and service planning in a fuzzy inventory model: A geometric programming approach, Applied Mathematical Modelling, 37 (2013), 6683-6694. doi: 10.1016/j.apm.2012.12.020.
[43]	B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, Applied Mathematics and Computation, 217 (2011), 6159-6167. doi: 10.1016/j.amc.2010.12.098.
[44]	B. Sarkar, S. S. Sana and K. 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. doi: 10.1016/j.eswa.2011.04.044.
[45]	B. R. Sarker, S. Mukherjee and C. V. Balan, An order-level lot size inventory model with inventory-level dependent demand and deterioration, Int. J. Prod. Eco., 48 (1997), 227-236. doi: 10.1016/S0925-5273(96)00107-7.
[46]	B. R. Sarker and H. Pan, Effects of inflation and time value of money on order quantity and allowable shortage, Internat. J. of Prod. Managem., 34 (1994), 65-72. doi: 10.1016/0925-5273(94)90047-7.
[47]	J. Shi, G. Zhang and K. K. Lai, Ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand, Optimization: A Journal of Mathematical Programming and Operations Research, 61 (2012), 151-162. doi: 10.1080/02331934.2011.590485.
[48]	J. Taheri-Tolgari, A. Mirzazadeh and F. Jolai, An inventory model for imperfect items under inflationary conditions with considering inspection errors, Computers and Mathematics with Applications, 63 (2012), 1007-1019. doi: 10.1016/j.camwa.2011.09.050.
[49]	Y. C. Tsao and G. J. Sheen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Comput. and Oper. Res., 35 (2008), 3562-3580. doi: 10.1016/j.cor.2007.01.024.
[50]	H. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Comp. Oper. Res., 22 (1995), 345-356.
[51]	H. M. Wee and S. T. Law, Replenishment and Pricing Policy for Deteriorating Items Taking into Account the Time Value of Money, Internat. J. Prod. Econ., 71 (2001), 213-220. doi: 10.1016/S0925-5273(00)00121-3.
[52]	K. S. Wu, L. Y. Ouyang and C. T. Yang, An optimal replenishment policy for non-instantaneous deteriorating items with stock dependent demand and partial backlogging, Internat. J. of Prod. Econ., 101 (2006), 369-384. doi: 10.1016/j.ijpe.2005.01.010.
[53]	C. T. Yang, L. Y. Quyang and H. H. Wu, Retailers optimal pricing and ordering policies for Non-instantaneous deteriorating items with price-dependent demand and partial backlogging, Math. Problems in Eng., 2009 (2009), Article ID 198305, 18 pages. doi: 10.1155/2009/198305.
[54]	J. Zhang, Z. Bai and W. Tang, Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277. doi: 10.3934/jimo.2014.10.1261.
[55]	S. X. Zhu, Joint pricing and inventory replenishment decisions with returns and expediting, Eur. J. Oper. Res., 216 (2012), 105-112. doi: 10.1016/j.ejor.2011.07.024.

show all references

References:

[1]	P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Managem. Sci., 42 (1996), 1093-1104. doi: 10.1287/mnsc.42.8.1093.
[2]	P. L. Abad, Optimal price and order size for a reseller under partial backordering, Comp. and Oper. Res., 28 (2001), 53-65. doi: 10.1016/S0305-0548(99)00086-6.
[3]	E. T. Anderson, K. Hansen, D. Simister and L. K. Wang, How are demand and returns related? Theory and empirical evidence, Working paper, Kellogg School of Management, Northwestern University, February 2006.
[4]	A. K. Bhunia, C. K. Jaggi, A. Sharma and R. Sharma, A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging, Applied Mathematics and Computation, 232 (2014), 1125-1137. doi: 10.1016/j.amc.2014.01.115.
[5]	J. A. Buzacott, Economic order quantity with inflation, Operational Quarterly, 26 (1975), 553-558. doi: 10.2307/3008214.
[6]	C. T. Chang, J. T. Teng and S. K. Goyal, Optimal replenishment policies for non instantaneous deteriorating items with stock-dependent demand. Internat, J. of Prod. Econ, 123 (2010), 62-68.
[7]	H. J. Chang, J. T. Teng, L. Y. Ouyang and C. Y. Dye, Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging, Eur. J. Oper. Res., 168 (2005), 51-64. doi: 10.1016/j.ejor.2004.05.003.
[8]	J. Chen and P. C. Bell, The impact of customer returns on pricing and order decisions, Eur. J. Oper. Res., 195 (2009), 280-295. doi: 10.1016/j.ejor.2008.01.030.
[9]	R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Trans., 5 (1973), 323-326. doi: 10.1080/05695557308974918.
