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January  2017, 13(1): 187-206. doi: 10.3934/jimo.2016011

Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages

1. 

Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 426 791, South Korea

2. 

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

* Corresponding author: ss.sumonsarkar@gmail.com, Office Ph. No. 031-436-8118

1Dr. Biswajit Sarkar is in leave on lien from Vidyasagar University.

Received  March 2015 Revised  December 2015 Published  March 2016

In literature, many inventory studies have been developed by assuming deterioration of items as either a variable or constant. But in real life situation, deterioration of goods can be reduced by adding some extra effective capital investment in preservation technology. In this paper, a deteriorating inventory model with ramp-type demand under stock-dependent consumption rate by assuming preservation technology cost as a decision variable is formulated. Shortages are allowed and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. The objective of this study is to obtain the optimal replenishment and preservation technology investment strategies so that the total profit per unit time is maximum. Further, the necessary and sufficient conditions are considered to prove the existence and uniqueness of the optimal solution. Some numerical examples along with graphical representations are provided to illustrate the proposed model. Sensitivity analysis of the optimal solution with respect to major parameters of the system has been carried out and the implications are discussed.

Citation: Biswajit Sarkar, Buddhadev Mandal, Sumon Sarkar. Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages. Journal of Industrial & Management Optimization, 2017, 13 (1) : 187-206. doi: 10.3934/jimo.2016011
References:
[1]

P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Management Science, 42 (1996), 1093-1104. doi: 10.1287/mnsc.42.8.1093. Google Scholar

[2]

H. K. Alfares, Inventory model with stock-level dependent demand rate and variable holding cost, International Journal of Production Economics, 108 (2007), 259-265. doi: 10.1016/j.ijpe.2006.12.013. Google Scholar

[3]

M. AsghariS. J. Abrishami and F. Mahdavi, Reverse logistics network design with incentive-dependent return, Industrial Engineering & Management Systems, 13 (2014), 383-397. doi: 10.7232/iems.2014.13.4.383. Google Scholar

[4]

A. Bouras and L. Tadj, Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy, Journal of Industrial and Management Optimization, 11 (2015), 1041-1058. doi: 10.3934/jimo.2015.11.1041. 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), 1176-1182. Google Scholar

[6]

C. J. Chung and H. M. Wee, Short life-cycle deteriorating product remanufacturing in a green supply chain inventory control system, International Journal of Production Economics, 129 (2011), 195-203. doi: 10.1016/j.ijpe.2010.09.033. Google Scholar

[7]

C. J. Chung and H. M. Wee, An integrated production-inventory deteriorating model for pricing policy considering imperfect production, inspection planning and warranty-period and stock-level-dependant demand, International Journal of System Science, 39 (2008), 823-837. doi: 10.1080/00207720801902598. Google Scholar

[8]

C. J. Chung and H. M. Wee, Scheduling and replenishment plan for an integrated deteriorating inventory model with stock-dependent selling rate, International Journal of Advanced Manufacturing Technology, 35 (2007), 665-679. doi: 10.1007/s00170-006-0744-7. Google Scholar

[9]

R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transaction, 5 (1973), 323-326. doi: 10.1080/05695557308974918. Google Scholar

[10]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Computers & Industrial Engineering, 57 (2009), 1105-1113. Google Scholar

[11]

L. E. Cárdenas-Barrón, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Applied Mathematical Modelling, 35 (2011), 2394-2407. doi: 10.1016/j.apm.2010.11.053. Google Scholar

[12]

K. K. Damghani and A. Shahrokh, Solving a new multi-period multi-objective multi-product aggregate production planning problem using fuzzy goal programming, Industrial Engineering & Management Systems, 13 (2014), 369-382. Google Scholar

[13]

C. Y. DyeH. J. Chang and J. T. Teng, A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging, European Journal of Operational Research, 172 (2006), 417-429. doi: 10.1016/j.ejor.2004.10.025. Google Scholar

[14]

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), 106-112. doi: 10.1016/j.ejor.2011.10.016. Google Scholar

[15]

C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880. doi: 10.1016/j.omega.2012.11.002. Google Scholar

[16]

C. Y. Dye and L. Y. Ouyang, An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging, European Journal of Operational Research, 163 (2005), 776-783. doi: 10.1016/j.ejor.2003.09.027. Google Scholar

