# American Institute of Mathematical Sciences

October  2013, 9(4): 855-883. doi: 10.3934/jimo.2013.9.855

## Two-warehouse inventory models for deteriorating products with ramp type demand rate

 1 Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece 2 Department of Business Administration, Business Excellence Laboratory, University of Macedonia, 156 Egnatia Street, 54006 Thessaloniki, Greece

Received  August 2011 Revised  February 2013 Published  August 2013

In today's business environment, there are various reasons, namely, bulk purchase discounts, seasonality of products, re-order costs, etc., which force the buyer to order more than the warehouse capacity (owned warehouse). Such reasons call for additional storage space to store the excess units purchased. This additional storage space is typically a rented warehouse. It is known that the demand of seasonal products increases at the beginning of the season up to a certain moment and then is stabilized to a constant rate for the remaining time of the season (ramp type demand rate). As a result, the buyer prefers to keep a higher inventory at the beginning of the season and so more units than can be stored in owned warehouse may be purchased. The excess quantities need additional storage space, which is facilitated by a rented warehouse.
In this study an order level two-warehouse inventory model for deteriorating seasonal products is studied. Shortages at the owned warehouse are allowed subject to partial backlogging. This two-warehouse inventory model is studied under two different policies. The first policy starts with an instant replenishment and ends with shortages and the second policy starts with shortages and ends without shortages. For each of the models, conditions for the existence and uniqueness of the optimal solution are derived and a simple procedure is developed to obtain the overall optimal replenishment policy. The dynamics of the model and the solution procedure have been illustrated with the help of a numerical example and a comprehensive sensitivity analysis, with respect to the most important parameters of the model, is considered.
Citation: Konstantina Skouri, Ioannis Konstantaras. Two-warehouse inventory models for deteriorating products with ramp type demand rate. Journal of Industrial & Management Optimization, 2013, 9 (4) : 855-883. doi: 10.3934/jimo.2013.9.855
##### References:
 [1] L. Benkherouf, A deterministic order level inventory model for deteriorating items with two storage facilities,, International Journal of Production Economics, 48 (1997), 167. Google Scholar [2] A. Goswami and K. S. Chaudhuri, An economic order quantity model for items with two levels of storage for a linear trend in demand,, Journal of the Operational Research Society, 43 (1992), 157. Google Scholar [3] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory,, European Journal of Operational Research, 134 (2001), 1. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar [4] R. M. Hill, Inventory model for increasing demand followed by level demand,, Journal of the Operational Research Society, 46 (1995), 1250. Google Scholar [5] T.-P. Hsieh, C.-Y. Dye and L.-Y. Ouyang, Determining optimal lot size for a two-warehouse system with deterioration and shortages using net present value,, European Journal of Operational Research, 191 (2008), 182. doi: 10.1016/j.ejor.2007.08.020. Google Scholar [6] M. D. Intriligator, "Mathematical Optimization and Economic Theory,", Philadelphia, (2002). doi: 10.1137/1.9780898719215. Google Scholar [7] S. Kar, A. K. Bhunia and M. Maiti, Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon,, Computers and Operations Research, 28 (2001), 1315. doi: 10.1016/S0305-0548(00)00042-3. Google Scholar [8] Y. Liang and F. Zhou, A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment,, Applied Mathematical Modelling, 35 (2011), 2221. doi: 10.1016/j.apm.2010.11.014. Google Scholar [9] 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 [10] T. P. M. Pakkala and K. K. Achary, A deterministic inventory model for deteriorating items with two-warehouses and finite replenishment rate,, European Journal of Operational Research, 57 (1992), 71. Google Scholar [11] F. Raafat, Survey of literature on continuously deteriorating inventory model,, Journal of the Operational Research Society, 42 (1991), 27. Google Scholar [12] L. Ruxian, L. Hongjie and J. R. Mawhinney, A review on deteriorating inventory study,, Journal of Service Science and Management, 3 (2010), 117. Google Scholar [13] K. V. S. Sarma, A deterministic order level inventory model for deteriorating items with two storage facilities,, European Journal of Operational Research, 29 (1987), 70. doi: 10.1016/0377-2217(87)90194-9. Google Scholar [14] N. H. Shah, Inventory model for deteriorating items and time value of money for a finite time horizon under the permissible delay in payments,, International Journal of Systems Science, 37 (2006), 9. doi: 10.1080/00207720500404334. Google Scholar [15] 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 [16] K. Skouri and I. Konstantaras, Order level inventory models for deteriorating seasonable/fashionable products with time-dependent demand and shortages,, Mathematical Problems in Engineering, 2009 (6797). doi: 10.1155/2009/679736. Google Scholar [17] J.-W. Wu, C. Lin, B. Tan and W.-C. Lee, An EOQ inventory model with ramp type demand rate for items with Weibull deterioration,, Information and Management Science, 10 (1999), 41. Google Scholar [18] K.-S. Wu, An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging,, Production Planning and Control, 12 (2001), 787. Google Scholar [19] K.-S. Wu and L.-Y. Ouyang, A replenishment policy for deteriorating items with ramp type demand rate,, Proc. Natl. Sci. Counc. ROC(A), 24 (2000), 279. Google Scholar [20] H.-L. Yang, Two-warehouse partial backlogging inventory models for deteriorating items under inflation,, International Journal of Production Economics, 103 (2006), 362. Google Scholar