[10]	T. K. Datta and A. K. Pal, Effects of inflation and time value of money on an inventory model with linear time-dependent demand rate and shortages, Eur. J. Oper. Res., 52 (1991), 326-333. doi: 10.1016/0377-2217(91)90167-T.
[11]	C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging, Omega, 35 (2007), 184-189. doi: 10.1016/j.omega.2005.05.002.
[12]	C. Y. Dye, L. Y. Quyang and T. P. Hsieh, Inventory and pricing strategy for deteriorating items with shortages: A discounted cash flow approach, Comput. and Industrial Engineering, 52 (2007), 29-40. doi: 10.1016/j.cie.2006.10.009.
[13]	K. V. Geetha and R. Uthayakumar, Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments, J. of Comp. and Appl. Math., 223 (2010), 2492-2505. doi: 10.1016/j.cam.2009.10.031.
[14]	P. M. Ghare and G. H. Schrader, A model for exponentially decaying inventory system, Internat. J. of Prod. Res., 21 (1963), 449-460.
[15]	A. Gholami-Qadikolaei, A. Mirzazadeh and R. Tavakkoli-Moghaddam, A stochastic multiobjective multiconstraint inventory model under inflationary condition and different inspection scenarios, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 227 (2013), 1057-1074. doi: 10.1177/0954405413481452.
[16]	M. Ghoreishi, A. Arshsadi-Khamseh and A. Mirzazadeh, Joint Optimal Pricing and Inventory Control for Deteriorating Items under Inflation and Customer Returns, Journal of Industrial Engineering, 2013 (2013), Article ID 709083, 7 pages. doi: 10.1155/2013/709083.
[17]	M. Ghoreishi, A. Mirzazadeh and G. W. Weber, Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns, Optimization, 63 (2014), 1785-1804. doi: 10.1080/02331934.2013.853059.
[18]	M. Ghoreishi, A. Mirzazadeh and I. Nakhai-Kamalabadi, Optimal pricing and lot-sizing policies for an economic production quantity model with non-instantaneous deteriorating items, permissible delay in payments, customer returns, and inflation, to appear in Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, (2014), doi: 10.1177/0954405414522215. doi: 10.1177/0954405414522215.
[19]	B. H. Gilding, Inflation and the optimal inventory replenishment schedule within a finite planning horizon, European Journal of Operational Research, 234 (2014), 683-693. doi: 10.1016/j.ejor.2013.11.001.
[20]	S. Goal, Y. P. Gupta and C. R. Bector, Impact of inflation on economic quantity discount schedules to increase vendor profits, Internat. J. of Systems Sci., 22 (1991), 197-207. doi: 10.1080/00207729108910600.
[21]	S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, Eur. J. Oper. Res., 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4.
[22]	A. Guria, B. Das, S. Mondal and M. Maiti, Inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment, Applied Mathematical Modeling, 37 (2013), 240-257. doi: 10.1016/j.apm.2012.02.010.
[23]	R. W. Hall, Price changes and order quantities: Impacts of discount rate and storage costs, IIE Trans., 24 (1992), 104-110. doi: 10.1080/07408179208964207.
[24]	M. A. Hariga, Optimal EOQ models for deteriorating items with time-varying demand, J. Oper. Res. Soc., 47 (1996), 1228-1246. doi: 10.2307/3010036.
[25]	M. A. Hariga and M. Ben-Daya, Optimal time varying lot sizing models under inflationary conditions, Eur. J. Oper. Res., 89 (1996), 313-325. doi: 10.1016/0377-2217(94)00256-8.
[26]	K. J. Heng, J. Labban and R. J. Linn, An order-level lot-size inventory model for deteriorating items with finite replenishment rate, Comp. Ind. Eng., 20 (1991), 187-197.
[27]	J. Hess and G. Mayhew, Modeling merchandise returns in direct marketing, J. of Direct Marketing, 11 (1997), 20-35. doi: 10.1002/(SICI)1522-7138(199721)11:2<20::AID-DIR4>3.3.CO;2-0.
[28]	I. Horowitz, EOQ and inflation uncertainty, International Journal of Prod. Econ., 65 (2000), 217-224. doi: 10.1016/S0925-5273(99)00034-1.
[29]	K. L. Hou and L. C. Lin, Optimal pricing and ordering policies for deteriorating items with multivariate demand under trade credit and inflation, OPSEARCH, 50 (2013), 404-417. doi: 10.1007/s12597-012-0115-0.
[30]	T. P. Hsieh and C. Y. Dye, Pricing and lot-sizing policies for deteriorating items with partial backlogging under inflation, Expert Syst. with Appl., 37 (2010), 7234-7242. doi: 10.1016/j.eswa.2010.04.004.