[17]

P. M. Ghare and G. F. Schrader, A model for exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar

[18]

P. HsuH. M. Wee and H. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394. doi: 10.1016/j.ijpe.2009.11.034. Google Scholar

[19]

E. Kusukawa, Supply chain coordination in 2-stage-ordering-production system with update of demand information, Industrial Engineering & Management Systems, 13 (2014), 304-318. doi: 10.7232/iems.2014.13.3.304. Google Scholar

[20]

W. S. LeeB. S. Kim and P. F. Opit, A stock pre-positioning model to maximize the total expected relief demand of disaster areas, Industrial Engineering & Management Systems, 13 (2014), 297-303. doi: 10.7232/iems.2014.13.3.297. Google Scholar

[21]

R. I. Levin, C. P. McLaughlin, R. P. Lemone and J. F. Kottas, Production/Operations Management: Contemporary Policy for Managing Operating Systems, 2nd edition, McGraw Hill, New York, 1972.Google Scholar

[22]

H. C. LiaoC. H. Tsai and C. T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63 (2000), 207-214. doi: 10.1016/S0925-5273(99)00015-8. Google Scholar

[23]

B. Mandal and A. K. Pal, Order level inventory system with ramp type demand rate for deteriorating items, Journal of Interdisciplinary Mathematics, 1 (1998), 49-66. doi: 10.1080/09720502.1998.10700243. Google Scholar

[24]

D. C. MontgomeryM. S. Bazaraa and A. K. Keswani, Inventory models with a mixture of backorders and lost sales, Naval Research Logistics Quarterly, 20 (1973), 255-263. doi: 10.1002/nav.3800200205. Google Scholar

[25]

B. Özyörük and N. Dönmeza, A fuzzy multi-objective linear programming model: A case study of an lpg distribution network, Industrial Engineering & Management Systems, 13 (2014), 319-329. Google Scholar

[26]

G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate, European Journal of Operational Research, 86 (1995), 281-292. doi: 10.1016/0377-2217(94)00103-J. Google Scholar

[27]

S. Papachristos and K. Skouri, An optimal replenishment policy for deteriorating items with time-varying demand and partial exponential type-backlogging, Operations Research Letters, 27 (2000), 175-184. doi: 10.1016/S0167-6377(00)00044-4. Google Scholar

[28]

K. S. Park, Inventory model with partial backorders, International Journal of System Science, 13 (1982), 1313-1317. doi: 10.1080/00207728208926430. Google Scholar

[29]

D. Rosenberg, A new analysis of a lot-size model with partial backordering, Naval Research Logistics Quarterly, 26 (1979), 349-353. Google Scholar

[30]

S. SanaS. K. Goyal and K. S. Chaudhuri, A production-inventory model for a deteriorating item with trended demand and shortages, European Journal of Operational Research, 157 (2004), 357-371. doi: 10.1016/S0377-2217(03)00222-4. Google Scholar

[31]

S. Sana, Optimal selling price and lotsize with time-varying deterioration and partial backlogging, Applied Mathematics and Computation, 217 (2010), 185-194. doi: 10.1016/j.amc.2010.05.040. Google Scholar

[32]

S. Sana and K. S. Chaudhuri, A deterministic EOQ model with delays in payments and price discount offers, European Journal of Operational Research, 184 (2008), 509-533. doi: 10.1016/j.ejor.2006.11.023. Google Scholar

[33]

S. Sana, An EOQ model for perishable item with stock-dependent demand and price discount rate, American Journal of Mathematical and Management, 30 (2012), 299-316. doi: 10.1080/01966324.2010.10737790. Google Scholar

[34]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55 (2012), 367-377. doi: 10.1016/j.mcm.2011.08.009. Google Scholar

[35]

B. Sarkar, An EOQ model with delay in payments and time-varying demand, Mathematical and Computer Modelling, 55 (2012), 367-377. doi: 10.1016/j.mcm.2011.08.009. Google Scholar

[36]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151. doi: 10.1016/j.apm.2012.07.026. Google Scholar

[37]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products, Mathematical Problems in Engineering, Available online, (2015).Google Scholar

[38]

B. Sarkar, An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production, Applied Mathematics and Computation, 218 (2012), 8295-8308. doi: 10.1016/j.amc.2012.01.053. Google Scholar