show all references

##### References:
 [1] L. Benkherouf, A deterministic order level inventory model for deteriorating items with two storage facilities,, International Journal of Production Economics, 48 (1997), 167. Google Scholar [2] A. Goswami and K. S. Chaudhuri, An economic order quantity model for items with two levels of storage for a linear trend in demand,, Journal of the Operational Research Society, 43 (1992), 157. Google Scholar [3] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory,, European Journal of Operational Research, 134 (2001), 1. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar [4] R. M. Hill, Inventory model for increasing demand followed by level demand,, Journal of the Operational Research Society, 46 (1995), 1250. Google Scholar [5] T.-P. Hsieh, C.-Y. Dye and L.-Y. Ouyang, Determining optimal lot size for a two-warehouse system with deterioration and shortages using net present value,, European Journal of Operational Research, 191 (2008), 182. doi: 10.1016/j.ejor.2007.08.020. Google Scholar [6] M. D. Intriligator, "Mathematical Optimization and Economic Theory,", Philadelphia, (2002). doi: 10.1137/1.9780898719215. Google Scholar [7] S. Kar, A. K. Bhunia and M. Maiti, Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon,, Computers and Operations Research, 28 (2001), 1315. doi: 10.1016/S0305-0548(00)00042-3. Google Scholar [8] Y. Liang and F. Zhou, A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment,, Applied Mathematical Modelling, 35 (2011), 2221. doi: 10.1016/j.apm.2010.11.014. Google Scholar [9] 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 [10] T. P. M. Pakkala and K. K. Achary, A deterministic inventory model for deteriorating items with two-warehouses and finite replenishment rate,, European Journal of Operational Research, 57 (1992), 71. Google Scholar [11] F. Raafat, Survey of literature on continuously deteriorating inventory model,, Journal of the Operational Research Society, 42 (1991), 27. Google Scholar [12] L. Ruxian, L. Hongjie and J. R. Mawhinney, A review on deteriorating inventory study,, Journal of Service Science and Management, 3 (2010), 117. Google Scholar [13] K. V. S. Sarma, A deterministic order level inventory model for deteriorating items with two storage facilities,, European Journal of Operational Research, 29 (1987), 70. doi: 10.1016/0377-2217(87)90194-9. Google Scholar [14] N. H. Shah, Inventory model for deteriorating items and time value of money for a finite time horizon under the permissible delay in payments,, International Journal of Systems Science, 37 (2006), 9. doi: 10.1080/00207720500404334. Google Scholar [15] 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 [16] K. Skouri and I. Konstantaras, Order level inventory models for deteriorating seasonable/fashionable products with time-dependent demand and shortages,, Mathematical Problems in Engineering, 2009 (6797). doi: 10.1155/2009/679736. Google Scholar [17] J.-W. Wu, C. Lin, B. Tan and W.-C. Lee, An EOQ inventory model with ramp type demand rate for items with Weibull deterioration,, Information and Management Science, 10 (1999), 41. Google Scholar [18] K.-S. Wu, An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging,, Production Planning and Control, 12 (2001), 787. Google Scholar [19] K.-S. Wu and L.-Y. Ouyang, A replenishment policy for deteriorating items with ramp type demand rate,, Proc. Natl. Sci. Counc. ROC(A), 24 (2000), 279. Google Scholar [20] H.-L. Yang, Two-warehouse partial backlogging inventory models for deteriorating items under inflation,, International Journal of Production Economics, 103 (2006), 362. Google Scholar