[31]	C. K. Jaggi, K. K. Aggarwal and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, Int. J. Prod. Econ., 103 (2006), 707-714. doi: 10.1016/j.ijpe.2006.01.004.
[32]	R. Maihami and I. Nakhai Kamalabadi, Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand, Int. J. Prod. Econ., 136 (2012), 116-122. doi: 10.1016/j.ijpe.2011.09.020.
[33]	R. Maihami and I. Nakhai Kamalabadi, Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging, Math. and Comp. Modelling, 55 (2012), 1722-1733. doi: 10.1016/j.mcm.2011.11.017.
[34]	A. Mirzazadeh, M. M. Seyed-Esfehani and S. M. T. Fatemi-Ghomi, An inventory model under uncertain inflationary conditions, finite production rate and inflation-dependent demand rate for deteriorating items with shortages, Internat. J. of Systems Sci., 40 (2009), 21-31. doi: 10.1080/00207720802088264.
[35]	R. B. Misra, A note on optimal inventory management under inflation, Naval Res. Logist. Quart., 26 (1979), 161-165. doi: 10.1002/nav.3800260116.
[36]	I. Moon and S. Lee, The effects of inflation and time value of money on an economic order quantity with a random product life cycle, Eur. J. Oper. Res., 125 (2000), 588-601. doi: 10.1016/S0377-2217(99)00270-2.
[37]	I. Moon, B. C. Giri and B. Ko, Order quantity models for ameliorating/deteriorating items under inflation and time discounting, Eur. J. Oper. Res., 162 (2005), 773-785. doi: 10.1016/j.ejor.2003.09.025.
[38]	A. Musa and B. Sani, Inventory ordering policies of delayed deteriorating items under permissible delay in payments, Internat. J. of Prod. Econ., 136 (2012), 75-83. doi: 10.1016/j.ijpe.2011.09.013.
[39]	L. Y. Ouyang, K. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Comp. and Indust. Eng., 51 (2006), 637-651. doi: 10.1016/j.cie.2006.07.012.
[40]	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), 437-454. doi: 10.3934/jimo.2013.9.437.
[41]	K. S. Park, Inflationary effect on EOQ under trade-credit financing, International Journal on Policy and Information, 10 (1986), 65-69.
[42]	F. Samadi, A. Mirzazadeh and M. M. Pedram, Marketing and service planning in a fuzzy inventory model: A geometric programming approach, Applied Mathematical Modelling, 37 (2013), 6683-6694. doi: 10.1016/j.apm.2012.12.020.
[43]	B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, Applied Mathematics and Computation, 217 (2011), 6159-6167. doi: 10.1016/j.amc.2010.12.098.
[44]	B. Sarkar, S. S. Sana and K. 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. doi: 10.1016/j.eswa.2011.04.044.
[45]	B. R. Sarker, S. Mukherjee and C. V. Balan, An order-level lot size inventory model with inventory-level dependent demand and deterioration, Int. J. Prod. Eco., 48 (1997), 227-236. doi: 10.1016/S0925-5273(96)00107-7.
[46]	B. R. Sarker and H. Pan, Effects of inflation and time value of money on order quantity and allowable shortage, Internat. J. of Prod. Managem., 34 (1994), 65-72. doi: 10.1016/0925-5273(94)90047-7.
[47]	J. Shi, G. Zhang and K. K. Lai, Ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand, Optimization: A Journal of Mathematical Programming and Operations Research, 61 (2012), 151-162. doi: 10.1080/02331934.2011.590485.
[48]	J. Taheri-Tolgari, A. Mirzazadeh and F. Jolai, An inventory model for imperfect items under inflationary conditions with considering inspection errors, Computers and Mathematics with Applications, 63 (2012), 1007-1019. doi: 10.1016/j.camwa.2011.09.050.
[49]	Y. C. Tsao and G. J. Sheen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Comput. and Oper. Res., 35 (2008), 3562-3580. doi: 10.1016/j.cor.2007.01.024.
[50]	H. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Comp. Oper. Res., 22 (1995), 345-356.
[51]	H. M. Wee and S. T. Law, Replenishment and Pricing Policy for Deteriorating Items Taking into Account the Time Value of Money, Internat. J. Prod. Econ., 71 (2001), 213-220. doi: 10.1016/S0925-5273(00)00121-3.
[52]	K. S. Wu, L. Y. Ouyang and C. T. Yang, An optimal replenishment policy for non-instantaneous deteriorating items with stock dependent demand and partial backlogging, Internat. J. of Prod. Econ., 101 (2006), 369-384. doi: 10.1016/j.ijpe.2005.01.010.
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