[39]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36. Google Scholar

[40]

B. Sarkar and S. Sarkar, Variable deterioration and demand-An inventory model, Economic Modelling, 31 (2013), 548-556. doi: 10.1016/j.econmod.2012.11.045. Google Scholar

[41]

M. Sarkar and B. Sarkar, An economic manufacturing quantity model with probabilistic deterioration in a production system, Economic Modelling, 31 (2013), 245-252. doi: 10.1016/j.econmod.2012.11.019. Google Scholar

[42]

B. SarkarS. Sarkar and W. Y. Yun, Retailer's optimal strategy for fixed lifetime products, International Journal of Machine Learning and Cybernetics, 7 (2016), 121-133. doi: 10.1007/s13042-015-0393-y. Google Scholar

[43]

B. Sarkar and S. Sarkar, An improved inventory model with partial backlogging, time-varying deterioration and stock-dependent demand, Economic Modelling, 30 (2013), 924-932. doi: 10.1016/j.econmod.2012.09.049. Google Scholar

[44]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 677-702. doi: 10.1007/s10479-014-1745-9. Google Scholar

[45]

B. SarkarB. K. SettA. Goswami and S. Sarkar, Mitigation of high-tech products with probabilistic deterioration and inflations, American Journal of Industrial and Business Management, 5 (2015), 73-89. doi: 10.4236/ajibm.2015.53009. Google Scholar

[46]

B. K. SettB. Sarkar and A. Goswami, A two-warehouse inventory model with increasing demand and time varying deterioration, Scientia Iranica, 19 (2012), 1969-1977. Google Scholar

[47]

E. A. Silver and R. Peterson, Decision Systems for Inventory Management and Production Planning, 2nd edition, Wiley, New York, 1985.Google Scholar

[48]

K. Skouri and S. Papachristos, A continuous review inventory model, with deteriorating items, time-varying demand, linear replenishment cost, partially time-varying backlogging, Applied Mathematical Modelling, 26 (2002), 603-617. doi: 10.1016/S0307-904X(01)00071-3. Google Scholar

[49]

K. SkouriI. KonstantarasS. Papachristos and I. Ganas, Inventory models with ramp-type demand rate, partial backlogging and weibull deterioration rate, European Journal of Operational Research, 192 (2009), 79-92. doi: 10.1016/j.ejor.2007.09.003. Google Scholar

[50]

T. Thongdee and R. Pitakaso, Differential evolution algorithms solving a multi-objective, source and stage location-allocation problem, Industrial Engineering & Management Systems, 14 (2015), 11-21. doi: 10.7232/iems.2015.14.1.011. Google Scholar

[51]

T. Watanabe and E. Kusukawa, Optimal operation for green supply chain considering demand information, collection incentive and quality of recycling parts, Industrial Engineering & Management Systems, 13 (2014), 129-147. doi: 10.7232/iems.2014.13.2.129. Google Scholar

[52]

H. M. WeeM. C. LeeJ. C. P. Yu and C. E. Wang, Optimal replenishment policy for a deteriorating green product:Lifecycle costing analysis, International Journal of Production Economics, 133 (2011), 603-611. Google Scholar

[53]

H. M. WeeY. D. HuangW. T. Wang and Y. L. Cheng, An EPQ model with partial backorders considering two backordering costs, Applied Mathematics and Computation, 232 (2014), 898-907. doi: 10.1016/j.amc.2014.01.106. Google Scholar

[54]

G. A. Widyadana and H. M. Wee, Optimal deteriorating items production inventory models with random machine breakdown and stochastic repair time, Applied Mathematical Modelling, 35 (2011), 3495-3508. doi: 10.1016/j.apm.2011.01.006. Google Scholar

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G. A. WidyadanaL. E. Cárdenas-Barrón and H. M. Wee, Economics order quantity model for deteriorating items with planned backorder level, Mathematical and Computer Modelling, 54 (2011), 1569-1575. doi: 10.1016/j.mcm.2011.04.028. Google Scholar

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K. S. Wu, An EOQ inventory model for items with Weibull distribution deterioration, ramp-type demand rate and partial backlogging, Production Planning & Control, 12 (2001), 787-793. doi: 10.1080/09537280110051819. Google Scholar

[57]

