# American Institute of Mathematical Sciences

October  2014, 10(4): 989-1000. doi: 10.3934/jimo.2014.10.989

## A deteriorating inventory model for an intermediary firm under return on inventory investment maximization

 1 Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan, Taiwan

Received  April 2012 Revised  September 2013 Published  February 2014

This paper investigates how the intermediary firms can optimally determine the purchasing cycle length of a deteriorating product under return on inventory investment (ROII) maximization criterion. There are three key features differentiating this paper from the extant literature and being considered simultaneously in this paper, which are: 1) an alternative performance measurement (i.e., ROII) is proposed to formulate the inventory system; 2) the decision maker in this paper is an intermediary firm instead of a retailer; and 3) the deteriorating nature of products is considered. By incorporating the deteriorating nature of products and the special structure of the intermediary firm environments into the traditional economic order quantity model, the inventory problem encountered by the intermediary firm is mathematically formulated as a non-linear programming problem. Several interesting properties of the proposed inventory problem are developed and an efficient iterative algorithm is provided to search for the optimal solution. Also, the convergence of the iterative algorithm developed in this paper is proved. Finally, a numerical example is presented to illustrate the features of the proposed problem and the convergent search algorithm.
Citation: Cheng-Kang Chen, Yi-Xiang Liao. A deteriorating inventory model for an intermediary firm under return on inventory investment maximization. Journal of Industrial & Management Optimization, 2014, 10 (4) : 989-1000. doi: 10.3934/jimo.2014.10.989
##### References:
 [1] R. L. Burden and J. D. Faires, Numerical Analysis,, 3rd edition, (2001). [2] C.-K. Chen and K. J. Min, Optimal selling quantity and purchasing price for intermediary firms,, International Journal of Operations & Production Management, 11 (1991), 64. doi: 10.1108/EUM0000000001291. [3] C.-K. Chen, Optimal determination of quality level, selling quantity and purchasing price for intermediate firms,, Production Planning & Control, 11 (2000), 706. doi: 10.1080/095372800432179. [4] C.-K. Chen and Y.-X. Liao, Optimal purchasing cycle length of a deteriorating product for intermediary firms,, Computational Optimization and Applications, 42 (2009), 289. doi: 10.1007/s10589-007-9080-6. [5] K.-J. Chung, P. Chu and S.-P. Lan, A note on EOQ models for deteriorating items under stock dependent selling rate,, European Journal of Operational Research, 124 (2000), 550. doi: 10.1016/S0377-2217(99)00203-9. [6] 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. doi: 10.1016/j.ejor.2003.09.027. [7] K. V. Geetha and R. Uthayakumar, Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments,, Computational and Applied Mathematics, 233 (2010), 2492. doi: 10.1016/j.cam.2009.10.031. [8] P. M. Ghare and G. F. Schrader, A model for an exponentially decaying inventory,, Journal of Industrial Engineering, 13 (1963), 238. [9] B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost,, European Journal of Operational Research, 105 (1998), 467. doi: 10.1016/S0377-2217(97)00086-6. [10] A. K. Jalan and K. S. Chaudhuri, Structural properties of an inventory system with deterioration and trended demand,, International Journal of Systems Science, 30 (1999), 627. doi: 10.1080/002077299292137. [11] S. Ladany and A. Sternlieb, The interaction of economic ordering quantities and marketing policies,, AIIE Transactions, 6 (1974), 35. doi: 10.1080/05695557408974930. [12] J. Li, K. J. Min, T. Otake and T. V. Voorhis, Inventory and investment in setup and quality operations under return on investment maximization,, European Journal of Operational Research, 185 (2008), 593. doi: 10.1016/j.ejor.2006.11.045. [13] G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate,, European Journal of Operational Research, 86 (1995), 281. doi: 10.1016/0377-2217(94)00103-J. [14] D. Rosenberg, Optimal price-inventory decisions: Profit vs. ROII,, IIE Transactions, 23 (1991), 17. doi: 10.1080/07408179108963837. [15] W. M. Smith, An investigation of some quantitative relationships between breakeven point analysis and economic lot size theory,, AIIE Transactions, 9 (1958), 52. [16] B. R. Sarker, S. Mukherjee and C. V. Balan, An order-level lot size inventory model with inventory-level dependent demand and deterioration,, International Journal of Production Economics, 48 (1997), 227. doi: 10.1016/S0925-5273(96)00107-7. [17] R. G. Schroeder and R. Krishnan, Return on investment as a criterion for inventory models,, Decision Sciences, 7 (1976), 697. doi: 10.1111/j.1540-5915.1976.tb00713.x. [18] E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling,, 3rd edition, (1998). [19] D. F. Spulber, Market Microstructure: Intermediaries and the Theory of the Firm,, Cambridge University Press, (1999). doi: 10.1017/CBO9780511625930. [20] O. Toshitsugu, K. J. Min and C.-K. Chen, Inventory and investmentin setup operations under return on investment maximization,, Computers & Operations Research, 26 (1999), 883. [21] H.-M. Wee and S.-T. Law, Economic production lot size for deteriorating items taking account of the time-value of money,, Computers & Operations Research, 26 (1999), 545. doi: 10.1016/S0305-0548(98)00078-1.