 [1] Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang, Yung-Fu Huang. Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012 [2] Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2018167 [3] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002 [4] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098 [5] Jia Shu, Zhengyi Li, Weijun Zhong. A market selection and inventory ordering problem under demand uncertainty. Journal of Industrial & Management Optimization, 2011, 7 (2) : 425-434. doi: 10.3934/jimo.2011.7.425 [6] Reza Lotfi, Gerhard-Wilhelm Weber, S. Mehdi Sajadifar, Nooshin Mardani. Interdependent demand in the two-period newsvendor problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-24. doi: 10.3934/jimo.2018143 [7] Wei Liu, Shiji Song, Cheng Wu. Single-period inventory model with discrete stochastic demand based on prospect theory. Journal of Industrial & Management Optimization, 2012, 8 (3) : 577-590. doi: 10.3934/jimo.2012.8.577 [8] Yanyi Xu, Arnab Bisi, Maqbool Dada. New structural properties of inventory models with Polya frequency distributed demand and fixed setup cost. Journal of Industrial & Management Optimization, 2017, 13 (2) : 931-945. doi: 10.3934/jimo.2016054 [9] Lizhao Yan, Fei Xu, Yongzeng Lai, Mingyong Lai. Stability strategies of manufacturing-inventory systems with unknown time-varying demand. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2033-2047. doi: 10.3934/jimo.2017030 [10] Zhijie Sasha Dong, Wei Chen, Qing Zhao, Jingquan Li. Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2018175 [11] Katherinne Salas Navarro, Jaime Acevedo Chedid, Whady F. Florez, Holman Ospina Mateus, Leopoldo Eduardo Cárdenas-Barrón, Shib Sankar Sana. A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019020 [12] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Deteriorating inventory with preservation technology under price- and stock-sensitive demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2019019 [13] Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019032 [14] Eungab Kim. On the admission control and demand management in a two-station tandem production system. Journal of Industrial & Management Optimization, 2011, 7 (1) : 1-18. doi: 10.3934/jimo.2011.7.1 [15] Jason Chao-Hsien Pan, Ku-Kuang Chang, Yu-Cheng Hsiao. Optimal inventory policies for serial-type and assembly-type supply chains with equal sized batch. Journal of Industrial & Management Optimization, 2015, 11 (3) : 1021-1040. doi: 10.3934/jimo.2015.11.1021 [16] Yanju Zhou, Zhen Shen, Renren Ying, Xuanhua Xu. A loss-averse two-product ordering model with information updating in two-echelon inventory system. Journal of Industrial & Management Optimization, 2018, 14 (2) : 687-705. doi: 10.3934/jimo.2017069 [17] 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 [18] Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial & Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609 [19] Vincent Choudri, Mathiyazhgan Venkatachalam, Sethuraman Panayappan. Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1153-1172. doi: 10.3934/jimo.2016.12.1153 [20] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. An integrated inventory model with variable holding cost under two levels of trade-credit policy. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 169-191. doi: 10.3934/naco.2018010

2018 Impact Factor: 1.025

## Metrics

• PDF downloads (10)
• HTML views (0)
• Cited by (3)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]