K. S. WuL. Y. Ouyang and C. T. Yang, Retailer's optimal ordering policy for deteriorating items with ramp-type demand under stock-dependent consumption rate, Information and Management Sciences, 19 (2008), 245-262. Google Scholar

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J. ZhangZ. 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. Google Scholar

show all references

References:
[1]

P. L. Abad, Optimal pricing and lot sizing under conditions of perishability and partial backordering, Management Science, 42 (1996), 1093-1104. doi: 10.1287/mnsc.42.8.1093. Google Scholar

[2]

H. K. Alfares, Inventory model with stock-level dependent demand rate and variable holding cost, International Journal of Production Economics, 108 (2007), 259-265. doi: 10.1016/j.ijpe.2006.12.013. Google Scholar

[3]

M. AsghariS. J. Abrishami and F. Mahdavi, Reverse logistics network design with incentive-dependent return, Industrial Engineering & Management Systems, 13 (2014), 383-397. doi: 10.7232/iems.2014.13.4.383. Google Scholar

[4]

A. Bouras and L. Tadj, Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy, Journal of Industrial and Management Optimization, 11 (2015), 1041-1058. doi: 10.3934/jimo.2015.11.1041. 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), 1176-1182. Google Scholar

[6]

C. J. Chung and H. M. Wee, Short life-cycle deteriorating product remanufacturing in a green supply chain inventory control system, International Journal of Production Economics, 129 (2011), 195-203. doi: 10.1016/j.ijpe.2010.09.033. Google Scholar

[7]

C. J. Chung and H. M. Wee, An integrated production-inventory deteriorating model for pricing policy considering imperfect production, inspection planning and warranty-period and stock-level-dependant demand, International Journal of System Science, 39 (2008), 823-837. doi: 10.1080/00207720801902598. Google Scholar

[8]

C. J. Chung and H. M. Wee, Scheduling and replenishment plan for an integrated deteriorating inventory model with stock-dependent selling rate, International Journal of Advanced Manufacturing Technology, 35 (2007), 665-679. doi: 10.1007/s00170-006-0744-7. Google Scholar

[9]

R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transaction, 5 (1973), 323-326. doi: 10.1080/05695557308974918. Google Scholar

[10]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Computers & Industrial Engineering, 57 (2009), 1105-1113. Google Scholar

[11]

L. E. Cárdenas-Barrón, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Applied Mathematical Modelling, 35 (2011), 2394-2407. doi: 10.1016/j.apm.2010.11.053. Google Scholar

[12]

K. K. Damghani and A. Shahrokh, Solving a new multi-period multi-objective multi-product aggregate production planning problem using fuzzy goal programming, Industrial Engineering & Management Systems, 13 (2014), 369-382. Google Scholar

[13]

C. Y. DyeH. J. Chang and J. T. Teng, A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging, European Journal of Operational Research, 172 (2006), 417-429. doi: 10.1016/j.ejor.2004.10.025. Google Scholar

[14]

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), 106-112. doi: 10.1016/j.ejor.2011.10.016. Google Scholar

[15]

C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880. doi: 10.1016/j.omega.2012.11.002. Google Scholar

[16]

C. Y. Dye and L. Y. Ouyang, An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging, European Journal of Operational Research, 163 (2005), 776-783. doi: 10.1016/j.ejor.2003.09.027. Google Scholar

[17]

P. M. Ghare and G. F. Schrader, A model for exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar

[18]

P. HsuH. M. Wee and H. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394. doi: 10.1016/j.ijpe.2009.11.034. Google Scholar

[19]

E. Kusukawa, Supply chain coordination in 2-stage-ordering-production system with update of demand information, Industrial Engineering & Management Systems, 13 (2014), 304-318. doi: 10.7232/iems.2014.13.3.304. Google Scholar

[20]

W. S. LeeB. S. Kim and P. F. Opit, A stock pre-positioning model to maximize the total expected relief demand of disaster areas, Industrial Engineering & Management Systems, 13 (2014), 297-303. doi: 10.7232/iems.2014.13.3.297. Google Scholar

[21]

R. I. Levin, C. P. McLaughlin, R. P. Lemone and J. F. Kottas, Production/Operations Management: Contemporary Policy for Managing Operating Systems, 2nd edition, McGraw Hill, New York, 1972.Google Scholar

[22]