show all references

##### References:
 [1] R. L. Burden and J. D. Faires, Numerical Analysis,, 3rd edition, (2001). [2] C.-K. Chen and K. J. Min, Optimal selling quantity and purchasing price for intermediary firms,, International Journal of Operations & Production Management, 11 (1991), 64. doi: 10.1108/EUM0000000001291. [3] C.-K. Chen, Optimal determination of quality level, selling quantity and purchasing price for intermediate firms,, Production Planning & Control, 11 (2000), 706. doi: 10.1080/095372800432179. [4] C.-K. Chen and Y.-X. Liao, Optimal purchasing cycle length of a deteriorating product for intermediary firms,, Computational Optimization and Applications, 42 (2009), 289. doi: 10.1007/s10589-007-9080-6. [5] K.-J. Chung, P. Chu and S.-P. Lan, A note on EOQ models for deteriorating items under stock dependent selling rate,, European Journal of Operational Research, 124 (2000), 550. doi: 10.1016/S0377-2217(99)00203-9. [6] 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. doi: 10.1016/j.ejor.2003.09.027. [7] K. V. Geetha and R. Uthayakumar, Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments,, Computational and Applied Mathematics, 233 (2010), 2492. doi: 10.1016/j.cam.2009.10.031. [8] P. M. Ghare and G. F. Schrader, A model for an exponentially decaying inventory,, Journal of Industrial Engineering, 13 (1963), 238. [9] B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost,, European Journal of Operational Research, 105 (1998), 467. doi: 10.1016/S0377-2217(97)00086-6. [10] A. K. Jalan and K. S. Chaudhuri, Structural properties of an inventory system with deterioration and trended demand,, International Journal of Systems Science, 30 (1999), 627. doi: 10.1080/002077299292137. [11] S. Ladany and A. Sternlieb, The interaction of economic ordering quantities and marketing policies,, AIIE Transactions, 6 (1974), 35. doi: 10.1080/05695557408974930. [12] J. Li, K. J. Min, T. Otake and T. V. Voorhis, Inventory and investment in setup and quality operations under return on investment maximization,, European Journal of Operational Research, 185 (2008), 593. doi: 10.1016/j.ejor.2006.11.045. [13] G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate,, European Journal of Operational Research, 86 (1995), 281. doi: 10.1016/0377-2217(94)00103-J. [14] D. Rosenberg, Optimal price-inventory decisions: Profit vs. ROII,, IIE Transactions, 23 (1991), 17. doi: 10.1080/07408179108963837. [15] W. M. Smith, An investigation of some quantitative relationships between breakeven point analysis and economic lot size theory,, AIIE Transactions, 9 (1958), 52. [16] B. R. Sarker, S. Mukherjee and C. V. Balan, An order-level lot size inventory model with inventory-level dependent demand and deterioration,, International Journal of Production Economics, 48 (1997), 227. doi: 10.1016/S0925-5273(96)00107-7. [17] R. G. Schroeder and R. Krishnan, Return on investment as a criterion for inventory models,, Decision Sciences, 7 (1976), 697. doi: 10.1111/j.1540-5915.1976.tb00713.x. [18] E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling,, 3rd edition, (1998). [19] D. F. Spulber, Market Microstructure: Intermediaries and the Theory of the Firm,, Cambridge University Press, (1999). doi: 10.1017/CBO9780511625930. [20] O. Toshitsugu, K. J. Min and C.-K. Chen, Inventory and investmentin setup operations under return on investment maximization,, Computers & Operations Research, 26 (1999), 883. [21] H.-M. Wee and S.-T. Law, Economic production lot size for deteriorating items taking account of the time-value of money,, Computers & Operations Research, 26 (1999), 545. doi: 10.1016/S0305-0548(98)00078-1.
 [1] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [2] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [3] Ruopeng Wang, Jinting Wang, Chang Sun. Optimal pricing and inventory management for a loss averse firm when facing strategic customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1521-1544. doi: 10.3934/jimo.2018019 [4] Jonas C. P. Yu, H. M. Wee, K. J. Wang. Supply chain partnership for Three-Echelon deteriorating inventory model. Journal of Industrial & Management Optimization, 2008, 4 (4) : 827-842. doi: 10.3934/jimo.2008.4.827 [5] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [6] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [7] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [8] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [9] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [10] Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial & Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19 [11] Po-Chung Yang, Hui-Ming Wee, Shen-Lian Chung, Yong-Yan Huang. Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand. Journal of Industrial & Management Optimization, 2013, 9 (4) : 769-787. doi: 10.3934/jimo.2013.9.769 [12] 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 [13] 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 [14] 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 [15] 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 [16] Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096 [17] 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 [18] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [19] Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645 [20] Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467

2018 Impact Factor: 1.025