H. C. LiaoC. H. Tsai and C. T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63 (2000), 207-214. doi: 10.1016/S0925-5273(99)00015-8. Google Scholar

[23]

B. Mandal and A. K. Pal, Order level inventory system with ramp type demand rate for deteriorating items, Journal of Interdisciplinary Mathematics, 1 (1998), 49-66. doi: 10.1080/09720502.1998.10700243. Google Scholar

[24]

D. C. MontgomeryM. S. Bazaraa and A. K. Keswani, Inventory models with a mixture of backorders and lost sales, Naval Research Logistics Quarterly, 20 (1973), 255-263. doi: 10.1002/nav.3800200205. Google Scholar

[25]

B. Özyörük and N. Dönmeza, A fuzzy multi-objective linear programming model: A case study of an lpg distribution network, Industrial Engineering & Management Systems, 13 (2014), 319-329. Google Scholar

[26]

G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate, European Journal of Operational Research, 86 (1995), 281-292. doi: 10.1016/0377-2217(94)00103-J. Google Scholar

[27]

S. Papachristos and K. Skouri, An optimal replenishment policy for deteriorating items with time-varying demand and partial exponential type-backlogging, Operations Research Letters, 27 (2000), 175-184. doi: 10.1016/S0167-6377(00)00044-4. Google Scholar

[28]

K. S. Park, Inventory model with partial backorders, International Journal of System Science, 13 (1982), 1313-1317. doi: 10.1080/00207728208926430. Google Scholar

[29]

D. Rosenberg, A new analysis of a lot-size model with partial backordering, Naval Research Logistics Quarterly, 26 (1979), 349-353. Google Scholar

[30]

S. SanaS. K. Goyal and K. S. Chaudhuri, A production-inventory model for a deteriorating item with trended demand and shortages, European Journal of Operational Research, 157 (2004), 357-371. doi: 10.1016/S0377-2217(03)00222-4. Google Scholar

[31]

S. Sana, Optimal selling price and lotsize with time-varying deterioration and partial backlogging, Applied Mathematics and Computation, 217 (2010), 185-194. doi: 10.1016/j.amc.2010.05.040. Google Scholar

[32]

S. Sana and K. S. Chaudhuri, A deterministic EOQ model with delays in payments and price discount offers, European Journal of Operational Research, 184 (2008), 509-533. doi: 10.1016/j.ejor.2006.11.023. Google Scholar

[33]

S. Sana, An EOQ model for perishable item with stock-dependent demand and price discount rate, American Journal of Mathematical and Management, 30 (2012), 299-316. doi: 10.1080/01966324.2010.10737790. Google Scholar

[34]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathematical and Computer Modelling, 55 (2012), 367-377. doi: 10.1016/j.mcm.2011.08.009. Google Scholar

[35]

B. Sarkar, An EOQ model with delay in payments and time-varying demand, Mathematical and Computer Modelling, 55 (2012), 367-377. doi: 10.1016/j.mcm.2011.08.009. Google Scholar

[36]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151. doi: 10.1016/j.apm.2012.07.026. Google Scholar

[37]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products, Mathematical Problems in Engineering, Available online, (2015).Google Scholar

[38]

B. Sarkar, An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production, Applied Mathematics and Computation, 218 (2012), 8295-8308. doi: 10.1016/j.amc.2012.01.053. Google Scholar

[39]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36. Google Scholar

[40]

B. Sarkar and S. Sarkar, Variable deterioration and demand-An inventory model, Economic Modelling, 31 (2013), 548-556. doi: 10.1016/j.econmod.2012.11.045. Google Scholar

[41]

M. Sarkar and B. Sarkar, An economic manufacturing quantity model with probabilistic deterioration in a production system, Economic Modelling, 31 (2013), 245-252. doi: 10.1016/j.econmod.2012.11.019. Google Scholar

[42]

B. SarkarS. Sarkar and W. Y. Yun, Retailer's optimal strategy for fixed lifetime products, International Journal of Machine Learning and Cybernetics, 7 (2016), 121-133. doi: 10.1007/s13042-015-0393-y. Google Scholar

[43]

B. Sarkar and S. Sarkar, An improved inventory model with partial backlogging, time-varying deterioration and stock-dependent demand, Economic Modelling, 30 (2013), 924-932. doi: 10.1016/j.econmod.2012.09.049. Google Scholar

[44]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 677-702. doi: 10.1007/s10479-014-1745-9. Google Scholar

[45]

B. SarkarB. K. SettA. Goswami and S. Sarkar, Mitigation of high-tech products with probabilistic deterioration and inflations, American Journal of Industrial and Business Management, 5 (2015), 73-89. doi: 10.4236/ajibm.2015.53009. Google Scholar

[46]

B. K. SettB. Sarkar and A. Goswami, A two-warehouse inventory model with increasing demand and time varying deterioration, Scientia Iranica, 19 (2012), 1969-1977. Google Scholar

[47]

E. A. Silver and R. Peterson, Decision Systems for Inventory Management and Production Planning, 2nd edition, Wiley, New York, 1985.Google Scholar

[48]

K. Skouri and S. Papachristos, A continuous review inventory model, with deteriorating items, time-varying demand, linear replenishment cost, partially time-varying backlogging, Applied Mathematical Modelling, 26 (2002), 603-617. doi: 10.1016/S0307-904X(01)00071-3. Google Scholar

[49]

K. SkouriI. KonstantarasS. Papachristos and I. Ganas, Inventory models with ramp-type demand rate, partial backlogging and weibull deterioration rate, European Journal of Operational Research, 192 (2009), 79-92. doi: 10.1016/j.ejor.2007.09.003. Google Scholar

[50]

T. Thongdee and R. Pitakaso, Differential evolution algorithms solving a multi-objective, source and stage location-allocation problem, Industrial Engineering & Management Systems, 14 (2015), 11-21. doi: 10.7232/iems.2015.14.1.011. Google Scholar

[51]

T. Watanabe and E. Kusukawa, Optimal operation for green supply chain considering demand information, collection incentive and quality of recycling parts, Industrial Engineering & Management Systems, 13 (2014), 129-147. doi: 10.7232/iems.2014.13.2.129. Google Scholar

[52]

H. M. WeeM. C. LeeJ. C. P. Yu and C. E. Wang, Optimal replenishment policy for a deteriorating green product:Lifecycle costing analysis, International Journal of Production Economics, 133 (2011), 603-611. Google Scholar

[53]

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Figure 1.  Graphical presentation of the inventory system (Case 1: μ>t1)
Figure 2.  Graphical presentation of the inventory system (Case 2: μ>t1)
Figure 3.  Graphical presentation of total profit function versus time and preservation technology cost (Example 1)
Figure 4.  Graphical presentation of total profit function versus time and preservation technology cost (Example 2)
Table 1.  Comparison between the contributions of different authors
Author (s) Ramp-type demand Deterio-ration Preser-vation Stock-dependent consumption rate Partial back-logging
Ghare and Schrader [17]
Covert and Philip [9]
Sana, Goyal, and Chaudhuri [30]
Dye, Chang, and Teng [13]
Chung and Wee [7]
Widyadana and Wee [54]
Chung and Wee [6]
Wee, Lee, Yu, and Wang [52]
Sarkar [34]
Sett, Sarkar, and Goswami [46]
Sarkar [35]
Sarkar and Sarkar [36]
Sarkar and Sarkar [40]
Sarkar, Sarkar, and Yun [41]
Sarkar [42]
Bouras and Tadj [4]
Sarkar, Saren, and
C$\acute{a}$rdenas-Barr$\acute{o}$n [44]
Hsu, Wee, and Teng [18]
Dye and Hsieh [14]
Dye [15]
Zhang, Bai, and Tang [58]
Montgomery, Bazaraa,
and Keswani [24]
Park [28]
Rosenberg [29]
Abad [1]
Chang and Dye [5]
Papachristos and Skouri [27]
Skouri and Papachristos [48]
C$\acute{a}$rdenas-Barr$\acute{o}$n [10]
Sana [31]
C$\acute{a}$rdenas-Barr$\acute{o}$n [11]
Widyadana, C$\acute{a}$rdenas-Barr$\acute{o}$n, and Wee [55]
Sarkar and Sarkar [43]
Wee, Huang,
Wang and Cheng [53]
Sarkar [37]
Sarkar, Mandal, and Sarkar [39]
Mandal and Pal [23]
Wu [56]
Wu, Ouyang, and Yang [57]
Skouri, Konstantaras,
Papachristos, and Ganas [49]
Sarkar, Sett, Goswami,
and Sarkar [45]
Levin, McLaughlin,
Lemone, and Kottas [21]
Silver and Peterson [47]
Padmanabhan and Vrat [26]
Liao, Tsai, and Su [22]
Dye and Ouyang [16]
Alfares [2]
Chung and Wee [8]
Sana and Chaudhuri [32]
Sana [33]
Sarkar [38]
Author (s) Ramp-type demand Deterio-ration Preser-vation Stock-dependent consumption rate Partial back-logging
Ghare and Schrader [17]
Covert and Philip [9]
Sana, Goyal, and Chaudhuri [30]
Dye, Chang, and Teng [13]
Chung and Wee [7]
Widyadana and Wee [54]
Chung and Wee [6]
Wee, Lee, Yu, and Wang [52]
Sarkar [34]
Sett, Sarkar, and Goswami [46]
Sarkar [35]
Sarkar and Sarkar [36]
Sarkar and Sarkar [40]
Sarkar, Sarkar, and Yun [41]
Sarkar [42]
Bouras and Tadj [4]
Sarkar, Saren, and
C$\acute{a}$rdenas-Barr$\acute{o}$n [44]
Hsu, Wee, and Teng [18]
Dye and Hsieh [14]
Dye [15]
Zhang, Bai, and Tang [58]
Montgomery, Bazaraa,
and Keswani [24]
Park [28]
Rosenberg [29]
Abad [1]
Chang and Dye [5]
Papachristos and Skouri [27]
Skouri and Papachristos [48]
C$\acute{a}$rdenas-Barr$\acute{o}$n [10]
Sana [31]
C$\acute{a}$rdenas-Barr$\acute{o}$n [11]
Widyadana, C$\acute{a}$rdenas-Barr$\acute{o}$n, and Wee [55]
Sarkar and Sarkar [43]
Wee, Huang,
Wang and Cheng [53]
Sarkar [37]
Sarkar, Mandal, and Sarkar [39]
Mandal and Pal [23]
Wu [56]
Wu, Ouyang, and Yang [57]
Skouri, Konstantaras,
Papachristos, and Ganas [49]
Sarkar, Sett, Goswami,
and Sarkar [45]
Levin, McLaughlin,
Lemone, and Kottas [21]
Silver and Peterson [47]
Padmanabhan and Vrat [26]
Liao, Tsai, and Su [22]
Dye and Ouyang [16]
Alfares [2]
Chung and Wee [8]
Sana and Chaudhuri [32]
Sana [33]
Sarkar [38]
Table 2.  Sensitivity analysis of the key parameters.
Parameters Changes (in %) Model 1 Model 2
-50% +00.27 +00.76
-25% +00.14 +00.38
A +25% -00.14 -00.38
+50% -00.27 -00.76
-50% -95.65 -97.17
-25% -48.39 -49.19
s +25% +49.34 +49.97
+50% +99.48 +100.52
-50% +42.97 +44.83
-25% +21.34 +22.18
p +25% -21.10 -21.84
+50% -41.96 -43.41
-50% +04.81 +03.57
-25% +02.12 +01.63
h +25% -01.69 -01.37
+50% -03.04 -02.53
-50% +00.22 +00.42
-25% +00.11 +00.18
b +25% -00.08 -00.14
+50% -00.15 -00.25
-50% +00.06 +00.11
-25% +00.03 +00.05
l +25% -00.03 -00.05
+50% -00.06 -00.09
Parameters Changes (in %) Model 1 Model 2
-50% +00.27 +00.76
-25% +00.14 +00.38
A +25% -00.14 -00.38
+50% -00.27 -00.76
-50% -95.65 -97.17
-25% -48.39 -49.19
s +25% +49.34 +49.97
+50% +99.48 +100.52
-50% +42.97 +44.83
-25% +21.34 +22.18
p +25% -21.10 -21.84
+50% -41.96 -43.41
-50% +04.81 +03.57
-25% +02.12 +01.63
h +25% -01.69 -01.37
+50% -03.04 -02.53
-50% +00.22 +00.42
-25% +00.11 +00.18
b +25% -00.08 -00.14
+50% -00.15 -00.25
-50% +00.06 +00.11
-25% +00.03 +00.05
l +25% -00.03 -00.05
+50% -00.06 -00.09
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